{"id":211,"date":"2019-08-20T17:04:07","date_gmt":"2019-08-20T21:04:07","guid":{"rendered":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/binomial-theorem\/"},"modified":"2022-06-01T10:39:40","modified_gmt":"2022-06-01T14:39:40","slug":"binomial-theorem","status":"publish","type":"chapter","link":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/binomial-theorem\/","title":{"raw":"Binomial Theorem","rendered":"Binomial Theorem"},"content":{"raw":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\nIn this section, you will:\n<ul>\n \t<li>Apply the Binomial Theorem.<\/li>\n<\/ul>\n<\/div>\nA polynomial with two terms is called a binomial. We have already learned to multiply binomials and to raise binomials to powers, but raising a binomial to a high power can be tedious and time-consuming. In this section, we will discuss a shortcut that will allow us to find[latex]\\,{\\left(x+y\\right)}^{n}\\,[\/latex]without multiplying the binomial by itself [latex]n[\/latex] times.\n<div id=\"fs-id1165135195368\" class=\"bc-section section\">\n<h3>Identifying Binomial Coefficients<\/h3>\n<p id=\"fs-id1165135181540\">In <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/chapter\/counting-principles\/\">Counting Principles<\/a>, we studied <span class=\"no-emphasis\">combinations<\/span>. In the shortcut to finding[latex]\\,{\\left(x+y\\right)}^{n},\\,[\/latex]we will need to use combinations to find the coefficients that will appear in the expansion of the binomial. In this case, we use the notation[latex]\\,\\left(\\begin{array}{c}n\\\\ r\\end{array}\\right)\\,[\/latex] instead of [latex]C\\left(n,r\\right),[\/latex] but it can be calculated in the same way. So<\/p>\n\n<div id=\"eip-id1165137656645\" class=\"unnumbered\">[latex]\\,\\left(\\begin{array}{c}n\\\\ r\\end{array}\\right)=C\\left(n,r\\right)=\\frac{n!}{r!\\left(n-r\\right)!}\\,[\/latex]<\/div>\n<p id=\"fs-id1165135692906\">The combination[latex]\\,\\left(\\begin{array}{c}n\\\\ r\\end{array}\\right)\\,[\/latex]is called a <strong>binomial coefficient<\/strong>. An example of a binomial coefficient is[latex]\\,\\left(\\begin{array}{c}5\\\\ 2\\end{array}\\right)=C\\left(5,2\\right)=10.\\,[\/latex]<\/p>\n\n<div id=\"fs-id1165137482950\" class=\"textbox key-takeaways\">\n<h3>Binomial Coefficients<\/h3>\n<p id=\"fs-id1165137472666\">If [latex]n[\/latex] and [latex]r[\/latex]are integers greater than or equal to 0 with [latex]n\\ge r,[\/latex] then the binomial coefficient is<\/p>\n\n<div id=\"fs-id1165135188055\" class=\"unnumbered aligncenter\">[latex]\\left(\\begin{array}{c}n\\\\ r\\end{array}\\right)=C\\left(n,r\\right)=\\frac{n!}{r!\\left(n-r\\right)!}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165135174912\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1165137386960\"><strong>Is a binomial coefficient always a whole number?<\/strong><\/p>\n<p id=\"fs-id1165137436611\"><em>Yes. Just as the number of combinations must always be a whole number, a binomial coefficient will always be a whole number. <\/em><\/p>\n\n<\/div>\n<div id=\"Example_11_06_01\" class=\"textbox examples\">\n<div id=\"fs-id1165137837055\">\n<div id=\"fs-id1165137714295\">\n<h3>Finding Binomial Coefficients<\/h3>\n<p id=\"fs-id1165135253860\">Find each binomial coefficient.<\/p>\n\n<ol id=\"fs-id1165137553883\" type=\"a\">\n \t<li>[latex]\\left(\\begin{array}{c}5\\\\ 3\\end{array}\\right)[\/latex]<\/li>\n \t<li>[latex]\\left(\\begin{array}{c}9\\\\ 2\\end{array}\\right)[\/latex]<\/li>\n \t<li>[latex]\\left(\\begin{array}{c}9\\\\ 7\\end{array}\\right)[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165137933188\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137933188\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137933188\"]Use the formula to calculate each binomial coefficient. You can also use the [latex]{n}_{}{C}_{r}[\/latex] function on your calculator.\n<div id=\"eip-id1165137755879\" class=\"unnumbered\">[latex]\\left(\\begin{array}{c}n\\\\ r\\end{array}\\right)=C\\left(n,r\\right)=\\frac{n!}{r!\\left(n-r\\right)!}[\/latex]<\/div>\n<ol type=\"a\">\n \t<li>[latex]\\left(\\begin{array}{c}5\\\\ 3\\end{array}\\right)=\\frac{5!}{3!\\left(5-3\\right)!}=\\frac{5\\cdot 4\\cdot 3!}{3!2!}=10[\/latex]<\/li>\n \t<li>[latex]\\left(\\begin{array}{c}9\\\\ 2\\end{array}\\right)=\\frac{9!}{2!\\left(9-2\\right)!}=\\frac{9\\cdot 8\\cdot 7!}{2!7!}=36[\/latex]<\/li>\n \t<li>[latex]\\left(\\begin{array}{c}9\\\\ 7\\end{array}\\right)=\\frac{9!}{7!\\left(9-7\\right)!}=\\frac{9\\cdot 8\\cdot 7!}{7!2!}=36[\/latex]<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<div id=\"fs-id1165135161493\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137749058\">Notice that we obtained the same result for parts (b) and (c). If you look closely at the solution for these two parts, you will see that you end up with the same two factorials in the denominator, but the order is reversed, just as with combinations.<\/p>\n\n<div id=\"eip-id1165137416511\" class=\"unnumbered\">[latex]\\left(\\begin{array}{c}n\\\\ r\\end{array}\\right)=\\left(\\begin{array}{c}n\\\\ n-r\\end{array}\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135309824\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137812282\">Find each binomial coefficient.<\/p>\n\n<div id=\"ti_11_06_01\">\n<div id=\"fs-id1165137640379\">\n<ol id=\"eip-id1165135332801\" type=\"a\">\n \t<li>[latex]\\,\\left(\\begin{array}{c}7\\\\ 3\\end{array}\\right)\\,[\/latex]<\/li>\n \t<li>[latex]\\,\\left(\\begin{array}{c}11\\\\ 4\\end{array}\\right)\\,[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165137653724\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137653724\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137653724\"]\n<ol id=\"eip-id1165135331689\" type=\"a\">\n \t<li>35<\/li>\n \t<li>330<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137722450\" class=\"bc-section section\">\n<h3>Using the Binomial Theorem<\/h3>\n<p id=\"fs-id1165137445287\">When we expand [latex]{\\left(x+y\\right)}^{n}[\/latex] by multiplying, the result is called a binomial expansion, and it includes binomial coefficients. If we wanted to expand [latex]{\\left(x+y\\right)}^{52},[\/latex] we might multiply [latex]\\left(x+y\\right)[\/latex] by itself fifty-two times. This could take hours! If we examine some simple binomial expansions, we can find patterns that will lead us to a shortcut for finding more complicated binomial expansions.<\/p>\n\n<div id=\"eip-id1165137387258\" class=\"unnumbered\">[latex]\\begin{array}{l}{\\left(x+y\\right)}^{2}={x}^{2}+2xy+{y}^{2}\\hfill \\\\ {\\left(x+y\\right)}^{3}={x}^{3}+3{x}^{2}y+3x{y}^{2}+{y}^{3}\\hfill \\\\ {\\left(x+y\\right)}^{4}={x}^{4}+4{x}^{3}y+6{x}^{2}{y}^{2}+4x{y}^{3}+{y}^{4}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137602015\">First, let\u2019s examine the exponents. With each successive term, the exponent for [latex]x[\/latex] decreases and the exponent for [latex]y[\/latex] increases. The sum of the two exponents is [latex]n[\/latex] for each term.<\/p>\n<p id=\"fs-id1165137738201\">Next, let\u2019s examine the coefficients. Notice that the coefficients increase and then decrease in a symmetrical pattern. The coefficients follow a pattern:<\/p>\n\n<div id=\"eip-id1165135173876\" class=\"unnumbered\">[latex]\\left(\\begin{array}{c}n\\\\ 0\\end{array}\\right),\\left(\\begin{array}{c}n\\\\ 1\\end{array}\\right),\\left(\\begin{array}{c}n\\\\ 2\\end{array}\\right),...,\\left(\\begin{array}{c}n\\\\ n\\end{array}\\right).[\/latex]<\/div>\n<p id=\"fs-id1165137666545\">These patterns lead us to the <strong>Binomial Theorem<\/strong>, which can be used to expand any binomial.<\/p>\n\n<div id=\"eip-id1165135194720\" class=\"unnumbered\">[latex]\\begin{array}{ll}{\\left(x+y\\right)}^{n}\\hfill &amp; =\\sum _{k=0}^{n}\\left(\\begin{array}{c}n\\\\ k\\end{array}\\right){x}^{n-k}{y}^{k}\\hfill \\\\ \\hfill &amp; ={x}^{n}+\\left(\\begin{array}{c}n\\\\ 1\\end{array}\\right){x}^{n-1}y+\\left(\\begin{array}{c}n\\\\ 2\\end{array}\\right){x}^{n-2}{y}^{2}+...