{"id":27,"date":"2019-08-20T17:01:15","date_gmt":"2019-08-20T21:01:15","guid":{"rendered":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/polynomials\/"},"modified":"2022-06-01T10:39:16","modified_gmt":"2022-06-01T14:39:16","slug":"polynomials","status":"publish","type":"chapter","link":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/polynomials\/","title":{"raw":"Polynomials","rendered":"Polynomials"},"content":{"raw":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\nIn this section students will:\n<ul>\n \t<li>Identify the degree and leading coefficient of polynomials.<\/li>\n \t<li>Add and subtract polynomials.<\/li>\n \t<li>Multiply polynomials.<\/li>\n \t<li>Use FOIL to multiply binomials.<\/li>\n \t<li>Perform operations with polynomials of several variables.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1167339150823\">Earl is building a doghouse, whose front is in the shape of a square topped with a triangle. There will be a rectangular door through which the dog can enter and exit the house. Earl wants to find the area of the front of the doghouse so that he can purchase the correct amount of paint. Using the measurements of the front of the house, shown in <a class=\"autogenerated-content\" href=\"#Figure_01_04_001\">(Figure)<\/a>, we can create an expression that combines several variable terms, allowing us to solve this problem and others like it.<\/p>\n\n<div id=\"Figure_01_04_001\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132101\/CNX_CAT_Figure_01_04_001.jpg\" alt=\"Sketch of a house formed by a square and a triangle based on the top of the square. A rectangle is placed at the bottom center of the square to mark a doorway. The height of the door is labeled: x and the width of the door is labeled: 1 foot. The side of the square is labeled: 2x. The height of the triangle is labeled: 3\/2 feet.\" width=\"487\" height=\"249\"> <strong>Figure 1.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1167339220668\">First find the area of the square in square feet.<\/p>\n\n<div id=\"fs-id1167339151023\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill A&amp; =&amp; {s}^{2}\\hfill \\\\ &amp; =&amp; {\\left(2x\\right)}^{2}\\hfill \\\\ &amp; =&amp; 4{x}^{2}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167339240684\">Then find the area of the triangle in square feet.<\/p>\n\n<div id=\"fs-id1167339432668\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill A&amp; =&amp; \\frac{1}{2}bh\\hfill \\\\ &amp; =&amp; \\text{ }\\frac{1}{2}\\left(2x\\right)\\left(\\frac{3}{2}\\right)\\hfill \\\\ &amp; =&amp; \\text{ }\\frac{3}{2}x\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167339173565\">Next find the area of the rectangular door in square feet.<\/p>\n\n<div id=\"fs-id1167339146588\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill A&amp; =&amp; lw\\hfill \\\\ &amp; =&amp; x\\cdot 1\\hfill \\\\ \\hfill &amp; =&amp; x\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167339157230\">The area of the front of the doghouse can be found by adding the areas of the square and the triangle, and then subtracting the area of the rectangle. When we do this, we get[latex]\\,4{x}^{2}+\\frac{3}{2}x-x\\,{\\text{ft}}^{2},[\/latex]or[latex]\\,4{x}^{2}+\\frac{1}{2}x\\,[\/latex]ft<sup>2<\/sup>.<\/p>\n<p id=\"fs-id1167339429031\">In this section, we will examine expressions such as this one, which combine several variable terms.<\/p>\n\n<div id=\"fs-id1167339226748\" class=\"bc-section section\">\n<h3>Identifying the Degree and Leading Coefficient of Polynomials<\/h3>\n<p id=\"fs-id1167339185162\">The formula just found is an example of a polynomial, which is a sum of or difference of terms, each consisting of a variable raised to a nonnegative integer power. A number multiplied by a variable raised to an exponent, such as[latex]\\,384\\pi ,[\/latex]is known as a coefficient. Coefficients can be positive, negative, or zero, and can be whole numbers, decimals, or fractions. Each product[latex]\\,{a}_{i}{x}^{i},[\/latex]such as[latex]\\,384\\pi w,[\/latex]is a term of a polynomial. If a term does not contain a variable, it is called a <em>constant<\/em>.<\/p>\n<p id=\"fs-id1167339138167\">A polynomial containing only one term, such as[latex]\\,5{x}^{4},[\/latex]is called a monomial. A polynomial containing two terms, such as[latex]\\,2x-9,[\/latex]is called a binomial. A polynomial containing three terms, such as[latex]\\,-3{x}^{2}+8x-7,[\/latex]is called a trinomial.<\/p>\n<p id=\"fs-id1167339222212\">We can find the degree of a polynomial by identifying the highest power of the variable that occurs in the polynomial. The term with the highest degree is called the leading term because it is usually written first. The coefficient of the leading term is called the leading coefficient. When a polynomial is written so that the powers are descending, we say that it is in standard form.<\/p>\n<span id=\"fs-id1167339243309\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132107\/CNX_CAT_Figure_01_04_002.jpg\" alt=\"A polynomial reading: a sub n times x to the nth power plus and so on plus a sub 2 times x squared plus a sub one times x plus a subzero is shown. The a in the term a sub n is labeled: leading coefficient. The n in the term x to the nth power is labeled: degree. Finally, the entire term is labeled as: Leading term.\"><\/span>\n<div id=\"fs-id1167339137964\" class=\"textbox key-takeaways\">\n<h3>Polynomials<\/h3>\n<p id=\"fs-id1167339155557\">A polynomial is an expression that can be written in the form<\/p>\n\n<div id=\"fs-id1167339177022\" class=\"unnumbered aligncenter\">[latex]{a}_{n}{x}^{n}+...+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/div>\n<p id=\"fs-id1167339220598\">Each real number <em>a<sub>i<\/sub><\/em>is called a coefficient. The number[latex]\\,{a}_{0}\\,[\/latex]that is not multiplied by a variable is called a <em>constant<\/em>. Each product[latex]\\,{a}_{i}{x}^{i}\\,[\/latex]is a term of a polynomial. The highest power of the variable that occurs in the polynomial is called the degree of a polynomial. The leading term is the term with the highest power, and its coefficient is called the leading coefficient.<\/p>\n\n<\/div>\n<div id=\"fs-id1167339231820\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1167339212575\"><strong>Given a polynomial expression, identify the degree and leading coefficient<\/strong>.<\/p>\n\n<ol id=\"fs-id1167339329242\" type=\"1\">\n \t<li>Find the highest power of <em>x<\/em> to determine the degree.<\/li>\n \t<li>Identify the term containing the highest power of <em>x<\/em> to find the leading term.<\/li>\n \t<li>Identify the coefficient of the leading term.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_04_01\" class=\"textbox examples\">\n<div id=\"fs-id1167339145967\">\n<div id=\"fs-id1167339145969\">\n<h3>Identifying the Degree and Leading Coefficient of a Polynomial<\/h3>\n<p id=\"fs-id1167339329112\">For the following polynomials, identify the degree, the leading term, and the leading coefficient.<\/p>\n\n<ol id=\"fs-id1167339329115\" type=\"a\">\n \t<li>[latex]3+2{x}^{2}-4{x}^{3}[\/latex]<\/li>\n \t<li>[latex]5{t}^{5}-2{t}^{3}+7t[\/latex]<\/li>\n \t<li>[latex]6p-{p}^{3}-2[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1167339276093\" class=\"solution textbox shaded\">\n[reveal-answer q=\"125678\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"125678\"]\n<ol id=\"fs-id1167339184064\" type=\"a\">\n \t<li>The highest power of <em>x<\/em> is 3, so the degree is 3. The leading term is the term containing that degree,[latex]\\,-4{x}^{3}.\\,[\/latex]The leading coefficient is the coefficient of that term,[latex]\\,-4.[\/latex]<\/li>\n \t<li>The highest power of <em>t<\/em> is[latex]\\,5,[\/latex]so the degree is[latex]\\,5.\\,[\/latex]The leading term is the term containing that degree,[latex]\\,5{t}^{5}.\\,[\/latex]The leading coefficient is the coefficient of that term,[latex]\\,5.[\/latex]<\/li>\n \t<li>The highest power of <em>p<\/em> is[latex]\\,3,[\/latex]so the degree is[latex]\\,3.\\,[\/latex]The leading term is the term containing that degree,[latex]\\,-{p}^{3},[\/latex]The leading coefficient is the coefficient of that term,[latex]\\,-1.[\/latex]<\/li>\n<\/ol>\n[\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339220571\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_04_01\">\n<div id=\"fs-id1167339300326\">\n<p id=\"fs-id1167339300327\">Identify the degree, leading term, and leading coefficient of the polynomial[latex]\\,4{x}^{2}-{x}^{6}+2x-6.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339196860\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339196860\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339196860\"]\n<p id=\"fs-id1167339196862\">The degree is 6, the leading term is[latex]\\,-{x}^{6},[\/latex]and the leading coefficient is[latex]\\,-1.[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339243781\" class=\"bc-section section\">\n<h3>Adding and Subtracting Polynomials<\/h3>\n<p id=\"fs-id1167339223237\">We can add and subtract polynomials by combining like terms, which are terms that contain the same variables raised to the same exponents. For example,[latex]\\,5{x}^{2}\\,[\/latex]and[latex]\\,-2{x}^{2}\\,[\/latex]are like terms, and can be added to get[latex]\\,3{x}^{2},[\/latex]but[latex]\\,3x\\,[\/latex]and[latex]\\,3{x}^{2}\\,[\/latex]are not like terms, and therefore cannot be added.<\/p>\n\n<div id=\"fs-id1167339290474\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1167339199765\"><strong>Given multiple polynomials, add or subtract them to simplify the expressions.\n<\/strong><\/p>\n\n<ol id=\"fs-id1167339199769\" type=\"1\">\n \t<li>Combine like terms.<\/li>\n \t<li>Simplify and write in standard form.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_04_02\" class=\"textbox examples\">\n<div id=\"fs-id1167339306483\">\n<div id=\"fs-id1167339306486\" class=\"textbox\">\n<h3>Adding Polynomials<\/h3>\n<p id=\"fs-id1167339226270\">Find the sum.<\/p>\n<p id=\"fs-id1167339149553\">[latex]\\left(12{x}^{2}+9x-21\\right)+\\left(4{x}^{3}+8{x}^{2}-5x+20\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339300037\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339300037\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339300037\"]\n<p id=\"fs-id1317912\">[latex]\\begin{array}{cc}4{x}^{3}+\\left(12{x}^{2}+8{x}^{2}\\right)+\\left(9x-5x\\right)+\\left(-21+20\\right) \\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{\u2003\u2003}\\text{Combine like terms}.\\hfill \\\\ 4{x}^{3}+20{x}^{2}+4x-1\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{\u2003\u2003}\\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div id=\"fs-id1167339213779\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1167339259571\">We can check our answers to these types of problems using a graphing calculator. To check, graph the problem as given along with the simplified answer. The two graphs should be equivalent. Be sure to use the same window to compare the graphs. Using different windows can make the expressions seem equivalent when they are not.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339243320\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_04_02\">\n<div id=\"fs-id1167339226215\">\n<p id=\"fs-id1167339226216\">Find the sum.<\/p>\n<p id=\"fs-id1167339330158\">[latex]\\left(2{x}^{3}+5{x}^{2}-x+1\\right)+\\left(2{x}^{2}-3x-4\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339223528\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339223528\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339223528\"]\n<p id=\"fs-id1167339223529\">[latex]2{x}^{3}+7{x}^{2}-4x-3[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_04_03\" class=\"textbox examples\">\n<div id=\"fs-id1167339428763\">\n<div id=\"fs-id1167339240332\">\n<h3>Subtracting Polynomials<\/h3>\n<p id=\"fs-id1167339240338\">Find the difference.<\/p>\n<p id=\"fs-id1167339432947\">[latex]\\left(7{x}^{4}-{x}^{2}+6x+1\\right)-\\left(5{x}^{3}-2{x}^{2}+3x+2\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339158579\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339158579\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339158579\"]\n<p id=\"fs-id1167339158581\">[latex]\\begin{array}{cc}7{x}^{4}-5{x}^{3}+\\left(-{x}^{2}+2{x}^{2}\\right)+\\left(6x-3x\\right)+\\left(1-2\\right)\\text{ }\\hfill &amp; \\phantom{\\rule{1em}{0ex}}\\text{Combine like terms}.\\hfill \\\\ 7{x}^{4}-5{x}^{3}+{x}^{2}+3x-1\\hfill &amp; \\phantom{\\rule{1em}{0ex}}\\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div id=\"fs-id1167339185145\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1167339185150\">Note that finding the difference between two polynomials is the same as adding the opposite of the second polynomial to the first.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339156471\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_04_03\">\n<div id=\"fs-id1167339317929\">\n<p id=\"fs-id1167339317930\">Find the difference.<\/p>\n<p id=\"fs-id1167339184123\">[latex]\\left(-7{x}^{3}-7{x}^{2}+6x-2\\right)-\\left(4{x}^{3}-6{x}^{2}-x+7\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339260393\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339260393\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339260393\"]\n<p id=\"fs-id1167339260394\">[latex]-11{x}^{3}-{x}^{2}+7x-9[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339214258\" class=\"bc-section section\">\n<h3>Multiplying Polynomials<\/h3>\n<p id=\"fs-id1167339139443\">Multiplying polynomials is a bit more challenging than adding and subtracting polynomials. We must use the distributive property to multiply each term in the first polynomial by each term in the second polynomial. We then combine like terms. We can also use a shortcut called the FOIL method when multiplying binomials. Certain special products follow patterns that we can memorize and use instead of multiplying the polynomials by hand each time. We will look at a variety of ways to multiply polynomials.<\/p>\n\n<div id=\"fs-id1167339198311\" class=\"bc-section section\">\n<h4>Multiplying Polynomials Using the Distributive Property<\/h4>\n<p id=\"fs-id1167339198316\">To multiply a number by a polynomial, we use the distributive property. The number must be distributed to each term of the polynomial. We can distribute the[latex]\\,2\\,[\/latex]in[latex]\\,2\\left(x+7\\right)\\,[\/latex]to obtain the equivalent expression[latex]\\,2x+14.\\,[\/latex]When multiplying polynomials, the distributive property allows us to multiply each term of the first polynomial by each term of the second. We then add the products together and combine like terms to simplify.<\/p>\n\n<div id=\"fs-id1167339260622\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1167339170371\"><strong>Given the multiplication of two polynomials, use the distributive property to simplify the expression.<\/strong><\/p>\n\n<ol id=\"fs-id1167339170375\" type=\"1\">\n \t<li>Multiply each term of the first polynomial by each term of the second.<\/li>\n \t<li>Combine like terms.<\/li>\n \t<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_04_04\" class=\"textbox examples\">\n<div id=\"fs-id1167339239993\">\n<div id=\"fs-id1167339239995\">\n<h3>Multiplying Polynomials Using the Distributive Property<\/h3>\n<p id=\"fs-id1167339158245\">Find the product.<\/p>\n<p id=\"fs-id1167339158248\">[latex]\\left(2x+1\\right)\\left(3{x}^{2}-x+4\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339306275\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339306275\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339306275\"]\n<p id=\"fs-id1167339306277\">[latex]\\begin{array}{cc}2x\\left(3{x}^{2}-x+4\\right)+1\\left(3{x}^{2}-x+4\\right) \\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{\u2003\u2003}\\text{Use the distributive property}.\\hfill \\\\ \\left(6{x}^{3}-2{x}^{2}+8x\\right)+\\left(3{x}^{2}-x+4\\right)\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{\u2003\u2003}\\text{Multiply}.\\hfill \\\\ 6{x}^{3}+\\left(-2{x}^{2}+3{x}^{2}\\right)+\\left(8x-x\\right)+4\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{\u2003\u2003}\\text{Combine like terms}.\\hfill \\\\ 6{x}^{3}+{x}^{2}+7x+4 \\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{\u2003\u2003}\\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div id=\"fs-id1167339344747\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1167339158177\">We can use a table to keep track of our work, as shown in <a class=\"autogenerated-content\" href=\"#Table_01_04_01\">(Figure)<\/a>. Write one polynomial across the top and the other down the side. For each box in the table, multiply the term for that row by the term for that column. Then add all of the terms together, combine like terms, and simplify.<\/p>\n\n<table id=\"Table_01_04_01\" summary=\"A table with 3 rows and 4 columns. The first entry of the first row is empty, the others are labeled: three times x squared, negative x, and positive four. The first entry of the second row is labeled: two times x. The second entry reads: six times x cubed. The third entry reads: negative two times x squared. The fourth entry reads: eight times x. The first entry of the third row reads: positive one. The second entry reads: three times x squared. The third entry reads: negative x. The fourth entry reads: four.\"><caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]3{x}^{2}[\/latex]<\/td>\n<td>[latex]-x[\/latex]<\/td>\n<td>[latex]+4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2x[\/latex]<\/td>\n<td>[latex]6{x}^{3}[\/latex]<\/td>\n<td>[latex]-2{x}^{2}[\/latex]<\/td>\n<td>[latex]8x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]+1[\/latex]<\/td>\n<td>[latex]3{x}^{2}[\/latex]<\/td>\n<td>[latex]-x[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339260552\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_04_04\">\n<div id=\"fs-id1167339150646\">\n<p id=\"fs-id1167339150648\">Find the product.<\/p>\n<p id=\"fs-id1167339150651\">[latex]\\left(3x+2\\right)\\left({x}^{3}-4{x}^{2}+7\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339223342\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339223342\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339223342\"]\n<p id=\"fs-id1167339223343\">[latex]3{x}^{4}-10{x}^{3}-8{x}^{2}+21x+14[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339243382\" class=\"bc-section section\">\n<h4>Using FOIL to Multiply Binomials<\/h4>\n<p id=\"fs-id1167339243388\">A shortcut called FOIL is sometimes used to find the product of two binomials. It is called FOIL because we multiply the <strong>f<\/strong>irst terms, the <strong>o<\/strong>uter terms, the <strong>i<\/strong>nner terms, and then the <strong>l<\/strong>ast terms of each binomial.<\/p>\n<span id=\"fs-id1167339168201\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132109\/CNX_CAT_Figure_01_04_003.jpg\" alt=\"Two quantities in parentheses are being multiplied, the first being: a times x plus b and the second being: c times x plus d. This expression equals ac times x squared plus ad times x plus bc times x plus bd. The terms ax and cx are labeled: First Terms. The terms ax and d are labeled: Outer Terms. The terms b and cx are labeled: Inner Terms. The terms b and d are labeled: Last Terms.\"><\/span>\n<p id=\"fs-id1167339429089\">The FOIL method arises out of the distributive property. We are simply multiplying each term of the first binomial by each term of the second binomial, and then combining like terms.<\/p>\n\n<div id=\"fs-id1167339429094\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1167339240108\"><strong>Given two binomials, use FOIL to simplify the expression.<\/strong><\/p>\n\n<ol id=\"fs-id1167339240112\" type=\"1\">\n \t<li>Multiply the first terms of each binomial.<\/li>\n \t<li>Multiply the outer terms of the binomials.<\/li>\n \t<li>Multiply the inner terms of the binomials.<\/li>\n \t<li>Multiply the last terms of each binomial.<\/li>\n \t<li>Add the products.<\/li>\n \t<li>Combine like terms and simplify.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_04_05\" class=\"textbox examples\">\n<div>\n<div>\n<h3>Using FOIL to Multiply Binomials<\/h3>\nUse FOIL to find the product.\n<p id=\"eip-id6327005\">[latex]\\left(2x-18\\right)\\left(3x+3\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"eip-id1167338039926\">\n\n[reveal-answer q=\"394130\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"394130\"]\n<p id=\"eip-id1673659\">Find the product of the first terms.<\/p>\n<span id=\"eip-id1673664\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132116\/CNX_CAT_Figure_01_04_004.jpg\" alt=\"\"><\/span>\n<p id=\"eip-id1673675\">Find the product of the outer terms.<\/p>\n<span id=\"eip-id1673679\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132124\/CNX_CAT_Figure_01_04_005.jpg\" alt=\"\"><\/span>\n<p id=\"eip-id1923767\">Find the product of the inner terms.<\/p>\n<span id=\"eip-id1923771\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132144\/CNX_CAT_Figure_01_04_006.jpg\" alt=\"\"><\/span>\n<p id=\"eip-id1923782\">Find the product of the last terms.<\/p>\n<span id=\"eip-id1923785\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132147\/CNX_CAT_Figure_01_04_007.jpg\" alt=\"\"><\/span>\n<p id=\"eip-id1167336078703\">[latex]\\begin{array}{cc}6{x}^{2}+6x-54x-54\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Add the products}.\\hfill \\\\ 6{x}^{2}+\\left(6x-54x\\right)-54\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Combine like terms}.\\hfill \\\\ 6{x}^{2}-48x-54\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\n<p id=\"eip-id1167336078703\">[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339223541\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_04_05\">\n<div id=\"fs-id1167339241487\">\n<p id=\"fs-id1167339241488\">Use FOIL to find the product.<\/p>\n<p id=\"fs-id1167339241491\">[latex]\\left(x+7\\right)\\left(3x-5\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339216120\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339216120\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339216120\"]\n<p id=\"fs-id1167339216121\">[latex]3{x}^{2}+16x-35[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339428910\" class=\"bc-section section\">\n<h4>Perfect Square Trinomials<\/h4>\n<p id=\"fs-id1167339432924\">Certain binomial products have special forms. When a binomial is squared, the result is called a perfect square trinomial. We can find the square by multiplying the binomial by itself. However, there is a special form that each of these perfect square trinomials takes, and memorizing the form makes squaring binomials much easier and faster. Let\u2019s look at a few perfect square trinomials to familiarize ourselves with the form.<\/p>\n\n<div id=\"fs-id1167339433593\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\text{ }{\\left(x+5\\right)}^{2}&amp; =&amp; {x}^{2}+10x+25\\hfill \\\\ \\hfill {\\left(x-3\\right)}^{2}&amp; =&amp; \\text{ }{x}^{2}-6x+9\\hfill \\\\ \\hfill {\\left(4x-1\\right)}^{2}&amp; =&amp; 16{x}^{2}-8x+1\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167339224293\">Notice that the first term of each trinomial is the square of the first term of the binomial and, similarly, the last term of each trinomial is the square of the last term of the binomial. The middle term is double the product of the two terms. Lastly, we see that the first sign of the trinomial is the same as the sign of the binomial.<\/p>\n\n<div id=\"fs-id1167339224300\" class=\"textbox key-takeaways\">\n<h3>Perfect Square Trinomials<\/h3>\n<p id=\"fs-id1167339330148\">When a binomial is squared, the result is the first term squared added to double the product of both terms and the last term squared.<\/p>\n\n<div id=\"Equation_01_04_01\">[latex]{\\left(x+a\\right)}^{2}=\\left(x+a\\right)\\left(x+a\\right)={x}^{2}+2ax+{a}^{2}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1167339220526\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1167339220533\"><strong>Given a binomial, square it using the formula for perfect square trinomials.\n<\/strong><\/p>\n\n<ol id=\"fs-id1167339231805\" type=\"1\">\n \t<li>Square the first term of the binomial.<\/li>\n \t<li>Square the last term of the binomial.<\/li>\n \t<li>For the middle term of the trinomial, double the product of the two terms.<\/li>\n \t<li>Add and simplify.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_04_06\" class=\"textbox examples\">\n<div id=\"fs-id1167339273765\">\n<div id=\"fs-id1167339344760\">\n<h3>Expanding Perfect Squares<\/h3>\n<p id=\"fs-id1167339344765\">Expand[latex]\\,{\\left(3x-8\\right)}^{2}.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339428957\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339428957\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339428957\"]\n<p id=\"fs-id1167339428959\">Begin by squaring the first term and the last term. For the middle term of the trinomial, double the product of the two terms.<\/p>\n\n<div id=\"fs-id1167339428964\" class=\"unnumbered aligncenter\">[latex]{\\left(3x\\right)}^{2}-2\\left(3x\\right)\\left(8\\right)+{\\left(-8\\right)}^{2}[\/latex][\/hidden-answer]<\/div>\n<p id=\"fs-id1167339299914\">Simplify<\/p>\n\n<div id=\"eip-id1171852212181\">[latex]\\,9{x}^{2}-48x+64.[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339328808\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_04_06\">\n<div id=\"fs-id1167339328819\">\n<p id=\"fs-id1167339328820\">Expand [latex]\\,{\\left(4x-1\\right)}^{2}.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339176899\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339176899\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339176899\"]\n<p id=\"fs-id1167339176900\">[latex]16{x}^{2}-8x+1[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339429838\" class=\"bc-section section\">\n<h4>Difference of Squares<\/h4>\n<p id=\"fs-id1167339188447\">Another special product is called the difference of squares, which occurs when we multiply a binomial by another binomial with the same terms but the opposite sign. Let\u2019s see what happens when we multiply[latex]\\,\\left(x+1\\right)\\left(x-1\\right)\\,[\/latex]using the FOIL method.<\/p>\n\n<div id=\"fs-id1167339428778\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\left(x+1\\right)\\left(x-1\\right)&amp; =&amp; {x}^{2}-x+x-1\\hfill \\\\ &amp; =&amp; {x}^{2}-1\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167339432898\">The middle term drops out, resulting in a difference of squares. Just as we did with the perfect squares, let\u2019s look at a few examples.<\/p>\n\n<div id=\"fs-id1167339432904\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\left(x+5\\right)\\left(x-5\\right)&amp; =&amp; {x}^{2}-25\\hfill \\\\ \\hfill \\left(x+11\\right)\\left(x-11\\right)&amp; =&amp; {x}^{2}-121\\hfill \\\\ \\hfill \\left(2x+3\\right)\\left(2x-3\\right)&amp; =&amp; 4{x}^{2}-9\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167339220812\">Because the sign changes in the second binomial, the outer and inner terms cancel each other out, and we are left only with the square of the first term minus the square of the last term.