+\\left(\\begin{array}{c}n\\\\ n-1\\end{array}\\right)x{y}^{n-1}+{y}^{n}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137602044\">Another way to see the coefficients is to examine the expansion of a binomial in general form,[latex]\\,x+y,\\,[\/latex]to successive powers 1, 2, 3, and 4.<\/p>\n\n<div id=\"eip-id1165137935737\" class=\"unnumbered\">[latex]\\begin{array}{l}{\\left(x+y\\right)}^{1}=x+y\\hfill \\\\ {\\left(x+y\\right)}^{2}={x}^{2}+2xy+{y}^{2}\\hfill \\\\ {\\left(x+y\\right)}^{3}={x}^{3}+3{x}^{2}y+3x{y}^{2}+{y}^{3}\\hfill \\\\ {\\left(x+y\\right)}^{4}={x}^{4}+4{x}^{3}y+6{x}^{2}{y}^{2}+4x{y}^{3}+{y}^{4}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137430560\">Can you guess the next expansion for the binomial[latex]\\,{\\left(x+y\\right)}^{5}?\\,[\/latex]<\/p>\n\n<div id=\"CNX_Precalc_Figure_11_06_002\" class=\"wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19155056\/CNX_Precalc_Figure_11_06_002.jpg\" alt=\"Graph of the function f_2.\" width=\"731\" height=\"413\"> <strong>Figure 1.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1165137443527\">See <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_11_06_002\">(Figure)<\/a>, which illustrates the following:<\/p>\n\n<ul id=\"fs-id1165137748570\">\n \t<li>There are [latex]n+1[\/latex] terms in the expansion of [latex]{\\left(x+y\\right)}^{n}.[\/latex]<\/li>\n \t<li>The degree (or sum of the exponents) for each term is [latex]n.[\/latex]<\/li>\n \t<li>The powers on [latex]x[\/latex] begin with [latex]n[\/latex] and decrease to 0.<\/li>\n \t<li>The powers on [latex]y[\/latex] begin with 0 and increase to [latex]n.[\/latex]<\/li>\n \t<li>The coefficients are symmetric.<\/li>\n<\/ul>\n<p id=\"fs-id1165137605699\">To determine the expansion on [latex]{\\left(x+y\\right)}^{5},[\/latex] we see [latex]n=5,[\/latex] thus, there will be 5+1 = 6 terms. Each term has a combined degree of 5. In descending order for powers of [latex]x,[\/latex] the pattern is as follows:<\/p>\n\n<ul id=\"fs-id1165137804247\">\n \t<li>Introduce [latex]{x}^{5},[\/latex] and then for each successive term reduce the exponent on [latex]x[\/latex] by 1 until [latex]{x}^{0}=1[\/latex] is reached.<\/li>\n \t<li>Introduce [latex]{y}^{0}=1,[\/latex] and then increase the exponent on [latex]y[\/latex] by 1 until [latex]{y}^{5}[\/latex] is reached.\n<div id=\"eip-id1165137643155\" class=\"unnumbered\">[latex]{x}^{5},\\,\\,{x}^{4}y,\\,\\,{x}^{3}{y}^{2},\\,\\,{x}^{2}{y}^{3},\\,\\,x{y}^{4},\\,\\,{y}^{5}[\/latex]<\/div><\/li>\n<\/ul>\n<p id=\"fs-id1165137445918\">The next expansion would be<\/p>\n\n<div id=\"eip-id1165137460981\" class=\"unnumbered\">[latex]{\\left(x+y\\right)}^{5}={x}^{5}+5{x}^{4}y+10{x}^{3}{y}^{2}+10{x}^{2}{y}^{3}+5x{y}^{4}+{y}^{5}.[\/latex]<\/div>\nBut where do those coefficients come from? The binomial coefficients are symmetric. We can see these coefficients in an array known as <span class=\"no-emphasis\">Pascal's Triangle<\/span>, shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_11_06_001\">(Figure)<\/a>.\n<div id=\"CNX_Precalc_Figure_11_06_001\" class=\"medium aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19155101\/CNX_Precalc_Figure_11_06_001.jpg\" alt=\"Pascal's Triangle\" width=\"731\" height=\"300\"> <strong>Figure 2.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1165137446339\">To generate Pascal\u2019s Triangle, we start by writing a 1. In the row below, row 2, we write two 1\u2019s. In the 3<sup>rd<\/sup> row, flank the ends of the rows with 1\u2019s, and add [latex]1+1[\/latex] to find the middle number, 2. In the [latex]n\\text{th}[\/latex] row, flank the ends of the row with 1\u2019s. Each element in the triangle is the sum of the two elements immediately above it.<\/p>\n<p id=\"fs-id1165137864198\">To see the connection between Pascal\u2019s Triangle and binomial coefficients, let us revisit the expansion of the binomials in general form.<\/p>\n<span id=\"fs-id1165137643968\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19155115\/CNX_Precalc_Figure_11_06_003.jpg\" alt=\"Pascal's Triangle expanded to show the values of the triangle as x and y terms with exponents\"><\/span>\n<div id=\"fs-id1165135187302\" class=\"textbox key-takeaways\">\n<h3>The Binomial Theorem<\/h3>\n<p id=\"fs-id1165137726456\">The Binomial Theorem is a formula that can be used to expand any binomial.<\/p>\n\n<div id=\"fs-id1165137828132\">[latex]\\begin{array}{ll}{\\left(x+y\\right)}^{n}\\hfill &amp; =\\sum _{k=0}^{n}\\left(\\begin{array}{c}n\\\\ k\\end{array}\\right){x}^{n-k}{y}^{k}\\hfill \\\\ \\hfill &amp; ={x}^{n}+\\left(\\begin{array}{c}n\\\\ 1\\end{array}\\right){x}^{n-1}y+\\left(\\begin{array}{c}n\\\\ 2\\end{array}\\right){x}^{n-2}{y}^{2}+...+\\left(\\begin{array}{c}n\\\\ n-1\\end{array}\\right)x{y}^{n-1}+{y}^{n}\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165135646154\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137634381\"><strong>Given a binomial, write it in expanded form.<\/strong><\/p>\n\n<ol id=\"fs-id1165135377104\" type=\"1\">\n \t<li>Determine the value of [latex]n[\/latex]according to the exponent.<\/li>\n \t<li>Evaluate the [latex]k=0[\/latex] through [latex]k=n[\/latex] using the Binomial Theorem formula.<\/li>\n \t<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_11_06_02\" class=\"textbox examples\">\n<div id=\"fs-id1165137843915\">\n<div id=\"fs-id1165137843917\">\n<h3>Expanding a Binomial<\/h3>\n<p id=\"fs-id1165137571473\">Write in expanded form.<\/p>\n\n<ol id=\"fs-id1165137571476\" type=\"a\">\n \t<li>[latex]\\,{\\left(x+y\\right)}^{5}\\,[\/latex]<\/li>\n \t<li>[latex]\\,{\\left(3x-y\\right)}^{4}\\,[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165137771774\" class=\"solution textbox shaded\">\n<div id=\"eip-id1165133341948\" class=\"unnumbered\">[reveal-answer q=\"221236\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"221236\"]\n<ol id=\"fs-id1165137804171\" type=\"a\">\n \t<li>Substitute [latex]n=5[\/latex] into the formula. Evaluate the [latex]k=0[\/latex] through [latex]k=5[\/latex] terms. Simplify.\n<div id=\"eip-id1165135600195\" class=\"unnumbered\">[latex]\\begin{array}{ll}{\\left(x+y\\right)}^{5}\\hfill &amp; =\\left(\\begin{array}{c}5\\\\ 0\\end{array}\\right){x}^{5}{y}^{0}+\\left(\\begin{array}{c}5\\\\ 1\\end{array}\\right){x}^{4}{y}^{1}+\\left(\\begin{array}{c}5\\\\ 2\\end{array}\\right){x}^{3}{y}^{2}+\\left(\\begin{array}{c}5\\\\ 3\\end{array}\\right){x}^{2}{y}^{3}+\\left(\\begin{array}{c}5\\\\ 4\\end{array}\\right){x}^{1}{y}^{4}+\\left(\\begin{array}{c}5\\\\ 5\\end{array}\\right){x}^{0}{y}^{5}\\hfill \\\\ {\\left(x+y\\right)}^{5}\\hfill &amp; ={x}^{5}+5{x}^{4}y+10{x}^{3}{y}^{2}+10{x}^{2}{y}^{3}+5x{y}^{4}+{y}^{5}\\hfill \\end{array}[\/latex]<\/div><\/li>\n \t<li>Substitute [latex]n=4[\/latex] into the formula. Evaluate the [latex]k=0[\/latex] through [latex]k=4[\/latex] terms. Notice that [latex]3x[\/latex] is in the place that was occupied by [latex]x[\/latex] and that [latex]\u2013y[\/latex] is in the place that was occupied by [latex]y.[\/latex] So we substitute them. Simplify.\n<div id=\"eip-id1165133341948\" class=\"unnumbered\">[latex]\\begin{array}{ll}{\\left(3x-y\\right)}^{4}\\hfill &amp; =\\left(\\begin{array}{c}4\\\\ 0\\end{array}\\right){\\left(3x\\right)}^{4}{\\left(-y\\right)}^{0}+\\left(\\begin{array}{c}4\\\\ 1\\end{array}\\right){\\left(3x\\right)}^{3}{\\left(-y\\right)}^{1}+\\left(\\begin{array}{c}4\\\\ 2\\end{array}\\right){\\left(3x\\right)}^{2}{\\left(-y\\right)}^{2}+\\left(\\begin{array}{c}4\\\\ 3\\end{array}\\right){\\left(3x\\right)}^{1}{\\left(-y\\right)}^{3}+\\left(\\begin{array}{c}4\\\\ 4\\end{array}\\right){\\left(3x\\right)}^{0}{\\left(-y\\right)}^{4}\\hfill \\\\ {\\left(3x-y\\right)}^{4}\\hfill &amp; =81{x}^{4}-108{x}^{3}y+54{x}^{2}{y}^{2}-12x{y}^{3}+{y}^{4}\\hfill \\end{array}[\/latex]<\/div><\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135191233\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165135296274\">Notice the alternating signs in part b. This happens because[latex]\\,\\left(-y\\right)\\,[\/latex]raised to odd powers is negative, but[latex]\\,\\left(-y\\right)\\,[\/latex]raised to even powers is positive. This will occur whenever the binomial contains a subtraction sign.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135501144\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_06_02\">\n<div id=\"fs-id1165137772198\">\n<p id=\"fs-id1165137772200\">Write in expanded form.<\/p>\n\n<ol id=\"eip-id1700717\" type=\"a\">\n \t<li>[latex]{\\left(x-y\\right)}^{5}[\/latex]<\/li>\n \t<li>[latex]{\\left(2x+5y\\right)}^{3}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165137635439\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137635439\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137635439\"]\n<ol id=\"eip-id1366342\" type=\"a\">\n \t<li>[latex]{x}^{5}-5{x}^{4}y+10{x}^{3}{y}^{2}-10{x}^{2}{y}^{3}+5x{y}^{4}-{y}^{5}[\/latex]<\/li>\n \t<li>[latex]8{x}^{3}+60{x}^{2}y+150x{y}^{2}+125{y}^{3}[\/latex]<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137658053\" class=\"bc-section section\">\n<h3>Using the Binomial Theorem to Find a Single Term<\/h3>\n<p id=\"fs-id1165137736641\">Expanding a binomial with a high exponent such as[latex]\\,{\\left(x+2y\\right)}^{16}\\,[\/latex]can be a lengthy process.<\/p>\n<p id=\"eip-164\">Sometimes we are interested only in a certain term of a binomial expansion. We do not need to fully expand a binomial to find a single specific term.<\/p>\n<p id=\"fs-id1165137834237\">Note the pattern of coefficients in the expansion of[latex]\\,{\\left(x+y\\right)}^{5}.[\/latex]<\/p>\n\n<div id=\"eip-405\" class=\"unnumbered aligncenter\">[latex]{\\left(x+y\\right)}^{5}={x}^{5}+\\left(\\begin{array}{c}5\\\\ 1\\end{array}\\right){x}^{4}y+\\left(\\begin{array}{c}5\\\\ 2\\end{array}\\right){x}^{3}{y}^{2}+\\left(\\begin{array}{c}5\\\\ 3\\end{array}\\right){x}^{2}{y}^{3}+\\left(\\begin{array}{c}5\\\\ 4\\end{array}\\right)x{y}^{4}+{y}^{5}[\/latex]<\/div>\n<p id=\"fs-id1165137855349\">The second term is[latex]\\,\\left(\\begin{array}{c}5\\\\ 1\\end{array}\\right){x}^{4}y.\\,[\/latex]The third term is[latex]\\,\\left(\\begin{array}{c}5\\\\ 2\\end{array}\\right){x}^{3}{y}^{2}.\\,[\/latex]We can generalize this result.<\/p>\n\n<div id=\"eip-415\" class=\"unnumbered aligncenter\">[latex]\\left(\\begin{array}{c}n\\\\ r\\end{array}\\right){x}^{n-r}{y}^{r}[\/latex]<\/div>\n<div id=\"fs-id1165137466315\" class=\"textbox key-takeaways\">\n<h3>The (r+1)th Term of a Binomial Expansion<\/h3>\n<p id=\"fs-id1165137628103\">The[latex]\\,\\left(r+1\\right)\\text{th}\\,[\/latex]term of the <span class=\"no-emphasis\">binomial expansion<\/span> of[latex]\\,{\\left(x+y\\right)}^{n}\\,[\/latex]is:<\/p>\n\n<div id=\"fs-id1165135203716\">[latex]\\left(\\begin{array}{c}n\\\\ r\\end{array}\\right){x}^{n-r}{y}^{r}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137400324\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<strong>Given a binomial, write a specific term without fully expanding.<\/strong>\n<ol id=\"fs-id1165137414548\" type=\"1\">\n \t<li>Determine the value of [latex]n[\/latex] according to the exponent.<\/li>\n \t<li>Determine [latex]\\left(r+1\\right).[\/latex]<\/li>\n \t<li>Determine [latex]r.[\/latex]<\/li>\n \t<li>Replace [latex]r[\/latex] in the formula for the [latex]\\left(r+1\\right)\\text{th}[\/latex] term of the binomial expansion.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_11_06_03\" class=\"textbox examples\">\n<div id=\"fs-id1165137605967\">\n<div id=\"fs-id1165137605969\">\n<h3>Writing a Given Term of a Binomial Expansion<\/h3>\nFind the tenth term of[latex]\\,{\\left(x+2y\\right)}^{16}\\,[\/latex]without fully expanding the binomial.\n\n<\/div>\n<div id=\"fs-id1165137834759\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137834759\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137834759\"]\n<p id=\"fs-id1165137737706\">Because we are looking for the tenth term, [latex]\\,r+1=10,\\,[\/latex] we will use [latex]\\,r=9[\/latex] in our calculations.<\/p>\n\n<div id=\"eip-id1165137464257\" class=\"unnumbered\">[latex]\\left(\\begin{array}{c}n\\\\ r\\end{array}\\right){x}^{n-r}{y}^{r}[\/latex]<\/div>\n<div id=\"eip-id1165137894367\" class=\"unnumbered\">[latex]\\left(\\begin{array}{c}16\\\\ 9\\end{array}\\right){x}^{16-9}{\\left(2y\\right)}^{9}=5\\text{,}857\\text{,}280{x}^{7}{y}^{9}[\/latex]<\/div>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137732277\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_06_03\">\n<div id=\"fs-id1165137657120\">\n<p id=\"fs-id1165137657121\">Find the sixth term of[latex]\\,{\\left(3x-y\\right)}^{9}\\,[\/latex]without fully expanding the binomial.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137758550\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137758550\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137758550\"]\n[latex]\\,-10,206{x}^{4}{y}^{5}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137862398\" class=\"precalculus media\">\n<p id=\"fs-id1165137611861\">Access these online resources for additional instruction and practice with binomial expansion.<\/p>\n\n<ul>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/binomialtheorem\">The Binomial Theorem<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/btexample\">Binomial Theorem Example<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135705092\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"eip-id1165134166609\" summary=\"..\">\n<tbody>\n<tr>\n<td>Binomial Theorem<\/td>\n<td>[latex]{\\left(x+y\\right)}^{n}=\\sum _{k-0}^{n}\\left(\\begin{array}{c}n\\\\ k\\end{array}\\right){x}^{n-k}{y}^{k}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left(r+1\\right)th\\,[\/latex]term of a binomial expansion<\/td>\n<td>[latex]\\left(\\begin{array}{c}n\\\\ r\\end{array}\\right){x}^{n-r}{y}^{r}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137469809\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165135187830\">\n \t<li>[latex]\\left(\\begin{array}{c}n\\\\ r\\end{array}\\right)\\,[\/latex]is called a binomial coefficient and is equal to [latex]C\\left(n,r\\right).\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_11_06_01\">(Figure)<\/a>.<\/li>\n \t<li>The Binomial Theorem allows us to expand binomials without multiplying. See <a class=\"autogenerated-content\" href=\"#Example_11_06_02\">(Figure)<\/a>.<\/li>\n \t<li>We can find a given term of a binomial expansion without fully expanding the binomial. See <a class=\"autogenerated-content\" href=\"#Example_11_06_03\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165134089502\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165134089506\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165137419807\">\n<div id=\"fs-id1165135181496\">\n<p id=\"fs-id1165135181499\">What is a binomial coefficient, and how it is calculated?<\/p>\n\n<\/div>\n<div id=\"fs-id1165135501149\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135501149\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135501149\"]A binomial coefficient is an alternative way of denoting the combination [latex]\\,C\\left(n,r\\right).\\,[\/latex]It is defined as[latex]\\,\\left(\\begin{array}{c}n\\\\ r\\end{array}\\right)=\\,C\\left(n,r\\right)\\,=\\frac{n!}{r!\\left(n-r\\right)!}.[\/latex][\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137442166\">\n<div id=\"fs-id1165137442168\">\n<p id=\"fs-id1165137442170\">What role do binomial coefficients play in a binomial expansion? Are they restricted to any type of number?<\/p>\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135190462\">\n<p id=\"fs-id1165135190464\">What is the Binomial Theorem and what is its use?