<\/p>\n\n<div id=\"fs-id1167339220817\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1167339220824\"><strong>Is there a special form for the sum of squares?<\/strong><\/p>\n<p id=\"fs-id1167339199657\"><em>No. The difference of squares occurs because the opposite signs of the binomials cause the middle terms to disappear. There are no two binomials that multiply to equal a sum of squares.<\/em><\/p>\n\n<\/div>\n<div id=\"fs-id1167339199666\" class=\"textbox key-takeaways\">\n<h3>Difference of Squares<\/h3>\n<p id=\"fs-id1167339199674\">When a binomial is multiplied by a binomial with the same terms separated by the opposite sign, the result is the square of the first term minus the square of the last term.<\/p>\n\n<div id=\"Equation_01_04_02\">[latex]\\left(a+b\\right)\\left(a-b\\right)={a}^{2}-{b}^{2}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1167339137913\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1167339273978\"><strong>Given a binomial multiplied by a binomial with the same terms but the opposite sign, find the difference of squares.<\/strong><\/p>\n\n<ol id=\"fs-id1167339273983\" type=\"1\">\n \t<li>Square the first term of the binomials.<\/li>\n \t<li>Square the last term of the binomials.<\/li>\n \t<li>Subtract the square of the last term from the square of the first term.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_04_07\" class=\"textbox examples\">\n<div id=\"fs-id1167339315486\">\n<div id=\"fs-id1167339315488\">\n<h3>Multiplying Binomials Resulting in a Difference of Squares<\/h3>\n<p id=\"fs-id1167339315493\">Multiply[latex]\\,\\left(9x+4\\right)\\left(9x-4\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339273948\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339273948\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339273948\"]\n<p id=\"fs-id1167339273950\">Square the first term to get[latex]\\,{\\left(9x\\right)}^{2}=81{x}^{2}.\\,[\/latex]Square the last term to get[latex]\\,{4}^{2}=16.\\,[\/latex]Subtract the square of the last term from the square of the first term to find the product of[latex]\\,81{x}^{2}-16.[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339138650\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_04_07\">\n<div id=\"fs-id1167339212421\">\n<p id=\"fs-id1167339212422\">Multiply[latex]\\,\\left(2x+7\\right)\\left(2x-7\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339318295\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339318295\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339318295\"]\n<p id=\"fs-id1167339318296\">[latex]4{x}^{2}-49[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339138125\" class=\"bc-section section\">\n<h3>Performing Operations with Polynomials of Several Variables<\/h3>\n<p id=\"fs-id1167339138130\">We have looked at polynomials containing only one variable. However, a polynomial can contain several variables. All of the same rules apply when working with polynomials containing several variables. Consider an example:<\/p>\n\n<div id=\"fs-id1167339429970\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{cc}\\left(a+2b\\right)\\left(4a-b-c\\right)\\hfill &amp; \\hfill \\\\ a\\left(4a-b-c\\right)+2b\\left(4a-b-c\\right)\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Use the distributive property}.\\hfill \\\\ 4{a}^{2}-ab-ac+8ab-2{b}^{2}-2bc\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Multiply}.\\hfill \\\\ 4{a}^{2}+\\left(-ab+8ab\\right)-ac-2{b}^{2}-2bc\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Combine like terms}.\\hfill \\\\ 4{a}^{2}+7ab-ac-2bc-2{b}^{2}\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Simplify}.\\hfill \\end{array}[\/latex]<\/div>\n<div id=\"Example_01_04_08\" class=\"textbox examples\">\n<div id=\"fs-id1167339318091\">\n<div id=\"fs-id1167339318093\">\n<h3>Multiplying Polynomials Containing Several Variables<\/h3>\n<p id=\"fs-id1167339315450\">Multiply[latex]\\,\\left(x+4\\right)\\left(3x-2y+5\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339299948\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339299948\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339299948\"]\n<p id=\"fs-id1167339299950\">Follow the same steps that we used to multiply polynomials containing only one variable.<\/p>\n\n<div id=\"fs-id1167339299954\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{cc}x\\left(3x-2y+5\\right)+4\\left(3x-2y+5\\right) \\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Use the distributive property}.\\hfill \\\\ 3{x}^{2}-2xy+5x+12x-8y+20\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Multiply}.\\hfill \\\\ 3{x}^{2}-2xy+\\left(5x+12x\\right)-8y+20\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Combine like terms}.\\hfill \\\\ 3{x}^{2}-2xy+17x-8y+20 \\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Simplify}.\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339240133\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_04_08\">\n<div id=\"fs-id1167339240144\">\n<p id=\"fs-id1167339240145\">Multiply [latex]\\left(3x-1\\right)\\left(2x+7y-9\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339240262\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339240262\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339240262\"]\n<p id=\"fs-id1167339240263\">[latex]\\,6{x}^{2}+21xy-29x-7y+9[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339315650\" class=\"precalculus media\">\n<p id=\"fs-id1167339219216\">Access these online resources for additional instruction and practice with polynomials.<\/p>\n\n<ul id=\"fs-id1167339219219\">\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/addsubpoly\">Adding and Subtracting Polynomials<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/multiplpoly\">Multiplying Polynomials<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/specialpolyprod\">Special Products of Polynomials<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339219247\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"fs-id1167339219254\" summary=\"..\">\n<tbody>\n<tr>\n<td>perfect square trinomial<\/td>\n<td>[latex]{\\left(x+a\\right)}^{2}=\\left(x+a\\right)\\left(x+a\\right)={x}^{2}+2ax+{a}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>difference of squares<\/td>\n<td>[latex]\\left(a+b\\right)\\left(a-b\\right)={a}^{2}-{b}^{2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1167339242305\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1167339242312\">\n \t<li>A polynomial is a sum of terms each consisting of a variable raised to a non-negative integer power. The degree is the highest power of the variable that occurs in the polynomial. The leading term is the term containing the highest degree, and the leading coefficient is the coefficient of that term. See <a class=\"autogenerated-content\" href=\"#Example_01_04_01\">(Figure)<\/a>.<\/li>\n \t<li>We can add and subtract polynomials by combining like terms. See <a class=\"autogenerated-content\" href=\"#Example_01_04_02\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_01_04_03\">(Figure)<\/a>.<\/li>\n \t<li>To multiply polynomials, use the distributive property to multiply each term in the first polynomial by each term in the second. Then add the products. See <a class=\"autogenerated-content\" href=\"#Example_01_04_04\">(Figure)<\/a>.<\/li>\n \t<li>FOIL (First, Outer, Inner, Last) is a shortcut that can be used to multiply binomials. See <a class=\"autogenerated-content\" href=\"#Example_01_04_05\">(Figure)<\/a>.<\/li>\n \t<li>Perfect square trinomials and difference of squares are special products. See <a class=\"autogenerated-content\" href=\"#Example_01_04_06\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_01_04_07\">(Figure)<\/a>.<\/li>\n \t<li>Follow the same rules to work with polynomials containing several variables. See <a class=\"autogenerated-content\" href=\"#Example_01_04_08\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1167339184236\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1167339184243\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1167339184249\">\n<div id=\"fs-id1167339184250\">\n<p id=\"fs-id1167339184251\">Evaluate the following statement: The degree of a polynomial in standard form is the exponent of the leading term. Explain why the statement is true or false.<\/p>\n\n<\/div>\n<div id=\"fs-id1167339216132\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339216132\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339216132\"]\n<p id=\"fs-id1167339216133\">The statement is true. In standard form, the polynomial with the highest value exponent is placed first and is the leading term. The degree of a polynomial is the value of the highest exponent, which in standard form is also the exponent of the leading term.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339216138\">\n<div id=\"fs-id1167339216140\">\n<p id=\"fs-id1167339216141\">Many times, multiplying two binomials with two variables results in a trinomial. This is not the case when there is a difference of two squares. Explain why the product in this case is also a binomial.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339216146\">\n<div id=\"fs-id1167339216147\">\n<p id=\"fs-id1167339216148\">You can multiply polynomials with any number of terms and any number of variables using four basic steps over and over until you reach the expanded polynomial. What are the four steps?<\/p>\n\n<\/div>\n<div id=\"fs-id1167339216153\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339216153\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339216153\"]\n<p id=\"fs-id1167339216154\">Use the distributive property, multiply, combine like terms, and simplify.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339216157\">\n<div id=\"fs-id1167339216158\">\n<p id=\"fs-id1167339216159\">State whether the following statement is true and explain why or why not: A trinomial is always a higher degree than a monomial.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339216165\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1167339216170\">For the following exercises, identify the degree of the polynomial.<\/p>\n\n<div id=\"fs-id1167339216173\">\n<div id=\"fs-id1167339216174\">\n<p id=\"fs-id1167339216175\">[latex]7x-2{x}^{2}+13[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339299770\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339299770\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339299770\"]\n<p id=\"fs-id1167339299771\">2<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339299774\">\n<div id=\"fs-id1167339299775\">\n<p id=\"fs-id1167339299776\">[latex]14{m}^{3}+{m}^{2}-16m+8[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339299820\">\n<div id=\"fs-id1167339299821\">\n<p id=\"fs-id1167339299822\">[latex]-625{a}^{8}+16{b}^{4}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339268841\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339268841\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339268841\"]\n<p id=\"fs-id1167339268842\">8<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339268846\">\n<div id=\"fs-id1167339268847\">\n<p id=\"fs-id1167339268848\">[latex]200p-30{p}^{2}m+40{m}^{3}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339223145\">\n<div id=\"fs-id1167339223146\">\n<p id=\"fs-id1167339223147\">[latex]{x}^{2}+4x+4[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339223177\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339223177\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339223177\"]\n<p id=\"fs-id1167339223178\">2<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339223181\">\n<div id=\"fs-id1167339223182\">\n<p id=\"fs-id1167339223184\">[latex]6{y}^{4}-{y}^{5}+3y-4[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1167339432583\">For the following exercises, find the sum or difference.<\/p>\n\n<div id=\"fs-id1167339432587\">\n<div id=\"fs-id1167339432588\">\n<p id=\"fs-id1167339432589\">[latex]\\left(12{x}^{2}+3x\\right)-\\left(8{x}^{2}-19\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339432656\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339432656\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339432656\"]\n<p id=\"fs-id1167339432657\">[latex]4{x}^{2}+3x+19[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339185745\">\n<div id=\"fs-id1167339185746\">\n<p id=\"fs-id1167339185747\">[latex]\\left(4{z}^{3}+8{z}^{2}-z\\right)+\\left(-2{z}^{2}+z+6\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339306370\">\n<div id=\"fs-id1167339306371\">\n<p id=\"fs-id1167339306372\">[latex]\\left(6{w}^{2}+24w+24\\right)-\\left(3w{}^{2}-6w+3\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339223800\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339223800\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339223800\"]\n<p id=\"fs-id1167339223801\">[latex]3{w}^{2}+30w+21[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339223833\">\n<div id=\"fs-id1167339223834\">\n<p id=\"fs-id1167339223835\">[latex]\\left(7{a}^{3}+6{a}^{2}-4a-13\\right)+\\left(-3{a}^{3}-4{a}^{2}+6a+17\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339262640\">\n<div id=\"fs-id1167339262641\">\n<p id=\"fs-id1167339262642\">[latex]\\left(11{b}^{4}-6{b}^{3}+18{b}^{2}-4b+8\\right)-\\left(3{b}^{3}+6{b}^{2}+3b\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339273678\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339273678\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339273678\"]\n<p id=\"fs-id1167339273679\">[latex]11{b}^{4}-9{b}^{3}+12{b}^{2}-7b+8[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339196660\">\n<div id=\"fs-id1167339196661\">\n<p id=\"fs-id1167339196662\">[latex]\\left(49{p}^{2}-25\\right)+\\left(16{p}^{4}-32{p}^{2}+16\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1167339196741\">For the following exercises, find the product.