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137452921\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137452921\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137452921\"]\n<p id=\"fs-id1165137452923\">The Binomial Theorem is defined as[latex]\\,{\\left(x+y\\right)}^{n}=\\sum _{k=0}^{n}\\left(\\begin{array}{c}n\\\\ k\\end{array}\\right){x}^{n-k}{y}^{k}\\,[\/latex]and can be used to expand any binomial.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137445986\">\n<div id=\"fs-id1165137445988\">\n<p id=\"fs-id1165137445990\">When is it an advantage to use the Binomial Theorem? Explain.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137408889\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1165135169145\">For the following exercises, evaluate the binomial coefficient.<\/p>\n\n<div id=\"fs-id1165135526110\">\n<div id=\"fs-id1165135526112\">\n<p id=\"fs-id1165135526114\">[latex]\\left(\\begin{array}{c}6\\\\ 2\\end{array}\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137583395\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137583395\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137583395\"]\n<p id=\"fs-id1165137583397\">15<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137455051\">\n<div id=\"fs-id1165137455053\">\n<p id=\"fs-id1165137655192\">[latex]\\left(\\begin{array}{c}5\\\\ 3\\end{array}\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135484521\">\n<div id=\"fs-id1165137541584\">\n<p id=\"fs-id1165137541586\">[latex]\\left(\\begin{array}{c}7\\\\ 4\\end{array}\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135484154\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135484154\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135484154\"]\n<p id=\"fs-id1165135484156\">35<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137772477\">\n<div>[latex]\\left(\\begin{array}{c}9\\\\ 7\\end{array}\\right)[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165135506400\">\n<div id=\"fs-id1165135506403\">\n<p id=\"fs-id1165135570073\">[latex]\\left(\\begin{array}{c}10\\\\ 9\\end{array}\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137480607\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137480607\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137480607\"]\n<p id=\"fs-id1165137480609\">10<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137656508\">\n<div id=\"fs-id1165137656510\">\n<p id=\"fs-id1165137530730\">[latex]\\left(\\begin{array}{c}25\\\\ 11\\end{array}\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135241276\">\n<div id=\"fs-id1165135424707\">\n<p id=\"fs-id1165135424709\">[latex]\\left(\\begin{array}{c}17\\\\ 6\\end{array}\\right)[\/latex]<\/p>\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"207307\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"207307\"]12,376[\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135193638\">\n<div id=\"fs-id1165135193640\">\n<p id=\"fs-id1165137409164\">[latex]\\left(\\begin{array}{c}200\\\\ 199\\end{array}\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137807329\">For the following exercises, use the Binomial Theorem to expand each binomial.<\/p>\n\n<div id=\"fs-id1165137442493\">\n<div id=\"fs-id1165137442495\">\n<p id=\"fs-id1165137724327\">[latex]{\\left(4a-b\\right)}^{3}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137831242\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137831242\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137831242\"]\n[latex]64{a}^{3}-48{a}^{2}b+12a{b}^{2}-{b}^{3}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137548190\">\n<div id=\"fs-id1165137548192\">\n<p id=\"fs-id1165137548194\">[latex]{\\left(5a+2\\right)}^{3}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137542019\">\n<div>[latex]{\\left(3a+2b\\right)}^{3}[\/latex]<\/div>\n<div id=\"fs-id1165135394319\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135394319\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135394319\"]\n<p id=\"fs-id1165135394321\">[latex]27{a}^{3}+54{a}^{2}b+36a{b}^{2}+8{b}^{3}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137414737\">\n<div id=\"fs-id1165137414739\">\n<p id=\"fs-id1165137647996\">[latex]{\\left(2x+3y\\right)}^{4}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1165137896227\">[latex]{\\left(4x+2y\\right)}^{5}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135516855\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135516855\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135516855\"]\n<p id=\"fs-id1165135516857\">[latex]1024{x}^{5}+2560{x}^{4}y+2560{x}^{3}{y}^{2}+1280{x}^{2}{y}^{3}+320x{y}^{4}+32{y}^{5}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137678200\">\n<div id=\"fs-id1165137678202\">\n<p id=\"fs-id1165137678204\">[latex]{\\left(3x-2y\\right)}^{4}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137667609\">\n<div id=\"fs-id1165137667611\">\n<p id=\"fs-id1165137667613\">[latex]{\\left(4x-3y\\right)}^{5}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137837121\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137837121\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137837121\"]\n<p id=\"fs-id1165137642794\">[latex]1024{x}^{5}-3840{x}^{4}y+5760{x}^{3}{y}^{2}-4320{x}^{2}{y}^{3}+1620x{y}^{4}-243{y}^{5}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137418542\">\n<div id=\"fs-id1165137418544\">\n<p id=\"fs-id1165137418546\">[latex]{\\left(\\frac{1}{x}+3y\\right)}^{5}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135526126\">\n<div id=\"fs-id1165135526128\">\n<p id=\"fs-id1165135526130\">[latex]{\\left({x}^{-1}+2{y}^{-1}\\right)}^{4}[\/latex]<\/p>\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165137805817\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137805817\"]\n<p id=\"fs-id1165137805817\">[latex]\\frac{1}{{x}^{4}}+\\frac{8}{{x}^{3}y}+\\frac{24}{{x}^{2}{y}^{2}}+\\frac{32}{x{y}^{3}}+\\frac{16}{{y}^{4}}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137828074\">\n<p id=\"fs-id1165137828076\">[latex]{\\left(\\sqrt{x}-\\sqrt{y}\\right)}^{5}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135149859\">For the following exercises, use the Binomial Theorem to write the first three terms of each binomial.<\/p>\n\n<div>\n<div>\n<p id=\"fs-id1165137565759\">[latex]{\\left(a+b\\right)}^{17}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137527392\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137527392\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137527392\"]\n<p id=\"fs-id1165137535275\">[latex]{a}^{17}+17{a}^{16}b+136{a}^{15}{b}^{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135182971\">\n<div id=\"fs-id1165137871899\">\n<p id=\"fs-id1165137871901\">[latex]{\\left(x-1\\right)}^{18}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137736266\">\n<div id=\"fs-id1165137736268\">\n<p id=\"fs-id1165137409564\">[latex]{\\left(a-2b\\right)}^{15}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135188234\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135188234\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135188234\"]\n[latex]{a}^{15}-30{a}^{14}b+420{a}^{13}{b}^{2}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137543302\">\n<div id=\"fs-id1165137543304\">\n<p id=\"fs-id1165137543306\">[latex]{\\left(x-2y\\right)}^{8}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137635106\">\n<div>\n<p id=\"fs-id1165137874644\">[latex]{\\left(3a+b\\right)}^{20}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137659105\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137659105\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137659105\"]\n<p id=\"fs-id1165137659107\">[latex]3,486,784,401{a}^{20}+23,245,229,340{a}^{19}b+73,609,892,910{a}^{18}{b}^{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137646617\">\n<div id=\"fs-id1165137526699\">\n<p id=\"fs-id1165137526701\">[latex]{\\left(2a+4b\\right)}^{7}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137732098\">\n<div id=\"fs-id1165137573195\">\n<p id=\"fs-id1165137573197\">[latex]{\\left({x}^{3}-\\sqrt{y}\\right)}^{8}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137433769\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137433769\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137433769\"]\n<p id=\"fs-id1165137414388\">[latex]{x}^{24}-8{x}^{21}\\sqrt{y}+28{x}^{18}y[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165134032296\">For the following exercises, find the indicated term of each binomial without fully expanding the binomial.