<\/p>\n\n<div id=\"fs-id1167339196745\">\n<div id=\"fs-id1167339196746\">\n<p id=\"fs-id1167339196747\">[latex]\\left(4x+2\\right)\\left(6x-4\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339259665\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339259665\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339259665\"]\n<p id=\"fs-id1167339259666\">[latex]24{x}^{2}-4x-8[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339259698\">\n<div id=\"fs-id1167339259699\">\n<p id=\"fs-id1167339259700\">[latex]\\left(14{c}^{2}+4c\\right)\\left(2{c}^{2}-3c\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339223391\">\n<div id=\"fs-id1167339223392\">\n<p id=\"fs-id1167339223394\">[latex]\\left(6{b}^{2}-6\\right)\\left(4{b}^{2}-4\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339223456\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339223456\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339223456\"]\n<p id=\"fs-id1167339223458\">[latex]24{b}^{4}-48{b}^{2}+24[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339223497\">\n<div id=\"fs-id1167339223498\">\n<p id=\"fs-id1167339223499\">[latex]\\left(3d-5\\right)\\left(2d+9\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339306553\">\n<div id=\"fs-id1167339306554\">\n<p id=\"fs-id1167339306555\">[latex]\\left(9v-11\\right)\\left(11v-9\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339306603\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339306603\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339306603\"]\n<p id=\"fs-id1167339306604\">[latex]99{v}^{2}-202v+99[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339286443\">\n<div id=\"fs-id1167339286444\">\n<p id=\"fs-id1167339286446\">[latex]\\left(4{t}^{2}+7t\\right)\\left(-3{t}^{2}+4\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339286513\">\n<div id=\"fs-id1167339286514\">\n<p id=\"fs-id1167339286515\">[latex]\\left(8n-4\\right)\\left({n}^{2}+9\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339286568\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339286568\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339286568\"]\n<p id=\"fs-id1167339286569\">[latex]8{n}^{3}-4{n}^{2}+72n-36[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1167339199464\">For the following exercises, expand the binomial.<\/p>\n\n<div id=\"fs-id1167339199468\">\n<div id=\"fs-id1167339199469\">\n<p id=\"fs-id1167339199470\">[latex]{\\left(4x+5\\right)}^{2}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339199508\">\n<div id=\"fs-id1167339199509\">\n<p id=\"fs-id1167339199510\">[latex]{\\left(3y-7\\right)}^{2}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339199549\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339199549\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339199549\"]\n<p id=\"fs-id1167339199550\">[latex]9{y}^{2}-42y+49[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339220230\">\n<div id=\"fs-id1167339220232\">\n<p id=\"fs-id1167339220233\">[latex]{\\left(12-4x\\right)}^{2}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339220271\">\n<div id=\"fs-id1167339220272\">[latex]{\\left(4p+9\\right)}^{2}[\/latex]<\/div>\n<div id=\"fs-id1167339220312\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339220312\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339220312\"]\n<p id=\"fs-id1167339220313\">[latex]16{p}^{2}+72p+81[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339220345\">\n<div id=\"fs-id1167339220346\">\n<p id=\"fs-id1167339220347\">[latex]{\\left(2m-3\\right)}^{2}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339226325\">\n<div id=\"fs-id1167339226326\">\n<p id=\"fs-id1167339226327\">[latex]{\\left(3y-6\\right)}^{2}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339226366\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339226366\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339226366\"]\n<p id=\"fs-id1167339226367\">[latex]9{y}^{2}-36y+36[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339226399\">\n<div id=\"fs-id1167339226400\">\n<p id=\"fs-id1167339226401\">[latex]{\\left(9b+1\\right)}^{2}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1167339196445\">For the following exercises, multiply the binomials.<\/p>\n\n<div id=\"fs-id1167339196448\">\n<div id=\"fs-id1167339196449\">\n<p id=\"fs-id1167339196450\">[latex]\\left(4c+1\\right)\\left(4c-1\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339196498\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339196498\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339196498\"]\n<p id=\"fs-id1167339196499\">[latex]16{c}^{2}-1[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339196525\">\n<div id=\"fs-id1167339196526\">\n<p id=\"fs-id1167339196527\">[latex]\\left(9a-4\\right)\\left(9a+4\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339196575\">\n<div id=\"fs-id1167339196576\">\n<p id=\"fs-id1167339196577\">[latex]\\left(15n-6\\right)\\left(15n+6\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339260447\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339260447\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339260447\"]\n<p id=\"fs-id1167339260448\">[latex]225{n}^{2}-36[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339260474\">\n<div id=\"fs-id1167339260475\">\n<p id=\"fs-id1167339260476\">[latex]\\left(25b+2\\right)\\left(25b-2\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339260524\">\n<div id=\"fs-id1167339260525\">\n<p id=\"fs-id1167339260526\">[latex]\\left(4+4m\\right)\\left(4-4m\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339239040\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339239040\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339239040\"]\n<p id=\"fs-id1167339239041\">[latex]-16{m}^{2}+16[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339239069\">\n<div id=\"fs-id1167339239070\">\n<p id=\"fs-id1167339239071\">[latex]\\left(14p+7\\right)\\left(14p-7\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339239119\">\n<div id=\"fs-id1167339239120\">\n<p id=\"fs-id1167339239121\">[latex]\\left(11q-10\\right)\\left(11q+10\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339239169\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339239169\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339239169\"]\n<p id=\"fs-id1167339321332\">[latex]121{q}^{2}-100[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1167339321358\">For the following exercises, multiply the polynomials.<\/p>\n\n<div id=\"fs-id1167339321361\">\n<div id=\"fs-id1167339321362\">\n<p id=\"fs-id1167339321363\">[latex]\\left(2{x}^{2}+2x+1\\right)\\left(4x-1\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339321425\">\n<div id=\"fs-id1167339321426\">\n<p id=\"fs-id1167339321427\">[latex]\\left(4{t}^{2}+t-7\\right)\\left(4{t}^{2}-1\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339138777\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339138777\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339138777\"]\n<p id=\"fs-id1167339138778\">[latex]16{t}^{4}+4{t}^{3}-32{t}^{2}-t+7[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339138836\">\n<div id=\"fs-id1167339138837\">\n<p id=\"fs-id1167339138838\">[latex]\\left(x-1\\right)\\left({x}^{2}-2x+1\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339138896\">\n<div id=\"fs-id1167339138897\">\n<p id=\"fs-id1167339138898\">[latex]\\left(y-2\\right)\\left({y}^{2}-4y-9\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339273787\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339273787\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339273787\"]\n<p id=\"fs-id1167339273788\">[latex]{y}^{3}-6{y}^{2}-y+18[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339273830\">\n<div id=\"fs-id1167339273831\">\n<p id=\"fs-id1167339273832\">[latex]\\left(6k-5\\right)\\left(6{k}^{2}+5k-1\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339273894\">\n<div id=\"fs-id1167339273895\">\n<p id=\"fs-id1167339273896\">[latex]\\left(3{p}^{2}+2p-10\\right)\\left(p-1\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339344407\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339344407\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339344407\"]\n<p id=\"fs-id1167339344408\">[latex]3{p}^{3}-{p}^{2}-12p+10[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339344452\">\n<div id=\"fs-id1167339344453\">\n<p id=\"fs-id1167339344454\">[latex]\\left(4m-13\\right)\\left(2{m}^{2}-7m+9\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339344516\">\n<div id=\"fs-id1167339344517\">\n<p id=\"fs-id1167339344518\">[latex]\\left(a+b\\right)\\left(a-b\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339344562\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339344562\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339344562\"]\n<p id=\"fs-id1167339213918\">[latex]{a}^{2}-{b}^{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339213948\">\n<div id=\"fs-id1167339213949\">\n<p id=\"fs-id1167339213950\">[latex]\\left(4x-6y\\right)\\left(6x-4y\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339214002\">\n<div id=\"fs-id1167339214003\">\n<p id=\"fs-id1167339214004\">[latex]{\\left(4t-5u\\right)}^{2}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339214045\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339214045\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339214045\"]\n<p id=\"fs-id1167339214046\">[latex]16{t}^{2}-40tu+25{u}^{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339214089\">\n<div id=\"fs-id1167339214090\">\n<p id=\"fs-id1167339214091\">[latex]\\left(9m+4n-1\\right)\\left(2m+8\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339222871\">\n<div id=\"fs-id1167339222872\">\n<p id=\"fs-id1167339222873\">[latex]\\left(4t-x\\right)\\left(t-x+1\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339222924\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339222924\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339222924\"]\n<p id=\"fs-id1167339222925\">[latex]4{t}^{2}+{x}^{2}+4t-5tx-x[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339222977\">\n<div id=\"fs-id1167339222978\">\n<p id=\"fs-id1167339222979\">[latex]\\left({b}^{2}-1\\right)\\left({a}^{2}+2ab+{b}^{2}\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339223053\">\n<div id=\"fs-id1167339223054\">\n<p id=\"fs-id1167339223055\">[latex]\\left(4r-d\\right)\\left(6r+7d\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339225614\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339225614\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339225614\"]\n<p id=\"fs-id1167339225615\">[latex]24{r}^{2}+22rd-7{d}^{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339225658\">\n<div id=\"fs-id1167339225659\">\n<p id=\"fs-id1167339225660\">[latex]\\left(x+y\\right)\\left({x}^{2}-xy+{y}^{2}\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339225725\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<div id=\"fs-id1167339225730\">\n<div id=\"fs-id1167339225731\">\n<p id=\"fs-id1167339437824\">A developer wants to purchase a plot of land to build a house. The area of the plot can be described by the following expression:[latex]\\,\\left(4x+1\\right)\\left(8x-3\\right)\\,[\/latex]where <em>x<\/em> is measured in meters. Multiply the binomials to find the area of the plot in standard form.<\/p>\n\n<\/div>\n<div id=\"fs-id1167339225791\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339225791\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339225791\"]\n<p id=\"fs-id1167339225792\">[latex]32{x}^{2}-4x-3\\,[\/latex]m<sup>2<\/sup><\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339242981\">\n<div id=\"fs-id1167339242982\">\n<p id=\"fs-id1167339242983\">A prospective buyer wants to know how much grain a specific silo can hold. The area of the floor of the silo is[latex]\\,{\\left(2x+9\\right)}^{2}.\\,[\/latex]The height of the silo is[latex]\\,10x+10,[\/latex]where <em>x<\/em> is measured in feet. Expand the square and multiply by the height to find the expression that shows how much grain the silo can hold.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339243059\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<p id=\"fs-id1167339243064\">For the following exercises, perform the given operations.<\/p>\n\n<div id=\"fs-id1167339243067\">\n<div id=\"fs-id1167339243068\">\n<p id=\"fs-id1167339243069\">[latex]{\\left(4t-7\\right)}^{2}\\left(2t+1\\right)-\\left(4{t}^{2}+2t+11\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339243163\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339243163\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339243163\"]\n<p id=\"fs-id1167339243164\">[latex]32{t}^{3}-100{t}^{2}+40t+38[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339230814\">\n<div id=\"fs-id1167339230815\">\n<p id=\"fs-id1167339230816\">[latex]\\left(3b+6\\right)\\left(3b-6\\right)\\left(9{b}^{2}-36\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1167339230891\">\n<div id=\"fs-id1167339230892\">\n<p id=\"fs-id1167339219515\">[latex]\\left({a}^{2}+4ac+4{c}^{2}\\right)\\left({a}^{2}-4{c}^{2}\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1167339230976\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1167339230976\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1167339230976\"]\n<p id=\"fs-id1167339230977\">[latex]{a}^{4}+4{a}^{3}c-16a{c}^{3}-16{c}^{4}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1167339280889\">\n \t<dt>binomial<\/dt>\n \t<dd id=\"fs-id1167339280892\">a polynomial containing two terms<\/dd>\n<\/dl>\n<dl id=\"fs-id1167339280895\">\n \t<dt>coefficient<\/dt>\n \t<dd id=\"fs-id1167339280898\">any real number[latex]\\,{a}_{i}\\,[\/latex]in a polynomial in the form[latex]\\,{a}_{n}{x}^{n}+...+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1167339281002\">\n \t<dt>degree<\/dt>\n \t<dd>the highest power of the variable that occurs in a polynomial<\/dd>\n<\/dl>\n<dl id=\"fs-id1167339281008\">\n \t<dt>difference of squares<\/dt>\n \t<dd id=\"fs-id1167339281011\">the binomial that results when a binomial is multiplied by a binomial with the same terms, but the opposite sign<\/dd>\n<\/dl>\n<dl id=\"fs-id1167339281016\">\n \t<dt>leading coefficient<\/dt>\n \t<dd id=\"fs-id1167339281019\">the coefficient of the leading term<\/dd>\n<\/dl>\n<dl id=\"fs-id1167339281022\">\n \t<dt>leading term<\/dt>\n \t<dd id=\"fs-id1167339281025\">the term containing the highest degree<\/dd>\n<\/dl>\n<dl id=\"fs-id1167339281028\">\n \t<dt>monomial<\/dt>\n \t<dd id=\"fs-id1167339281032\">a polynomial containing one term<\/dd>\n<\/dl>\n<dl id=\"fs-id1167339281035\">\n \t<dt>perfect square trinomial<\/dt>\n \t<dd id=\"fs-id1167339281038\">the trinomial that results when a binomial is squared<\/dd>\n<\/dl>\n<dl id=\"fs-id1167339281041\">\n \t<dt>polynomial<\/dt>\n \t<dd id=\"fs-id1167339281044\">a sum of terms each consisting of a variable raised to a nonnegative integer power<\/dd>\n<\/dl>\n<dl id=\"fs-id1167339281048\">\n \t<dt>term of a polynomial<\/dt>\n \t<dd id=\"fs-id1167339281051\">any[latex]\\,{a}_{i}{x}^{i}\\,[\/latex]of a polynomial in the form[latex]\\,{a}_{n}{x}^{n}+...+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1167339317674\">\n \t<dt>trinomial<\/dt>\n \t<dd id=\"fs-id1167339317677\">a polynomial containing three terms<\/dd>\n<\/dl>\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>In this section students will:<\/p>\n<ul>\n<li>Identify the degree and leading coefficient of polynomials.<\/li>\n<li>Add and subtract polynomials.<\/li>\n<li>Multiply polynomials.<\/li>\n<li>Use FOIL to multiply binomials.<\/li>\n<li>Perform operations with polynomials of several variables.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1167339150823\">Earl is building a doghouse, whose front is in the shape of a square topped with a triangle. There will be a rectangular door through which the dog can enter and exit the house. Earl wants to find the area of the front of the doghouse so that he can purchase the correct amount of paint. Using the measurements of the front of the house, shown in <a class=\"autogenerated-content\" href=\"#Figure_01_04_001\">(Figure)<\/a>, we can create an expression that combines several variable terms, allowing us to solve this problem and others like it.<\/p>\n<div id=\"Figure_01_04_001\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132101\/CNX_CAT_Figure_01_04_001.jpg\" alt=\"Sketch of a house formed by a square and a triangle based on the top of the square. A rectangle is placed at the bottom center of the square to mark a doorway. The height of the door is labeled: x and the width of the door is labeled: 1 foot. The side of the square is labeled: 2x. The height of the triangle is labeled: 3\/2 feet.\" width=\"487\" height=\"249\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 1.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1167339220668\">First find the area of the square in square feet.<\/p>\n<div id=\"fs-id1167339151023\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill A& =& {s}^{2}\\hfill \\\\ & =& {\\left(2x\\right)}^{2}\\hfill \\\\ & =& 4{x}^{2}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167339240684\">Then find the area of the triangle in square feet.<\/p>\n<div id=\"fs-id1167339432668\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill A& =& \\frac{1}{2}bh\\hfill \\\\ & =& \\text{ }\\frac{1}{2}\\left(2x\\right)\\left(\\frac{3}{2}\\right)\\hfill \\\\ & =& \\text{ }\\frac{3}{2}x\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167339173565\">Next find the area of the rectangular door in square feet.<\/p>\n<div id=\"fs-id1167339146588\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill A& =& lw\\hfill \\\\ & =& x\\cdot 1\\hfill \\\\ \\hfill & =& x\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167339157230\">The area of the front of the doghouse can be found by adding the areas of the square and the triangle, and then subtracting the area of the rectangle. When we do this, we get[latex]\\,4{x}^{2}+\\frac{3}{2}x-x\\,{\\text{ft}}^{2},[\/latex]or[latex]\\,4{x}^{2}+\\frac{1}{2}x\\,[\/latex]ft<sup>2<\/sup>.<\/p>\n<p id=\"fs-id1167339429031\">In this section, we will examine expressions such as this one, which combine several variable terms.<\/p>\n<div id=\"fs-id1167339226748\" class=\"bc-section section\">\n<h3>Identifying the Degree and Leading Coefficient of Polynomials<\/h3>\n<p id=\"fs-id1167339185162\">The formula just found is an example of a polynomial, which is a sum of or difference of terms, each consisting of a variable raised to a nonnegative integer power. A number multiplied by a variable raised to an exponent, such as[latex]\\,384\\pi ,[\/latex]is known as a coefficient. Coefficients can be positive, negative, or zero, and can be whole numbers, decimals, or fractions. Each product[latex]\\,{a}_{i}{x}^{i},[\/latex]such as[latex]\\,384\\pi w,[\/latex]is a term of a polynomial. If a term does not contain a variable, it is called a <em>constant<\/em>.<\/p>\n<p id=\"fs-id1167339138167\">A polynomial containing only one term, such as[latex]\\,5{x}^{4},[\/latex]is called a monomial. A polynomial containing two terms, such as[latex]\\,2x-9,[\/latex]is called a binomial. A polynomial containing three terms, such as[latex]\\,-3{x}^{2}+8x-7,[\/latex]is called a trinomial.<\/p>\n<p id=\"fs-id1167339222212\">We can find the degree of a polynomial by identifying the highest power of the variable that occurs in the polynomial. The term with the highest degree is called the leading term because it is usually written first. The coefficient of the leading term is called the leading coefficient. When a polynomial is written so that the powers are descending, we say that it is in standard form.<\/p>\n<p><span id=\"fs-id1167339243309\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132107\/CNX_CAT_Figure_01_04_002.jpg\" alt=\"A polynomial reading: a sub n times x to the nth power plus and so on plus a sub 2 times x squared plus a sub one times x plus a subzero is shown. The a in the term a sub n is labeled: leading coefficient. The n in the term x to the nth power is labeled: degree. Finally, the entire term is labeled as: Leading term.\" \/><\/span><\/p>\n<div id=\"fs-id1167339137964\" class=\"textbox key-takeaways\">\n<h3>Polynomials<\/h3>\n<p id=\"fs-id1167339155557\">A polynomial is an expression that can be written in the form<\/p>\n<div id=\"fs-id1167339177022\" class=\"unnumbered aligncenter\">[latex]{a}_{n}{x}^{n}+...+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/div>\n<p id=\"fs-id1167339220598\">Each real number <em>a<sub>i<\/sub><\/em>is called a coefficient. The number[latex]\\,{a}_{0}\\,[\/latex]that is not multiplied by a variable is called a <em>constant<\/em>. Each product[latex]\\,{a}_{i}{x}^{i}\\,[\/latex]is a term of a polynomial. The highest power of the variable that occurs in the polynomial is called the degree of a polynomial. The leading term is the term with the highest power, and its coefficient is called the leading coefficient.<\/p>\n<\/div>\n<div id=\"fs-id1167339231820\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1167339212575\"><strong>Given a polynomial expression, identify the degree and leading coefficient<\/strong>.<\/p>\n<ol id=\"fs-id1167339329242\" type=\"1\">\n<li>Find the highest power of <em>x<\/em> to determine the degree.<\/li>\n<li>Identify the term containing the highest power of <em>x<\/em> to find the leading term.<\/li>\n<li>Identify the coefficient of the leading term.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_04_01\" class=\"textbox examples\">\n<div id=\"fs-id1167339145967\">\n<div id=\"fs-id1167339145969\">\n<h3>Identifying the Degree and Leading Coefficient of a Polynomial<\/h3>\n<p id=\"fs-id1167339329112\">For the following polynomials, identify the degree, the leading term, and the leading coefficient.<\/p>\n<ol id=\"fs-id1167339329115\" type=\"a\">\n<li>[latex]3+2{x}^{2}-4{x}^{3}[\/latex]<\/li>\n<li>[latex]5{t}^{5}-2{t}^{3}+7t[\/latex]<\/li>\n<li>[latex]6p-{p}^{3}-2[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1167339276093\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<ol id=\"fs-id1167339184064\" type=\"a\">\n<li>The highest power of <em>x<\/em> is 3, so the degree is 3. The leading term is the term containing that degree,[latex]\\,-4{x}^{3}.\\,[\/latex]The leading coefficient is the coefficient of that term,[latex]\\,-4.[\/latex]<\/li>\n<li>The highest power of <em>t<\/em> is[latex]\\,5,[\/latex]so the degree is[latex]\\,5.\\,[\/latex]The leading term is the term containing that degree,[latex]\\,5{t}^{5}.\\,[\/latex]The leading coefficient is the coefficient of that term,[latex]\\,5.[\/latex]<\/li>\n<li>The highest power of <em>p<\/em> is[latex]\\,3,[\/latex]so the degree is[latex]\\,3.\\,[\/latex]The leading term is the term containing that degree,[latex]\\,-{p}^{3},[\/latex]The leading coefficient is the coefficient of that term,[latex]\\,-1.[\/latex]<\/li>\n<\/ol>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339220571\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_04_01\">\n<div id=\"fs-id1167339300326\">\n<p id=\"fs-id1167339300327\">Identify the degree, leading term, and leading coefficient of the polynomial[latex]\\,4{x}^{2}-{x}^{6}+2x-6.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339196860\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339196862\">The degree is 6, the leading term is[latex]\\,-{x}^{6},[\/latex]and the leading coefficient is[latex]\\,-1.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339243781\" class=\"bc-section section\">\n<h3>Adding and Subtracting Polynomials<\/h3>\n<p id=\"fs-id1167339223237\">We can add and subtract polynomials by combining like terms, which are terms that contain the same variables raised to the same exponents. For example,[latex]\\,5{x}^{2}\\,[\/latex]and[latex]\\,-2{x}^{2}\\,[\/latex]are like terms, and can be added to get[latex]\\,3{x}^{2},[\/latex]but[latex]\\,3x\\,[\/latex]and[latex]\\,3{x}^{2}\\,[\/latex]are not like terms, and therefore cannot be added.<\/p>\n<div id=\"fs-id1167339290474\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1167339199765\"><strong>Given multiple polynomials, add or subtract them to simplify the expressions.<br \/>\n<\/strong><\/p>\n<ol id=\"fs-id1167339199769\" type=\"1\">\n<li>Combine like terms.<\/li>\n<li>Simplify and write in standard form.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_04_02\" class=\"textbox examples\">\n<div id=\"fs-id1167339306483\">\n<div id=\"fs-id1167339306486\" class=\"textbox\">\n<h3>Adding Polynomials<\/h3>\n<p id=\"fs-id1167339226270\">Find the sum.<\/p>\n<p id=\"fs-id1167339149553\">[latex]\\left(12{x}^{2}+9x-21\\right)+\\left(4{x}^{3}+8{x}^{2}-5x+20\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339300037\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1317912\">[latex]\\begin{array}{cc}4{x}^{3}+\\left(12{x}^{2}+8{x}^{2}\\right)+\\left(9x-5x\\right)+\\left(-21+20\\right) \\hfill & \\phantom{\\rule{2em}{0ex}}\\text{\u2003\u2003}\\text{Combine like terms}.\\hfill \\\\ 4{x}^{3}+20{x}^{2}+4x-1\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{\u2003\u2003}\\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<div id=\"fs-id1167339213779\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1167339259571\">We can check our answers to these types of problems using a graphing calculator. To check, graph the problem as given along with the simplified answer. The two graphs should be equivalent. Be sure to use the same window to compare the graphs. Using different windows can make the expressions seem equivalent when they are not.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339243320\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_04_02\">\n<div id=\"fs-id1167339226215\">\n<p id=\"fs-id1167339226216\">Find the sum.<\/p>\n<p id=\"fs-id1167339330158\">[latex]\\left(2{x}^{3}+5{x}^{2}-x+1\\right)+\\left(2{x}^{2}-3x-4\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339223528\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339223529\">[latex]2{x}^{3}+7{x}^{2}-4x-3[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_04_03\" class=\"textbox examples\">\n<div id=\"fs-id1167339428763\">\n<div id=\"fs-id1167339240332\">\n<h3>Subtracting Polynomials<\/h3>\n<p id=\"fs-id1167339240338\">Find the difference.<\/p>\n<p id=\"fs-id1167339432947\">[latex]\\left(7{x}^{4}-{x}^{2}+6x+1\\right)-\\left(5{x}^{3}-2{x}^{2}+3x+2\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339158579\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339158581\">[latex]\\begin{array}{cc}7{x}^{4}-5{x}^{3}+\\left(-{x}^{2}+2{x}^{2}\\right)+\\left(6x-3x\\right)+\\left(1-2\\right)\\text{ }\\hfill & \\phantom{\\rule{1em}{0ex}}\\text{Combine like terms}.\\hfill \\\\ 7{x}^{4}-5{x}^{3}+{x}^{2}+3x-1\\hfill & \\phantom{\\rule{1em}{0ex}}\\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<div id=\"fs-id1167339185145\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1167339185150\">Note that finding the difference between two polynomials is the same as adding the opposite of the second polynomial to the first.