<\/p>\n\n<div id=\"fs-id1165135344957\">\n<div id=\"fs-id1165135344959\">\n<p id=\"fs-id1165135344961\">The fourth term of[latex]\\,{\\left(2x-3y\\right)}^{4}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137696650\">\n<div id=\"fs-id1165137696653\">\n<p id=\"fs-id1165135193960\">The fourth term of[latex]\\,{\\left(3x-2y\\right)}^{5}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137827785\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137827785\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137827785\"]\n<p id=\"fs-id1165137827787\">[latex]-720{x}^{2}{y}^{3}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137898985\">\n<div id=\"fs-id1165137898987\">\n<p id=\"fs-id1165137898989\">The third term of[latex]\\,{\\left(6x-3y\\right)}^{7}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135195054\">\n<div id=\"fs-id1165135195056\">\n<p id=\"fs-id1165137737799\">The eighth term of[latex]\\,{\\left(7+5y\\right)}^{14}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137549805\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137549805\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137549805\"]\n<p id=\"fs-id1165137659226\">[latex]220,812,466,875,000{y}^{7}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137466522\">\n<div id=\"fs-id1165137466524\">\n<p id=\"fs-id1165137466526\">The seventh term of[latex]\\,{\\left(a+b\\right)}^{11}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135173932\">\n<div id=\"fs-id1165135173934\">\n<p id=\"fs-id1165135173936\">The fifth term of[latex]\\,{\\left(x-y\\right)}^{7}[\/latex]<\/p>\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165137673496\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137673496\"]\n<p id=\"fs-id1165137673496\">[latex]35{x}^{3}{y}^{4}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137768225\">\n<p id=\"fs-id1165137768227\">The tenth term of[latex]\\,{\\left(x-1\\right)}^{12}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135541625\">\n<div id=\"fs-id1165135541627\">\n<p id=\"fs-id1165135541629\">The ninth term of[latex]\\,{\\left(a-3{b}^{2}\\right)}^{11}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137423625\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137423625\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137423625\"]\n[latex]1,082,565{a}^{3}{b}^{16}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137723425\">\n<div id=\"fs-id1165137536035\">\n<p id=\"fs-id1165137536037\">The fourth term of[latex]\\,{\\left({x}^{3}-\\frac{1}{2}\\right)}^{10}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135152237\">\n<div id=\"fs-id1165135152240\">\n<p id=\"fs-id1165137445389\">The eighth term of[latex]\\,{\\left(\\frac{y}{2}+\\frac{2}{x}\\right)}^{9}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137692780\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137692780\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137692780\"]\n<p id=\"fs-id1165137692782\">[latex]\\frac{1152{y}^{2}}{{x}^{7}}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137653220\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1165137635092\">For the following exercises, use the Binomial Theorem to expand the binomial [latex]f\\left(x\\right)={\\left(x+3\\right)}^{4}.[\/latex] Then find and graph each indicated sum on one set of axes.<\/p>\n\n<div id=\"fs-id1165137407882\">\n<div id=\"fs-id1165137407884\">\n<p id=\"fs-id1165137693683\">Find and graph[latex]\\,{f}_{1}\\left(x\\right),\\,[\/latex]such that[latex]\\,{f}_{1}\\left(x\\right)\\,[\/latex]is the first term of the expansion.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137925296\">\n<div id=\"fs-id1165137925298\">\n<p id=\"fs-id1165137925300\">Find and graph[latex]\\,{f}_{2}\\left(x\\right),\\,[\/latex]such that[latex]\\,{f}_{2}\\left(x\\right)\\,[\/latex]is the sum of the first two terms of the expansion.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137575802\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137575802\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137575802\"]\n<p id=\"fs-id1165137575804\">[latex]{f}_{2}\\left(x\\right)={x}^{4}+12{x}^{3}[\/latex]<\/p>\n<span id=\"fs-id1165137726516\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19155129\/CNX_Precalc_Figure_11_06_202.jpg\" alt=\"Graph of the function f_2.\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137812692\">\n<div id=\"fs-id1165137812694\">\n<p id=\"fs-id1165137812696\">Find and graph[latex]\\,{f}_{3}\\left(x\\right),\\,[\/latex]such that[latex]\\,{f}_{3}\\left(x\\right)\\,[\/latex] is the sum of the first three terms of the expansion.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137723651\">\n<div id=\"fs-id1165137723653\">\n<p id=\"fs-id1165137723655\">Find and graph[latex]\\,{f}_{4}\\left(x\\right),\\,[\/latex]such that[latex]\\,{f}_{4}\\left(x\\right)\\,[\/latex]is the sum of the first four terms of the expansion.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135408467\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135408467\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135408467\"]\n<p id=\"fs-id1165135408469\">[latex]{f}_{4}\\left(x\\right)={x}^{4}+12{x}^{3}+54{x}^{2}+108x[\/latex]<\/p>\n<span id=\"fs-id1165137482837\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19155139\/CNX_Precalc_Figure_11_06_204.jpg\" alt=\"Graph of the function f_4.\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137557834\">\n<div id=\"fs-id1165137557836\">\n<p id=\"fs-id1165137557838\">Find and graph[latex]\\,{f}_{5}\\left(x\\right),\\,[\/latex]such that[latex]\\,{f}_{5}\\left(x\\right)\\,[\/latex]is the sum of the first five terms of the expansion.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137836668\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<div id=\"fs-id1165137836673\">\n<div id=\"fs-id1165137836675\">\n<p id=\"fs-id1165137932593\">In the expansion of[latex]\\,{\\left(5x+3y\\right)}^{n},\\,[\/latex]each term has the form[latex]\\,\\left(\\begin{array}{c}n\\\\ k\\end{array}\\right){a}^{n\u2013k}{b}^{k}, \\text{where} k\\,[\/latex]successively takes on the value[latex]\\,0,1,2,\\,...,\\,n.[\/latex]If[latex]\\,\\left(\\begin{array}{c}n\\\\ k\\end{array}\\right)=\\left(\\begin{array}{c}7\\\\ 2\\end{array}\\right),\\,[\/latex]what is the corresponding term?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137871611\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137871611\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137871611\"]\n<p id=\"fs-id1165137476761\">[latex]590,625{x}^{5}{y}^{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137441704\">\n<div id=\"fs-id1165137441706\">\n\nIn the expansion of[latex]\\,{\\left(a+b\\right)}^{n},\\,[\/latex]the coefficient of[latex]\\,{a}^{n-k}{b}^{k}\\,[\/latex]is the same as the coefficient of which other term?\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137651016\">\n<div id=\"fs-id1165137651019\">\n<p id=\"fs-id1165135209329\">Consider the expansion of[latex]\\,{\\left(x+b\\right)}^{40}.\\,[\/latex]What is the exponent of [latex]b[\/latex] in the [latex]k\\text{th}[\/latex] term?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137731710\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137731710\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137731710\"]\n<p id=\"fs-id1165137731712\">[latex]k-1[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137755502\">\n<div id=\"fs-id1165137755504\">\n\nFind[latex]\\,\\left(\\begin{array}{c}n\\\\ k-1\\end{array}\\right)+\\left(\\begin{array}{c}n\\\\ k\\end{array}\\right)\\,[\/latex]and write the answer as a binomial coefficient in the form[latex]\\,\\left(\\begin{array}{c}n\\\\ k\\end{array}\\right).