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339156471\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_04_03\">\n<div id=\"fs-id1167339317929\">\n<p id=\"fs-id1167339317930\">Find the difference.<\/p>\n<p id=\"fs-id1167339184123\">[latex]\\left(-7{x}^{3}-7{x}^{2}+6x-2\\right)-\\left(4{x}^{3}-6{x}^{2}-x+7\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339260393\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339260394\">[latex]-11{x}^{3}-{x}^{2}+7x-9[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339214258\" class=\"bc-section section\">\n<h3>Multiplying Polynomials<\/h3>\n<p id=\"fs-id1167339139443\">Multiplying polynomials is a bit more challenging than adding and subtracting polynomials. We must use the distributive property to multiply each term in the first polynomial by each term in the second polynomial. We then combine like terms. We can also use a shortcut called the FOIL method when multiplying binomials. Certain special products follow patterns that we can memorize and use instead of multiplying the polynomials by hand each time. We will look at a variety of ways to multiply polynomials.<\/p>\n<div id=\"fs-id1167339198311\" class=\"bc-section section\">\n<h4>Multiplying Polynomials Using the Distributive Property<\/h4>\n<p id=\"fs-id1167339198316\">To multiply a number by a polynomial, we use the distributive property. The number must be distributed to each term of the polynomial. We can distribute the[latex]\\,2\\,[\/latex]in[latex]\\,2\\left(x+7\\right)\\,[\/latex]to obtain the equivalent expression[latex]\\,2x+14.\\,[\/latex]When multiplying polynomials, the distributive property allows us to multiply each term of the first polynomial by each term of the second. We then add the products together and combine like terms to simplify.<\/p>\n<div id=\"fs-id1167339260622\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1167339170371\"><strong>Given the multiplication of two polynomials, use the distributive property to simplify the expression.<\/strong><\/p>\n<ol id=\"fs-id1167339170375\" type=\"1\">\n<li>Multiply each term of the first polynomial by each term of the second.<\/li>\n<li>Combine like terms.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_04_04\" class=\"textbox examples\">\n<div id=\"fs-id1167339239993\">\n<div id=\"fs-id1167339239995\">\n<h3>Multiplying Polynomials Using the Distributive Property<\/h3>\n<p id=\"fs-id1167339158245\">Find the product.<\/p>\n<p id=\"fs-id1167339158248\">[latex]\\left(2x+1\\right)\\left(3{x}^{2}-x+4\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339306275\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339306277\">[latex]\\begin{array}{cc}2x\\left(3{x}^{2}-x+4\\right)+1\\left(3{x}^{2}-x+4\\right) \\hfill & \\phantom{\\rule{2em}{0ex}}\\text{\u2003\u2003}\\text{Use the distributive property}.\\hfill \\\\ \\left(6{x}^{3}-2{x}^{2}+8x\\right)+\\left(3{x}^{2}-x+4\\right)\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{\u2003\u2003}\\text{Multiply}.\\hfill \\\\ 6{x}^{3}+\\left(-2{x}^{2}+3{x}^{2}\\right)+\\left(8x-x\\right)+4\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{\u2003\u2003}\\text{Combine like terms}.\\hfill \\\\ 6{x}^{3}+{x}^{2}+7x+4 \\hfill & \\phantom{\\rule{2em}{0ex}}\\text{\u2003\u2003}\\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<div id=\"fs-id1167339344747\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1167339158177\">We can use a table to keep track of our work, as shown in <a class=\"autogenerated-content\" href=\"#Table_01_04_01\">(Figure)<\/a>. Write one polynomial across the top and the other down the side. For each box in the table, multiply the term for that row by the term for that column. Then add all of the terms together, combine like terms, and simplify.<\/p>\n<table id=\"Table_01_04_01\" summary=\"A table with 3 rows and 4 columns. The first entry of the first row is empty, the others are labeled: three times x squared, negative x, and positive four. The first entry of the second row is labeled: two times x. The second entry reads: six times x cubed. The third entry reads: negative two times x squared. The fourth entry reads: eight times x. The first entry of the third row reads: positive one. The second entry reads: three times x squared. The third entry reads: negative x. The fourth entry reads: four.\">\n<caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]3{x}^{2}[\/latex]<\/td>\n<td>[latex]-x[\/latex]<\/td>\n<td>[latex]+4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2x[\/latex]<\/td>\n<td>[latex]6{x}^{3}[\/latex]<\/td>\n<td>[latex]-2{x}^{2}[\/latex]<\/td>\n<td>[latex]8x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]+1[\/latex]<\/td>\n<td>[latex]3{x}^{2}[\/latex]<\/td>\n<td>[latex]-x[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339260552\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_04_04\">\n<div id=\"fs-id1167339150646\">\n<p id=\"fs-id1167339150648\">Find the product.<\/p>\n<p id=\"fs-id1167339150651\">[latex]\\left(3x+2\\right)\\left({x}^{3}-4{x}^{2}+7\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339223342\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339223343\">[latex]3{x}^{4}-10{x}^{3}-8{x}^{2}+21x+14[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339243382\" class=\"bc-section section\">\n<h4>Using FOIL to Multiply Binomials<\/h4>\n<p id=\"fs-id1167339243388\">A shortcut called FOIL is sometimes used to find the product of two binomials. It is called FOIL because we multiply the <strong>f<\/strong>irst terms, the <strong>o<\/strong>uter terms, the <strong>i<\/strong>nner terms, and then the <strong>l<\/strong>ast terms of each binomial.<\/p>\n<p><span id=\"fs-id1167339168201\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132109\/CNX_CAT_Figure_01_04_003.jpg\" alt=\"Two quantities in parentheses are being multiplied, the first being: a times x plus b and the second being: c times x plus d. This expression equals ac times x squared plus ad times x plus bc times x plus bd. The terms ax and cx are labeled: First Terms. The terms ax and d are labeled: Outer Terms. The terms b and cx are labeled: Inner Terms. The terms b and d are labeled: Last Terms.\" \/><\/span><\/p>\n<p id=\"fs-id1167339429089\">The FOIL method arises out of the distributive property. We are simply multiplying each term of the first binomial by each term of the second binomial, and then combining like terms.<\/p>\n<div id=\"fs-id1167339429094\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1167339240108\"><strong>Given two binomials, use FOIL to simplify the expression.<\/strong><\/p>\n<ol id=\"fs-id1167339240112\" type=\"1\">\n<li>Multiply the first terms of each binomial.<\/li>\n<li>Multiply the outer terms of the binomials.<\/li>\n<li>Multiply the inner terms of the binomials.<\/li>\n<li>Multiply the last terms of each binomial.<\/li>\n<li>Add the products.<\/li>\n<li>Combine like terms and simplify.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_04_05\" class=\"textbox examples\">\n<div>\n<div>\n<h3>Using FOIL to Multiply Binomials<\/h3>\n<p>Use FOIL to find the product.<\/p>\n<p id=\"eip-id6327005\">[latex]\\left(2x-18\\right)\\left(3x+3\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"eip-id1167338039926\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"eip-id1673659\">Find the product of the first terms.<\/p>\n<p><span id=\"eip-id1673664\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132116\/CNX_CAT_Figure_01_04_004.jpg\" alt=\"\" \/><\/span><\/p>\n<p id=\"eip-id1673675\">Find the product of the outer terms.<\/p>\n<p><span id=\"eip-id1673679\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132124\/CNX_CAT_Figure_01_04_005.jpg\" alt=\"\" \/><\/span><\/p>\n<p id=\"eip-id1923767\">Find the product of the inner terms.<\/p>\n<p><span id=\"eip-id1923771\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132144\/CNX_CAT_Figure_01_04_006.jpg\" alt=\"\" \/><\/span><\/p>\n<p id=\"eip-id1923782\">Find the product of the last terms.<\/p>\n<p><span id=\"eip-id1923785\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19132147\/CNX_CAT_Figure_01_04_007.jpg\" alt=\"\" \/><\/span><\/p>\n<p id=\"eip-id1167336078703\">[latex]\\begin{array}{cc}6{x}^{2}+6x-54x-54\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Add the products}.\\hfill \\\\ 6{x}^{2}+\\left(6x-54x\\right)-54\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Combine like terms}.\\hfill \\\\ 6{x}^{2}-48x-54\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\n<p id=\"eip-id1167336078703\"><\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339223541\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_04_05\">\n<div id=\"fs-id1167339241487\">\n<p id=\"fs-id1167339241488\">Use FOIL to find the product.<\/p>\n<p id=\"fs-id1167339241491\">[latex]\\left(x+7\\right)\\left(3x-5\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339216120\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339216121\">[latex]3{x}^{2}+16x-35[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339428910\" class=\"bc-section section\">\n<h4>Perfect Square Trinomials<\/h4>\n<p id=\"fs-id1167339432924\">Certain binomial products have special forms. When a binomial is squared, the result is called a perfect square trinomial. We can find the square by multiplying the binomial by itself. However, there is a special form that each of these perfect square trinomials takes, and memorizing the form makes squaring binomials much easier and faster. Let\u2019s look at a few perfect square trinomials to familiarize ourselves with the form.<\/p>\n<div id=\"fs-id1167339433593\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\text{ }{\\left(x+5\\right)}^{2}& =& {x}^{2}+10x+25\\hfill \\\\ \\hfill {\\left(x-3\\right)}^{2}& =& \\text{ }{x}^{2}-6x+9\\hfill \\\\ \\hfill {\\left(4x-1\\right)}^{2}& =& 16{x}^{2}-8x+1\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167339224293\">Notice that the first term of each trinomial is the square of the first term of the binomial and, similarly, the last term of each trinomial is the square of the last term of the binomial. The middle term is double the product of the two terms. Lastly, we see that the first sign of the trinomial is the same as the sign of the binomial.<\/p>\n<div id=\"fs-id1167339224300\" class=\"textbox key-takeaways\">\n<h3>Perfect Square Trinomials<\/h3>\n<p id=\"fs-id1167339330148\">When a binomial is squared, the result is the first term squared added to double the product of both terms and the last term squared.<\/p>\n<div id=\"Equation_01_04_01\">[latex]{\\left(x+a\\right)}^{2}=\\left(x+a\\right)\\left(x+a\\right)={x}^{2}+2ax+{a}^{2}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1167339220526\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1167339220533\"><strong>Given a binomial, square it using the formula for perfect square trinomials.<br \/>\n<\/strong><\/p>\n<ol id=\"fs-id1167339231805\" type=\"1\">\n<li>Square the first term of the binomial.<\/li>\n<li>Square the last term of the binomial.<\/li>\n<li>For the middle term of the trinomial, double the product of the two terms.<\/li>\n<li>Add and simplify.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_04_06\" class=\"textbox examples\">\n<div id=\"fs-id1167339273765\">\n<div id=\"fs-id1167339344760\">\n<h3>Expanding Perfect Squares<\/h3>\n<p id=\"fs-id1167339344765\">Expand[latex]\\,{\\left(3x-8\\right)}^{2}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339428957\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339428959\">Begin by squaring the first term and the last term. For the middle term of the trinomial, double the product of the two terms.<\/p>\n<div id=\"fs-id1167339428964\" class=\"unnumbered aligncenter\">[latex]{\\left(3x\\right)}^{2}-2\\left(3x\\right)\\left(8\\right)+{\\left(-8\\right)}^{2}[\/latex]<\/details>\n<\/div>\n<p id=\"fs-id1167339299914\">Simplify<\/p>\n<div id=\"eip-id1171852212181\">[latex]\\,9{x}^{2}-48x+64.[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339328808\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_04_06\">\n<div id=\"fs-id1167339328819\">\n<p id=\"fs-id1167339328820\">Expand [latex]\\,{\\left(4x-1\\right)}^{2}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339176899\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339176900\">[latex]16{x}^{2}-8x+1[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339429838\" class=\"bc-section section\">\n<h4>Difference of Squares<\/h4>\n<p id=\"fs-id1167339188447\">Another special product is called the difference of squares, which occurs when we multiply a binomial by another binomial with the same terms but the opposite sign. Let\u2019s see what happens when we multiply[latex]\\,\\left(x+1\\right)\\left(x-1\\right)\\,[\/latex]using the FOIL method.<\/p>\n<div id=\"fs-id1167339428778\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\left(x+1\\right)\\left(x-1\\right)& =& {x}^{2}-x+x-1\\hfill \\\\ & =& {x}^{2}-1\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167339432898\">The middle term drops out, resulting in a difference of squares. Just as we did with the perfect squares, let\u2019s look at a few examples.<\/p>\n<div id=\"fs-id1167339432904\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\left(x+5\\right)\\left(x-5\\right)& =& {x}^{2}-25\\hfill \\\\ \\hfill \\left(x+11\\right)\\left(x-11\\right)& =& {x}^{2}-121\\hfill \\\\ \\hfill \\left(2x+3\\right)\\left(2x-3\\right)& =& 4{x}^{2}-9\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167339220812\">Because the sign changes in the second binomial, the outer and inner terms cancel each other out, and we are left only with the square of the first term minus the square of the last term.<\/p>\n<div id=\"fs-id1167339220817\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1167339220824\"><strong>Is there a special form for the sum of squares?<\/strong><\/p>\n<p id=\"fs-id1167339199657\"><em>No. The difference of squares occurs because the opposite signs of the binomials cause the middle terms to disappear. There are no two binomials that multiply to equal a sum of squares.<\/em><\/p>\n<\/div>\n<div id=\"fs-id1167339199666\" class=\"textbox key-takeaways\">\n<h3>Difference of Squares<\/h3>\n<p id=\"fs-id1167339199674\">When a binomial is multiplied by a binomial with the same terms separated by the opposite sign, the result is the square of the first term minus the square of the last term.<\/p>\n<div id=\"Equation_01_04_02\">[latex]\\left(a+b\\right)\\left(a-b\\right)={a}^{2}-{b}^{2}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1167339137913\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1167339273978\"><strong>Given a binomial multiplied by a binomial with the same terms but the opposite sign, find the difference of squares.<\/strong><\/p>\n<ol id=\"fs-id1167339273983\" type=\"1\">\n<li>Square the first term of the binomials.<\/li>\n<li>Square the last term of the binomials.<\/li>\n<li>Subtract the square of the last term from the square of the first term.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_04_07\" class=\"textbox examples\">\n<div id=\"fs-id1167339315486\">\n<div id=\"fs-id1167339315488\">\n<h3>Multiplying Binomials Resulting in a Difference of Squares<\/h3>\n<p id=\"fs-id1167339315493\">Multiply[latex]\\,\\left(9x+4\\right)\\left(9x-4\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339273948\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339273950\">Square the first term to get[latex]\\,{\\left(9x\\right)}^{2}=81{x}^{2}.\\,[\/latex]Square the last term to get[latex]\\,{4}^{2}=16.\\,[\/latex]Subtract the square of the last term from the square of the first term to find the product of[latex]\\,81{x}^{2}-16.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339138650\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_04_07\">\n<div id=\"fs-id1167339212421\">\n<p id=\"fs-id1167339212422\">Multiply[latex]\\,\\left(2x+7\\right)\\left(2x-7\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339318295\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339318296\">[latex]4{x}^{2}-49[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339138125\" class=\"bc-section section\">\n<h3>Performing Operations with Polynomials of Several Variables<\/h3>\n<p id=\"fs-id1167339138130\">We have looked at polynomials containing only one variable. However, a polynomial can contain several variables. All of the same rules apply when working with polynomials containing several variables. Consider an example:<\/p>\n<div id=\"fs-id1167339429970\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{cc}\\left(a+2b\\right)\\left(4a-b-c\\right)\\hfill & \\hfill \\\\ a\\left(4a-b-c\\right)+2b\\left(4a-b-c\\right)\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Use the distributive property}.\\hfill \\\\ 4{a}^{2}-ab-ac+8ab-2{b}^{2}-2bc\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Multiply}.\\hfill \\\\ 4{a}^{2}+\\left(-ab+8ab\\right)-ac-2{b}^{2}-2bc\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Combine like terms}.\\hfill \\\\ 4{a}^{2}+7ab-ac-2bc-2{b}^{2}\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Simplify}.\\hfill \\end{array}[\/latex]<\/div>\n<div id=\"Example_01_04_08\" class=\"textbox examples\">\n<div id=\"fs-id1167339318091\">\n<div id=\"fs-id1167339318093\">\n<h3>Multiplying Polynomials Containing Several Variables<\/h3>\n<p id=\"fs-id1167339315450\">Multiply[latex]\\,\\left(x+4\\right)\\left(3x-2y+5\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339299948\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339299950\">Follow the same steps that we used to multiply polynomials containing only one variable.<\/p>\n<div id=\"fs-id1167339299954\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{cc}x\\left(3x-2y+5\\right)+4\\left(3x-2y+5\\right) \\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Use the distributive property}.\\hfill \\\\ 3{x}^{2}-2xy+5x+12x-8y+20\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Multiply}.\\hfill \\\\ 3{x}^{2}-2xy+\\left(5x+12x\\right)-8y+20\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Combine like terms}.\\hfill \\\\ 3{x}^{2}-2xy+17x-8y+20 \\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Simplify}.\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339240133\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_01_04_08\">\n<div id=\"fs-id1167339240144\">\n<p id=\"fs-id1167339240145\">Multiply [latex]\\left(3x-1\\right)\\left(2x+7y-9\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339240262\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339240263\">[latex]\\,6{x}^{2}+21xy-29x-7y+9[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339315650\" class=\"precalculus media\">\n<p id=\"fs-id1167339219216\">Access these online resources for additional instruction and practice with polynomials.<\/p>\n<ul id=\"fs-id1167339219219\">\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/addsubpoly\">Adding and Subtracting Polynomials<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/multiplpoly\">Multiplying Polynomials<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/specialpolyprod\">Special Products of Polynomials<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339219247\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"fs-id1167339219254\" summary=\"..\">\n<tbody>\n<tr>\n<td>perfect square trinomial<\/td>\n<td>[latex]{\\left(x+a\\right)}^{2}=\\left(x+a\\right)\\left(x+a\\right)={x}^{2}+2ax+{a}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>difference of squares<\/td>\n<td>[latex]\\left(a+b\\right)\\left(a-b\\right)={a}^{2}-{b}^{2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1167339242305\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1167339242312\">\n<li>A polynomial is a sum of terms each consisting of a variable raised to a non-negative integer power. The degree is the highest power of the variable that occurs in the polynomial. The leading term is the term containing the highest degree, and the leading coefficient is the coefficient of that term. See <a class=\"autogenerated-content\" href=\"#Example_01_04_01\">(Figure)<\/a>.<\/li>\n<li>We can add and subtract polynomials by combining like terms. See <a class=\"autogenerated-content\" href=\"#Example_01_04_02\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_01_04_03\">(Figure)<\/a>.<\/li>\n<li>To multiply polynomials, use the distributive property to multiply each term in the first polynomial by each term in the second. Then add the products. See <a class=\"autogenerated-content\" href=\"#Example_01_04_04\">(Figure)<\/a>.<\/li>\n<li>FOIL (First, Outer, Inner, Last) is a shortcut that can be used to multiply binomials. See <a class=\"autogenerated-content\" href=\"#Example_01_04_05\">(Figure)<\/a>.<\/li>\n<li>Perfect square trinomials and difference of squares are special products. See <a class=\"autogenerated-content\" href=\"#Example_01_04_06\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_01_04_07\">(Figure)<\/a>.<\/li>\n<li>Follow the same rules to work with polynomials containing several variables. See <a class=\"autogenerated-content\" href=\"#Example_01_04_08\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1167339184236\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1167339184243\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1167339184249\">\n<div id=\"fs-id1167339184250\">\n<p id=\"fs-id1167339184251\">Evaluate the following statement: The degree of a polynomial in standard form is the exponent of the leading term. Explain why the statement is true or false.<\/p>\n<\/div>\n<div id=\"fs-id1167339216132\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339216133\">The statement is true. In standard form, the polynomial with the highest value exponent is placed first and is the leading term. The degree of a polynomial is the value of the highest exponent, which in standard form is also the exponent of the leading term.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339216138\">\n<div id=\"fs-id1167339216140\">\n<p id=\"fs-id1167339216141\">Many times, multiplying two binomials with two variables results in a trinomial. This is not the case when there is a difference of two squares. Explain why the product in this case is also a binomial.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339216146\">\n<div id=\"fs-id1167339216147\">\n<p id=\"fs-id1167339216148\">You can multiply polynomials with any number of terms and any number of variables using four basic steps over and over until you reach the expanded polynomial. What are the four steps?<\/p>\n<\/div>\n<div id=\"fs-id1167339216153\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339216154\">Use the distributive property, multiply, combine like terms, and simplify.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339216157\">\n<div id=\"fs-id1167339216158\">\n<p id=\"fs-id1167339216159\">State whether the following statement is true and explain why or why not: A trinomial is always a higher degree than a monomial.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339216165\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1167339216170\">For the following exercises, identify the degree of the polynomial.<\/p>\n<div id=\"fs-id1167339216173\">\n<div id=\"fs-id1167339216174\">\n<p id=\"fs-id1167339216175\">[latex]7x-2{x}^{2}+13[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339299770\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339299771\">2<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339299774\">\n<div id=\"fs-id1167339299775\">\n<p id=\"fs-id1167339299776\">[latex]14{m}^{3}+{m}^{2}-16m+8[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339299820\">\n<div id=\"fs-id1167339299821\">\n<p id=\"fs-id1167339299822\">[latex]-625{a}^{8}+16{b}^{4}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339268841\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339268842\">8<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339268846\">\n<div id=\"fs-id1167339268847\">\n<p id=\"fs-id1167339268848\">[latex]200p-30{p}^{2}m+40{m}^{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339223145\">\n<div id=\"fs-id1167339223146\">\n<p id=\"fs-id1167339223147\">[latex]{x}^{2}+4x+4[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339223177\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339223178\">2<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339223181\">\n<div id=\"fs-id1167339223182\">\n<p id=\"fs-id1167339223184\">[latex]6{y}^{4}-{y}^{5}+3y-4[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167339432583\">For the following exercises, find the sum or difference.<\/p>\n<div id=\"fs-id1167339432587\">\n<div id=\"fs-id1167339432588\">\n<p id=\"fs-id1167339432589\">[latex]\\left(12{x}^{2}+3x\\right)-\\left(8{x}^{2}-19\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339432656\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339432657\">[latex]4{x}^{2}+3x+19[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339185745\">\n<div id=\"fs-id1167339185746\">\n<p id=\"fs-id1167339185747\">[latex]\\left(4{z}^{3}+8{z}^{2}-z\\right)+\\left(-2{z}^{2}+z+6\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339306370\">\n<div id=\"fs-id1167339306371\">\n<p id=\"fs-id1167339306372\">[latex]\\left(6{w}^{2}+24w+24\\right)-\\left(3w{}^{2}-6w+3\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339223800\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339223801\">[latex]3{w}^{2}+30w+21[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339223833\">\n<div id=\"fs-id1167339223834\">\n<p id=\"fs-id1167339223835\">[latex]\\left(7{a}^{3}+6{a}^{2}-4a-13\\right)+\\left(-3{a}^{3}-4{a}^{2}+6a+17\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339262640\">\n<div id=\"fs-id1167339262641\">\n<p id=\"fs-id1167339262642\">[latex]\\left(11{b}^{4}-6{b}^{3}+18{b}^{2}-4b+8\\right)-\\left(3{b}^{3}+6{b}^{2}+3b\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339273678\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339273679\">[latex]11{b}^{4}-9{b}^{3}+12{b}^{2}-7b+8[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339196660\">\n<div id=\"fs-id1167339196661\">\n<p id=\"fs-id1167339196662\">[latex]\\left(49{p}^{2}-25\\right)+\\left(16{p}^{4}-32{p}^{2}+16\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167339196741\">For the following exercises, find the product.