\\,[\/latex]Prove it. <em>Hint:<\/em> Use the fact that, for any integer[latex]\\,p,\\,[\/latex]such that[latex]\\,p\\ge 1,\\,p!=p\\left(p-1\\right)!\\text{.}[\/latex]\n\n<\/div>\n<div id=\"fs-id1165137692126\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137692126\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137692126\"]\n<p id=\"fs-id1165137692128\">[latex]\\left(\\begin{array}{c}n\\\\ k-1\\end{array}\\right)+\\left(\\begin{array}{l}n\\\\ k\\end{array}\\right)=\\left(\\begin{array}{c}n+1\\\\ k\\end{array}\\right);\\,[\/latex]Proof:<\/p>\n<p id=\"fs-id1165137759673\">[latex]\\begin{array}{}\\\\ \\\\ \\\\ \\,\\,\\,\\,\\,\\left(\\begin{array}{c}n\\\\ k-1\\end{array}\\right)+\\left(\\begin{array}{l}n\\\\ k\\end{array}\\right)\\\\ =\\frac{n!}{k!\\left(n-k\\right)!}+\\frac{n!}{\\left(k-1\\right)!\\left(n-\\left(k-1\\right)\\right)!}\\\\ =\\frac{n!}{k!\\left(n-k\\right)!}+\\frac{n!}{\\left(k-1\\right)!\\left(n-k+1\\right)!}\\\\ =\\frac{\\left(n-k+1\\right)n!}{\\left(n-k+1\\right)k!\\left(n-k\\right)!}+\\frac{kn!}{k\\left(k-1\\right)!\\left(n-k+1\\right)!}\\\\ =\\frac{\\left(n-k+1\\right)n!+kn!}{k!\\left(n-k+1\\right)!}\\\\ =\\frac{\\left(n+1\\right)n!}{k!\\left(\\left(n+1\\right)-k\\right)!}\\\\ =\\frac{\\left(n+1\\right)!}{k!\\left(\\left(n+1\\right)-k\\right)!}\\\\ =\\left(\\begin{array}{c}n+1\\\\ k\\end{array}\\right)\\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137757975\">\n<div id=\"fs-id1165137757977\">\n\nWhich expression cannot be expanded using the Binomial Theorem? Explain.\n<ul id=\"eip-id1165137732312\">\n \t<li>[latex]\\left({x}^{2}-2x+1\\right)[\/latex]<\/li>\n \t<li>[latex]{\\left(\\sqrt{a}+4\\sqrt{a}-5\\right)}^{8}[\/latex]<\/li>\n \t<li>[latex]{\\left({x}^{3}+2{y}^{2}-z\\right)}^{5}[\/latex]<\/li>\n \t<li>[latex]{\\left(3{x}^{2}-\\sqrt{2{y}^{3}}\\right)}^{12}[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165135176542\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135176542\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135176542\"]\n<p id=\"fs-id1165135176544\">The expression[latex]\\,{\\left({x}^{3}+2{y}^{2}-z\\right)}^{5}\\,[\/latex]cannot be expanded using the Binomial Theorem because it cannot be rewritten as a binomial.<\/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-id1165137673583\">\n \t<dt>binomial coefficient<\/dt>\n \t<dd id=\"fs-id1165137673588\">the number of ways to choose<em> r<\/em> objects from <em>n<\/em> objects where order does not matter; equivalent to[latex]\\,C\\left(n,r\\right),\\,[\/latex]denoted[latex]\\,\\left(\\begin{array}{c}n\\\\ r\\end{array}\\right)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137812031\">\n \t<dt>binomial expansion<\/dt>\n \t<dd id=\"fs-id1165137812163\">the result of expanding[latex]\\,{\\left(x+y\\right)}^{n}\\,[\/latex]by multiplying<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135161212\">\n \t<dt>Binomial Theorem<\/dt>\n \t<dd id=\"fs-id1165135161217\">a formula that can be used to expand any binomial<\/dd>\n<\/dl>\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>In this section, you will:<\/p>\n<ul>\n<li>Apply the Binomial Theorem.<\/li>\n<\/ul>\n<\/div>\n<p>A polynomial with two terms is called a binomial. We have already learned to multiply binomials and to raise binomials to powers, but raising a binomial to a high power can be tedious and time-consuming. In this section, we will discuss a shortcut that will allow us to find[latex]\\,{\\left(x+y\\right)}^{n}\\,[\/latex]without multiplying the binomial by itself [latex]n[\/latex] times.<\/p>\n<div id=\"fs-id1165135195368\" class=\"bc-section section\">\n<h3>Identifying Binomial Coefficients<\/h3>\n<p id=\"fs-id1165135181540\">In <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/chapter\/counting-principles\/\">Counting Principles<\/a>, we studied <span class=\"no-emphasis\">combinations<\/span>. In the shortcut to finding[latex]\\,{\\left(x+y\\right)}^{n},\\,[\/latex]we will need to use combinations to find the coefficients that will appear in the expansion of the binomial. In this case, we use the notation[latex]\\,\\left(\\begin{array}{c}n\\\\ r\\end{array}\\right)\\,[\/latex] instead of [latex]C\\left(n,r\\right),[\/latex] but it can be calculated in the same way. So<\/p>\n<div id=\"eip-id1165137656645\" class=\"unnumbered\">[latex]\\,\\left(\\begin{array}{c}n\\\\ r\\end{array}\\right)=C\\left(n,r\\right)=\\frac{n!}{r!\\left(n-r\\right)!}\\,[\/latex]<\/div>\n<p id=\"fs-id1165135692906\">The combination[latex]\\,\\left(\\begin{array}{c}n\\\\ r\\end{array}\\right)\\,[\/latex]is called a <strong>binomial coefficient<\/strong>. An example of a binomial coefficient is[latex]\\,\\left(\\begin{array}{c}5\\\\ 2\\end{array}\\right)=C\\left(5,2\\right)=10.\\,[\/latex]<\/p>\n<div id=\"fs-id1165137482950\" class=\"textbox key-takeaways\">\n<h3>Binomial Coefficients<\/h3>\n<p id=\"fs-id1165137472666\">If [latex]n[\/latex] and [latex]r[\/latex]are integers greater than or equal to 0 with [latex]n\\ge r,[\/latex] then the binomial coefficient is<\/p>\n<div id=\"fs-id1165135188055\" class=\"unnumbered aligncenter\">[latex]\\left(\\begin{array}{c}n\\\\ r\\end{array}\\right)=C\\left(n,r\\right)=\\frac{n!}{r!\\left(n-r\\right)!}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165135174912\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1165137386960\"><strong>Is a binomial coefficient always a whole number?<\/strong><\/p>\n<p id=\"fs-id1165137436611\"><em>Yes. Just as the number of combinations must always be a whole number, a binomial coefficient will always be a whole number. <\/em><\/p>\n<\/div>\n<div id=\"Example_11_06_01\" class=\"textbox examples\">\n<div id=\"fs-id1165137837055\">\n<div id=\"fs-id1165137714295\">\n<h3>Finding Binomial Coefficients<\/h3>\n<p id=\"fs-id1165135253860\">Find each binomial coefficient.<\/p>\n<ol id=\"fs-id1165137553883\" type=\"a\">\n<li>[latex]\\left(\\begin{array}{c}5\\\\ 3\\end{array}\\right)[\/latex]<\/li>\n<li>[latex]\\left(\\begin{array}{c}9\\\\ 2\\end{array}\\right)[\/latex]<\/li>\n<li>[latex]\\left(\\begin{array}{c}9\\\\ 7\\end{array}\\right)[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165137933188\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>Use the formula to calculate each binomial coefficient. You can also use the [latex]{n}_{}{C}_{r}[\/latex] function on your calculator.<\/p>\n<div id=\"eip-id1165137755879\" class=\"unnumbered\">[latex]\\left(\\begin{array}{c}n\\\\ r\\end{array}\\right)=C\\left(n,r\\right)=\\frac{n!}{r!\\left(n-r\\right)!}[\/latex]<\/div>\n<ol type=\"a\">\n<li>[latex]\\left(\\begin{array}{c}5\\\\ 3\\end{array}\\right)=\\frac{5!}{3!\\left(5-3\\right)!}=\\frac{5\\cdot 4\\cdot 3!}{3!2!}=10[\/latex]<\/li>\n<li>[latex]\\left(\\begin{array}{c}9\\\\ 2\\end{array}\\right)=\\frac{9!}{2!\\left(9-2\\right)!}=\\frac{9\\cdot 8\\cdot 7!}{2!7!}=36[\/latex]<\/li>\n<li>[latex]\\left(\\begin{array}{c}9\\\\ 7\\end{array}\\right)=\\frac{9!}{7!\\left(9-7\\right)!}=\\frac{9\\cdot 8\\cdot 7!}{7!2!}=36[\/latex]<\/li>\n<\/ol>\n<\/details>\n<\/div>\n<div id=\"fs-id1165135161493\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137749058\">Notice that we obtained the same result for parts (b) and (c). If you look closely at the solution for these two parts, you will see that you end up with the same two factorials in the denominator, but the order is reversed, just as with combinations.<\/p>\n<div id=\"eip-id1165137416511\" class=\"unnumbered\">[latex]\\left(\\begin{array}{c}n\\\\ r\\end{array}\\right)=\\left(\\begin{array}{c}n\\\\ n-r\\end{array}\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135309824\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165137812282\">Find each binomial coefficient.