<\/p>\n<div id=\"fs-id1167339196745\">\n<div id=\"fs-id1167339196746\">\n<p id=\"fs-id1167339196747\">[latex]\\left(4x+2\\right)\\left(6x-4\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339259665\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339259666\">[latex]24{x}^{2}-4x-8[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339259698\">\n<div id=\"fs-id1167339259699\">\n<p id=\"fs-id1167339259700\">[latex]\\left(14{c}^{2}+4c\\right)\\left(2{c}^{2}-3c\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339223391\">\n<div id=\"fs-id1167339223392\">\n<p id=\"fs-id1167339223394\">[latex]\\left(6{b}^{2}-6\\right)\\left(4{b}^{2}-4\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339223456\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339223458\">[latex]24{b}^{4}-48{b}^{2}+24[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339223497\">\n<div id=\"fs-id1167339223498\">\n<p id=\"fs-id1167339223499\">[latex]\\left(3d-5\\right)\\left(2d+9\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339306553\">\n<div id=\"fs-id1167339306554\">\n<p id=\"fs-id1167339306555\">[latex]\\left(9v-11\\right)\\left(11v-9\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339306603\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339306604\">[latex]99{v}^{2}-202v+99[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339286443\">\n<div id=\"fs-id1167339286444\">\n<p id=\"fs-id1167339286446\">[latex]\\left(4{t}^{2}+7t\\right)\\left(-3{t}^{2}+4\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339286513\">\n<div id=\"fs-id1167339286514\">\n<p id=\"fs-id1167339286515\">[latex]\\left(8n-4\\right)\\left({n}^{2}+9\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339286568\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339286569\">[latex]8{n}^{3}-4{n}^{2}+72n-36[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1167339199464\">For the following exercises, expand the binomial.<\/p>\n<div id=\"fs-id1167339199468\">\n<div id=\"fs-id1167339199469\">\n<p id=\"fs-id1167339199470\">[latex]{\\left(4x+5\\right)}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339199508\">\n<div id=\"fs-id1167339199509\">\n<p id=\"fs-id1167339199510\">[latex]{\\left(3y-7\\right)}^{2}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339199549\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339199550\">[latex]9{y}^{2}-42y+49[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339220230\">\n<div id=\"fs-id1167339220232\">\n<p id=\"fs-id1167339220233\">[latex]{\\left(12-4x\\right)}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339220271\">\n<div id=\"fs-id1167339220272\">[latex]{\\left(4p+9\\right)}^{2}[\/latex]<\/div>\n<div id=\"fs-id1167339220312\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339220313\">[latex]16{p}^{2}+72p+81[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339220345\">\n<div id=\"fs-id1167339220346\">\n<p id=\"fs-id1167339220347\">[latex]{\\left(2m-3\\right)}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339226325\">\n<div id=\"fs-id1167339226326\">\n<p id=\"fs-id1167339226327\">[latex]{\\left(3y-6\\right)}^{2}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339226366\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339226367\">[latex]9{y}^{2}-36y+36[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339226399\">\n<div id=\"fs-id1167339226400\">\n<p id=\"fs-id1167339226401\">[latex]{\\left(9b+1\\right)}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167339196445\">For the following exercises, multiply the binomials.<\/p>\n<div id=\"fs-id1167339196448\">\n<div id=\"fs-id1167339196449\">\n<p id=\"fs-id1167339196450\">[latex]\\left(4c+1\\right)\\left(4c-1\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339196498\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339196499\">[latex]16{c}^{2}-1[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339196525\">\n<div id=\"fs-id1167339196526\">\n<p id=\"fs-id1167339196527\">[latex]\\left(9a-4\\right)\\left(9a+4\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339196575\">\n<div id=\"fs-id1167339196576\">\n<p id=\"fs-id1167339196577\">[latex]\\left(15n-6\\right)\\left(15n+6\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339260447\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339260448\">[latex]225{n}^{2}-36[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339260474\">\n<div id=\"fs-id1167339260475\">\n<p id=\"fs-id1167339260476\">[latex]\\left(25b+2\\right)\\left(25b-2\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339260524\">\n<div id=\"fs-id1167339260525\">\n<p id=\"fs-id1167339260526\">[latex]\\left(4+4m\\right)\\left(4-4m\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339239040\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339239041\">[latex]-16{m}^{2}+16[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339239069\">\n<div id=\"fs-id1167339239070\">\n<p id=\"fs-id1167339239071\">[latex]\\left(14p+7\\right)\\left(14p-7\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339239119\">\n<div id=\"fs-id1167339239120\">\n<p id=\"fs-id1167339239121\">[latex]\\left(11q-10\\right)\\left(11q+10\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339239169\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339321332\">[latex]121{q}^{2}-100[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1167339321358\">For the following exercises, multiply the polynomials.<\/p>\n<div id=\"fs-id1167339321361\">\n<div id=\"fs-id1167339321362\">\n<p id=\"fs-id1167339321363\">[latex]\\left(2{x}^{2}+2x+1\\right)\\left(4x-1\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339321425\">\n<div id=\"fs-id1167339321426\">\n<p id=\"fs-id1167339321427\">[latex]\\left(4{t}^{2}+t-7\\right)\\left(4{t}^{2}-1\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339138777\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339138778\">[latex]16{t}^{4}+4{t}^{3}-32{t}^{2}-t+7[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339138836\">\n<div id=\"fs-id1167339138837\">\n<p id=\"fs-id1167339138838\">[latex]\\left(x-1\\right)\\left({x}^{2}-2x+1\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339138896\">\n<div id=\"fs-id1167339138897\">\n<p id=\"fs-id1167339138898\">[latex]\\left(y-2\\right)\\left({y}^{2}-4y-9\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339273787\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339273788\">[latex]{y}^{3}-6{y}^{2}-y+18[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339273830\">\n<div id=\"fs-id1167339273831\">\n<p id=\"fs-id1167339273832\">[latex]\\left(6k-5\\right)\\left(6{k}^{2}+5k-1\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339273894\">\n<div id=\"fs-id1167339273895\">\n<p id=\"fs-id1167339273896\">[latex]\\left(3{p}^{2}+2p-10\\right)\\left(p-1\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339344407\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339344408\">[latex]3{p}^{3}-{p}^{2}-12p+10[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339344452\">\n<div id=\"fs-id1167339344453\">\n<p id=\"fs-id1167339344454\">[latex]\\left(4m-13\\right)\\left(2{m}^{2}-7m+9\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339344516\">\n<div id=\"fs-id1167339344517\">\n<p id=\"fs-id1167339344518\">[latex]\\left(a+b\\right)\\left(a-b\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339344562\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339213918\">[latex]{a}^{2}-{b}^{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339213948\">\n<div id=\"fs-id1167339213949\">\n<p id=\"fs-id1167339213950\">[latex]\\left(4x-6y\\right)\\left(6x-4y\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339214002\">\n<div id=\"fs-id1167339214003\">\n<p id=\"fs-id1167339214004\">[latex]{\\left(4t-5u\\right)}^{2}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339214045\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339214046\">[latex]16{t}^{2}-40tu+25{u}^{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339214089\">\n<div id=\"fs-id1167339214090\">\n<p id=\"fs-id1167339214091\">[latex]\\left(9m+4n-1\\right)\\left(2m+8\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339222871\">\n<div id=\"fs-id1167339222872\">\n<p id=\"fs-id1167339222873\">[latex]\\left(4t-x\\right)\\left(t-x+1\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339222924\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339222925\">[latex]4{t}^{2}+{x}^{2}+4t-5tx-x[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339222977\">\n<div id=\"fs-id1167339222978\">\n<p id=\"fs-id1167339222979\">[latex]\\left({b}^{2}-1\\right)\\left({a}^{2}+2ab+{b}^{2}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339223053\">\n<div id=\"fs-id1167339223054\">\n<p id=\"fs-id1167339223055\">[latex]\\left(4r-d\\right)\\left(6r+7d\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339225614\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339225615\">[latex]24{r}^{2}+22rd-7{d}^{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339225658\">\n<div id=\"fs-id1167339225659\">\n<p id=\"fs-id1167339225660\">[latex]\\left(x+y\\right)\\left({x}^{2}-xy+{y}^{2}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339225725\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<div id=\"fs-id1167339225730\">\n<div id=\"fs-id1167339225731\">\n<p id=\"fs-id1167339437824\">A developer wants to purchase a plot of land to build a house. The area of the plot can be described by the following expression:[latex]\\,\\left(4x+1\\right)\\left(8x-3\\right)\\,[\/latex]where <em>x<\/em> is measured in meters. Multiply the binomials to find the area of the plot in standard form.<\/p>\n<\/div>\n<div id=\"fs-id1167339225791\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339225792\">[latex]32{x}^{2}-4x-3\\,[\/latex]m<sup>2<\/sup><\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339242981\">\n<div id=\"fs-id1167339242982\">\n<p id=\"fs-id1167339242983\">A prospective buyer wants to know how much grain a specific silo can hold. The area of the floor of the silo is[latex]\\,{\\left(2x+9\\right)}^{2}.\\,[\/latex]The height of the silo is[latex]\\,10x+10,[\/latex]where <em>x<\/em> is measured in feet. Expand the square and multiply by the height to find the expression that shows how much grain the silo can hold.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339243059\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<p id=\"fs-id1167339243064\">For the following exercises, perform the given operations.<\/p>\n<div id=\"fs-id1167339243067\">\n<div id=\"fs-id1167339243068\">\n<p id=\"fs-id1167339243069\">[latex]{\\left(4t-7\\right)}^{2}\\left(2t+1\\right)-\\left(4{t}^{2}+2t+11\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339243163\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339243164\">[latex]32{t}^{3}-100{t}^{2}+40t+38[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339230814\">\n<div id=\"fs-id1167339230815\">\n<p id=\"fs-id1167339230816\">[latex]\\left(3b+6\\right)\\left(3b-6\\right)\\left(9{b}^{2}-36\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1167339230891\">\n<div id=\"fs-id1167339230892\">\n<p id=\"fs-id1167339219515\">[latex]\\left({a}^{2}+4ac+4{c}^{2}\\right)\\left({a}^{2}-4{c}^{2}\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1167339230976\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1167339230977\">[latex]{a}^{4}+4{a}^{3}c-16a{c}^{3}-16{c}^{4}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1167339280889\">\n<dt>binomial<\/dt>\n<dd id=\"fs-id1167339280892\">a polynomial containing two terms<\/dd>\n<\/dl>\n<dl id=\"fs-id1167339280895\">\n<dt>coefficient<\/dt>\n<dd id=\"fs-id1167339280898\">any real number[latex]\\,{a}_{i}\\,[\/latex]in a polynomial in the form[latex]\\,{a}_{n}{x}^{n}+...+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1167339281002\">\n<dt>degree<\/dt>\n<dd>the highest power of the variable that occurs in a polynomial<\/dd>\n<\/dl>\n<dl id=\"fs-id1167339281008\">\n<dt>difference of squares<\/dt>\n<dd id=\"fs-id1167339281011\">the binomial that results when a binomial is multiplied by a binomial with the same terms, but the opposite sign<\/dd>\n<\/dl>\n<dl id=\"fs-id1167339281016\">\n<dt>leading coefficient<\/dt>\n<dd id=\"fs-id1167339281019\">the coefficient of the leading term<\/dd>\n<\/dl>\n<dl id=\"fs-id1167339281022\">\n<dt>leading term<\/dt>\n<dd id=\"fs-id1167339281025\">the term containing the highest degree<\/dd>\n<\/dl>\n<dl id=\"fs-id1167339281028\">\n<dt>monomial<\/dt>\n<dd id=\"fs-id1167339281032\">a polynomial containing one term<\/dd>\n<\/dl>\n<dl id=\"fs-id1167339281035\">\n<dt>perfect square trinomial<\/dt>\n<dd id=\"fs-id1167339281038\">the trinomial that results when a binomial is squared<\/dd>\n<\/dl>\n<dl id=\"fs-id1167339281041\">\n<dt>polynomial<\/dt>\n<dd id=\"fs-id1167339281044\">a sum of terms each consisting of a variable raised to a nonnegative integer power<\/dd>\n<\/dl>\n<dl id=\"fs-id1167339281048\">\n<dt>term of a polynomial<\/dt>\n<dd id=\"fs-id1167339281051\">any[latex]\\,{a}_{i}{x}^{i}\\,[\/latex]of a polynomial in the form[latex]\\,{a}_{n}{x}^{n}+...+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1167339317674\">\n<dt>trinomial<\/dt>\n<dd id=\"fs-id1167339317677\">a polynomial containing three terms<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":291,"menu_order":5,"template":"","meta":{"pb_show_title":null,"pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-27","chapter","type-chapter","status-publish","hentry"],"part":18,"_links":{"self":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/27","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/users\/291"}],"version-history":[{"count":1,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/27\/revisions"}],"predecessor-version":[{"id":28,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/27\/revisions\/28"}],"part":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/parts\/18"}],"metadata":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/27\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/media?parent=27"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapter-type?post=27"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/contributor?post=27"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/license?post=27"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}