<\/p>\n<div id=\"ti_11_06_01\">\n<div id=\"fs-id1165137640379\">\n<ol id=\"eip-id1165135332801\" type=\"a\">\n<li>[latex]\\,\\left(\\begin{array}{c}7\\\\ 3\\end{array}\\right)\\,[\/latex]<\/li>\n<li>[latex]\\,\\left(\\begin{array}{c}11\\\\ 4\\end{array}\\right)\\,[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165137653724\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<ol id=\"eip-id1165135331689\" type=\"a\">\n<li>35<\/li>\n<li>330<\/li>\n<\/ol>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137722450\" class=\"bc-section section\">\n<h3>Using the Binomial Theorem<\/h3>\n<p id=\"fs-id1165137445287\">When we expand [latex]{\\left(x+y\\right)}^{n}[\/latex] by multiplying, the result is called a binomial expansion, and it includes binomial coefficients. If we wanted to expand [latex]{\\left(x+y\\right)}^{52},[\/latex] we might multiply [latex]\\left(x+y\\right)[\/latex] by itself fifty-two times. This could take hours! If we examine some simple binomial expansions, we can find patterns that will lead us to a shortcut for finding more complicated binomial expansions.<\/p>\n<div id=\"eip-id1165137387258\" class=\"unnumbered\">[latex]\\begin{array}{l}{\\left(x+y\\right)}^{2}={x}^{2}+2xy+{y}^{2}\\hfill \\\\ {\\left(x+y\\right)}^{3}={x}^{3}+3{x}^{2}y+3x{y}^{2}+{y}^{3}\\hfill \\\\ {\\left(x+y\\right)}^{4}={x}^{4}+4{x}^{3}y+6{x}^{2}{y}^{2}+4x{y}^{3}+{y}^{4}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137602015\">First, let\u2019s examine the exponents. With each successive term, the exponent for [latex]x[\/latex] decreases and the exponent for [latex]y[\/latex] increases. The sum of the two exponents is [latex]n[\/latex] for each term.<\/p>\n<p id=\"fs-id1165137738201\">Next, let\u2019s examine the coefficients. Notice that the coefficients increase and then decrease in a symmetrical pattern. The coefficients follow a pattern:<\/p>\n<div id=\"eip-id1165135173876\" class=\"unnumbered\">[latex]\\left(\\begin{array}{c}n\\\\ 0\\end{array}\\right),\\left(\\begin{array}{c}n\\\\ 1\\end{array}\\right),\\left(\\begin{array}{c}n\\\\ 2\\end{array}\\right),...,\\left(\\begin{array}{c}n\\\\ n\\end{array}\\right).[\/latex]<\/div>\n<p id=\"fs-id1165137666545\">These patterns lead us to the <strong>Binomial Theorem<\/strong>, which can be used to expand any binomial.<\/p>\n<div id=\"eip-id1165135194720\" class=\"unnumbered\">[latex]\\begin{array}{ll}{\\left(x+y\\right)}^{n}\\hfill & =\\sum _{k=0}^{n}\\left(\\begin{array}{c}n\\\\ k\\end{array}\\right){x}^{n-k}{y}^{k}\\hfill \\\\ \\hfill & ={x}^{n}+\\left(\\begin{array}{c}n\\\\ 1\\end{array}\\right){x}^{n-1}y+\\left(\\begin{array}{c}n\\\\ 2\\end{array}\\right){x}^{n-2}{y}^{2}+...+\\left(\\begin{array}{c}n\\\\ n-1\\end{array}\\right)x{y}^{n-1}+{y}^{n}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137602044\">Another way to see the coefficients is to examine the expansion of a binomial in general form,[latex]\\,x+y,\\,[\/latex]to successive powers 1, 2, 3, and 4.<\/p>\n<div id=\"eip-id1165137935737\" class=\"unnumbered\">[latex]\\begin{array}{l}{\\left(x+y\\right)}^{1}=x+y\\hfill \\\\ {\\left(x+y\\right)}^{2}={x}^{2}+2xy+{y}^{2}\\hfill \\\\ {\\left(x+y\\right)}^{3}={x}^{3}+3{x}^{2}y+3x{y}^{2}+{y}^{3}\\hfill \\\\ {\\left(x+y\\right)}^{4}={x}^{4}+4{x}^{3}y+6{x}^{2}{y}^{2}+4x{y}^{3}+{y}^{4}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137430560\">Can you guess the next expansion for the binomial[latex]\\,{\\left(x+y\\right)}^{5}?\\,[\/latex]<\/p>\n<div id=\"CNX_Precalc_Figure_11_06_002\" class=\"wp-caption aligncenter\">\n<figure style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19155056\/CNX_Precalc_Figure_11_06_002.jpg\" alt=\"Graph of the function f_2.\" width=\"731\" height=\"413\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 1.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165137443527\">See <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_11_06_002\">(Figure)<\/a>, which illustrates the following:<\/p>\n<ul id=\"fs-id1165137748570\">\n<li>There are [latex]n+1[\/latex] terms in the expansion of [latex]{\\left(x+y\\right)}^{n}.[\/latex]<\/li>\n<li>The degree (or sum of the exponents) for each term is [latex]n.[\/latex]<\/li>\n<li>The powers on [latex]x[\/latex] begin with [latex]n[\/latex] and decrease to 0.<\/li>\n<li>The powers on [latex]y[\/latex] begin with 0 and increase to [latex]n.[\/latex]<\/li>\n<li>The coefficients are symmetric.<\/li>\n<\/ul>\n<p id=\"fs-id1165137605699\">To determine the expansion on [latex]{\\left(x+y\\right)}^{5},[\/latex] we see [latex]n=5,[\/latex] thus, there will be 5+1 = 6 terms. Each term has a combined degree of 5. In descending order for powers of [latex]x,[\/latex] the pattern is as follows:<\/p>\n<ul id=\"fs-id1165137804247\">\n<li>Introduce [latex]{x}^{5},[\/latex] and then for each successive term reduce the exponent on [latex]x[\/latex] by 1 until [latex]{x}^{0}=1[\/latex] is reached.<\/li>\n<li>Introduce [latex]{y}^{0}=1,[\/latex] and then increase the exponent on [latex]y[\/latex] by 1 until [latex]{y}^{5}[\/latex] is reached.\n<div id=\"eip-id1165137643155\" class=\"unnumbered\">[latex]{x}^{5},\\,\\,{x}^{4}y,\\,\\,{x}^{3}{y}^{2},\\,\\,{x}^{2}{y}^{3},\\,\\,x{y}^{4},\\,\\,{y}^{5}[\/latex]<\/div>\n<\/li>\n<\/ul>\n<p id=\"fs-id1165137445918\">The next expansion would be<\/p>\n<div id=\"eip-id1165137460981\" class=\"unnumbered\">[latex]{\\left(x+y\\right)}^{5}={x}^{5}+5{x}^{4}y+10{x}^{3}{y}^{2}+10{x}^{2}{y}^{3}+5x{y}^{4}+{y}^{5}.[\/latex]<\/div>\n<p>But where do those coefficients come from? The binomial coefficients are symmetric. We can see these coefficients in an array known as <span class=\"no-emphasis\">Pascal&#8217;s Triangle<\/span>, shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_11_06_001\">(Figure)<\/a>.<\/p>\n<div id=\"CNX_Precalc_Figure_11_06_001\" class=\"medium aligncenter\">\n<figure style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19155101\/CNX_Precalc_Figure_11_06_001.jpg\" alt=\"Pascal's Triangle\" width=\"731\" height=\"300\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 2.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165137446339\">To generate Pascal\u2019s Triangle, we start by writing a 1. In the row below, row 2, we write two 1\u2019s. In the 3<sup>rd<\/sup> row, flank the ends of the rows with 1\u2019s, and add [latex]1+1[\/latex] to find the middle number, 2. In the [latex]n\\text{th}[\/latex] row, flank the ends of the row with 1\u2019s. Each element in the triangle is the sum of the two elements immediately above it.<\/p>\n<p id=\"fs-id1165137864198\">To see the connection between Pascal\u2019s Triangle and binomial coefficients, let us revisit the expansion of the binomials in general form.<\/p>\n<p><span id=\"fs-id1165137643968\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19155115\/CNX_Precalc_Figure_11_06_003.jpg\" alt=\"Pascal's Triangle expanded to show the values of the triangle as x and y terms with exponents\" \/><\/span><\/p>\n<div id=\"fs-id1165135187302\" class=\"textbox key-takeaways\">\n<h3>The Binomial Theorem<\/h3>\n<p id=\"fs-id1165137726456\">The Binomial Theorem is a formula that can be used to expand any binomial.<\/p>\n<div id=\"fs-id1165137828132\">[latex]\\begin{array}{ll}{\\left(x+y\\right)}^{n}\\hfill & =\\sum _{k=0}^{n}\\left(\\begin{array}{c}n\\\\ k\\end{array}\\right){x}^{n-k}{y}^{k}\\hfill \\\\ \\hfill & ={x}^{n}+\\left(\\begin{array}{c}n\\\\ 1\\end{array}\\right){x}^{n-1}y+\\left(\\begin{array}{c}n\\\\ 2\\end{array}\\right){x}^{n-2}{y}^{2}+...+\\left(\\begin{array}{c}n\\\\ n-1\\end{array}\\right)x{y}^{n-1}+{y}^{n}\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165135646154\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137634381\"><strong>Given a binomial, write it in expanded form.<\/strong><\/p>\n<ol id=\"fs-id1165135377104\" type=\"1\">\n<li>Determine the value of [latex]n[\/latex]according to the exponent.<\/li>\n<li>Evaluate the [latex]k=0[\/latex] through [latex]k=n[\/latex] using the Binomial Theorem formula.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_11_06_02\" class=\"textbox examples\">\n<div id=\"fs-id1165137843915\">\n<div id=\"fs-id1165137843917\">\n<h3>Expanding a Binomial<\/h3>\n<p id=\"fs-id1165137571473\">Write in expanded form.<\/p>\n<ol id=\"fs-id1165137571476\" type=\"a\">\n<li>[latex]\\,{\\left(x+y\\right)}^{5}\\,[\/latex]<\/li>\n<li>[latex]\\,{\\left(3x-y\\right)}^{4}\\,[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165137771774\" class=\"solution textbox shaded\">\n<div id=\"eip-id1165133341948\" class=\"unnumbered\">\n<details>\n<summary>Show Solution<\/summary>\n<ol id=\"fs-id1165137804171\" type=\"a\">\n<li>Substitute [latex]n=5[\/latex] into the formula. Evaluate the [latex]k=0[\/latex] through [latex]k=5[\/latex] terms. Simplify.\n<div id=\"eip-id1165135600195\" class=\"unnumbered\">[latex]\\begin{array}{ll}{\\left(x+y\\right)}^{5}\\hfill & =\\left(\\begin{array}{c}5\\\\ 0\\end{array}\\right){x}^{5}{y}^{0}+\\left(\\begin{array}{c}5\\\\ 1\\end{array}\\right){x}^{4}{y}^{1}+\\left(\\begin{array}{c}5\\\\ 2\\end{array}\\right){x}^{3}{y}^{2}+\\left(\\begin{array}{c}5\\\\ 3\\end{array}\\right){x}^{2}{y}^{3}+\\left(\\begin{array}{c}5\\\\ 4\\end{array}\\right){x}^{1}{y}^{4}+\\left(\\begin{array}{c}5\\\\ 5\\end{array}\\right){x}^{0}{y}^{5}\\hfill \\\\ {\\left(x+y\\right)}^{5}\\hfill & ={x}^{5}+5{x}^{4}y+10{x}^{3}{y}^{2}+10{x}^{2}{y}^{3}+5x{y}^{4}+{y}^{5}\\hfill \\end{array}[\/latex]<\/div>\n<\/li>\n<li>Substitute [latex]n=4[\/latex] into the formula. Evaluate the [latex]k=0[\/latex] through [latex]k=4[\/latex] terms. Notice that [latex]3x[\/latex] is in the place that was occupied by [latex]x[\/latex] and that [latex]\u2013y[\/latex] is in the place that was occupied by [latex]y.[\/latex] So we substitute them. Simplify.\n<div id=\"eip-id1165133341948\" class=\"unnumbered\">[latex]\\begin{array}{ll}{\\left(3x-y\\right)}^{4}\\hfill & =\\left(\\begin{array}{c}4\\\\ 0\\end{array}\\right){\\left(3x\\right)}^{4}{\\left(-y\\right)}^{0}+\\left(\\begin{array}{c}4\\\\ 1\\end{array}\\right){\\left(3x\\right)}^{3}{\\left(-y\\right)}^{1}+\\left(\\begin{array}{c}4\\\\ 2\\end{array}\\right){\\left(3x\\right)}^{2}{\\left(-y\\right)}^{2}+\\left(\\begin{array}{c}4\\\\ 3\\end{array}\\right){\\left(3x\\right)}^{1}{\\left(-y\\right)}^{3}+\\left(\\begin{array}{c}4\\\\ 4\\end{array}\\right){\\left(3x\\right)}^{0}{\\left(-y\\right)}^{4}\\hfill \\\\ {\\left(3x-y\\right)}^{4}\\hfill & =81{x}^{4}-108{x}^{3}y+54{x}^{2}{y}^{2}-12x{y}^{3}+{y}^{4}\\hfill \\end{array}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135191233\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165135296274\">Notice the alternating signs in part b. This happens because[latex]\\,\\left(-y\\right)\\,[\/latex]raised to odd powers is negative, but[latex]\\,\\left(-y\\right)\\,[\/latex]raised to even powers is positive. This will occur whenever the binomial contains a subtraction sign.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135501144\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_06_02\">\n<div id=\"fs-id1165137772198\">\n<p id=\"fs-id1165137772200\">Write in expanded form.<\/p>\n<ol id=\"eip-id1700717\" type=\"a\">\n<li>[latex]{\\left(x-y\\right)}^{5}[\/latex]<\/li>\n<li>[latex]{\\left(2x+5y\\right)}^{3}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165137635439\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<ol id=\"eip-id1366342\" type=\"a\">\n<li>[latex]{x}^{5}-5{x}^{4}y+10{x}^{3}{y}^{2}-10{x}^{2}{y}^{3}+5x{y}^{4}-{y}^{5}[\/latex]<\/li>\n<li>[latex]8{x}^{3}+60{x}^{2}y+150x{y}^{2}+125{y}^{3}[\/latex]<\/li>\n<\/ol>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137658053\" class=\"bc-section section\">\n<h3>Using the Binomial Theorem to Find a Single Term<\/h3>\n<p id=\"fs-id1165137736641\">Expanding a binomial with a high exponent such as[latex]\\,{\\left(x+2y\\right)}^{16}\\,[\/latex]can be a lengthy process.<\/p>\n<p id=\"eip-164\">Sometimes we are interested only in a certain term of a binomial expansion. We do not need to fully expand a binomial to find a single specific term.<\/p>\n<p id=\"fs-id1165137834237\">Note the pattern of coefficients in the expansion of[latex]\\,{\\left(x+y\\right)}^{5}.[\/latex]<\/p>\n<div id=\"eip-405\" class=\"unnumbered aligncenter\">[latex]{\\left(x+y\\right)}^{5}={x}^{5}+\\left(\\begin{array}{c}5\\\\ 1\\end{array}\\right){x}^{4}y+\\left(\\begin{array}{c}5\\\\ 2\\end{array}\\right){x}^{3}{y}^{2}+\\left(\\begin{array}{c}5\\\\ 3\\end{array}\\right){x}^{2}{y}^{3}+\\left(\\begin{array}{c}5\\\\ 4\\end{array}\\right)x{y}^{4}+{y}^{5}[\/latex]<\/div>\n<p id=\"fs-id1165137855349\">The second term is[latex]\\,\\left(\\begin{array}{c}5\\\\ 1\\end{array}\\right){x}^{4}y.\\,[\/latex]The third term is[latex]\\,\\left(\\begin{array}{c}5\\\\ 2\\end{array}\\right){x}^{3}{y}^{2}.\\,[\/latex]We can generalize this result.<\/p>\n<div id=\"eip-415\" class=\"unnumbered aligncenter\">[latex]\\left(\\begin{array}{c}n\\\\ r\\end{array}\\right){x}^{n-r}{y}^{r}[\/latex]<\/div>\n<div id=\"fs-id1165137466315\" class=\"textbox key-takeaways\">\n<h3>The (r+1)th Term of a Binomial Expansion<\/h3>\n<p id=\"fs-id1165137628103\">The[latex]\\,\\left(r+1\\right)\\text{th}\\,[\/latex]term of the <span class=\"no-emphasis\">binomial expansion<\/span> of[latex]\\,{\\left(x+y\\right)}^{n}\\,[\/latex]is:<\/p>\n<div id=\"fs-id1165135203716\">[latex]\\left(\\begin{array}{c}n\\\\ r\\end{array}\\right){x}^{n-r}{y}^{r}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137400324\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p><strong>Given a binomial, write a specific term without fully expanding.<\/strong><\/p>\n<ol id=\"fs-id1165137414548\" type=\"1\">\n<li>Determine the value of [latex]n[\/latex] according to the exponent.<\/li>\n<li>Determine [latex]\\left(r+1\\right).[\/latex]<\/li>\n<li>Determine [latex]r.[\/latex]<\/li>\n<li>Replace [latex]r[\/latex] in the formula for the [latex]\\left(r+1\\right)\\text{th}[\/latex] term of the binomial expansion.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_11_06_03\" class=\"textbox examples\">\n<div id=\"fs-id1165137605967\">\n<div id=\"fs-id1165137605969\">\n<h3>Writing a Given Term of a Binomial Expansion<\/h3>\n<p>Find the tenth term of[latex]\\,{\\left(x+2y\\right)}^{16}\\,[\/latex]without fully expanding the binomial.<\/p>\n<\/div>\n<div id=\"fs-id1165137834759\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137737706\">Because we are looking for the tenth term, [latex]\\,r+1=10,\\,[\/latex] we will use [latex]\\,r=9[\/latex] in our calculations.<\/p>\n<div id=\"eip-id1165137464257\" class=\"unnumbered\">[latex]\\left(\\begin{array}{c}n\\\\ r\\end{array}\\right){x}^{n-r}{y}^{r}[\/latex]<\/div>\n<div id=\"eip-id1165137894367\" class=\"unnumbered\">[latex]\\left(\\begin{array}{c}16\\\\ 9\\end{array}\\right){x}^{16-9}{\\left(2y\\right)}^{9}=5\\text{,}857\\text{,}280{x}^{7}{y}^{9}[\/latex]<\/div>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137732277\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_11_06_03\">\n<div id=\"fs-id1165137657120\">\n<p id=\"fs-id1165137657121\">Find the sixth term of[latex]\\,{\\left(3x-y\\right)}^{9}\\,[\/latex]without fully expanding the binomial.<\/p>\n<\/div>\n<div id=\"fs-id1165137758550\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>[latex]\\,-10,206{x}^{4}{y}^{5}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137862398\" class=\"precalculus media\">\n<p id=\"fs-id1165137611861\">Access these online resources for additional instruction and practice with binomial expansion.<\/p>\n<ul>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/binomialtheorem\">The Binomial Theorem<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/btexample\">Binomial Theorem Example<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135705092\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"eip-id1165134166609\" summary=\"..\">\n<tbody>\n<tr>\n<td>Binomial Theorem<\/td>\n<td>[latex]{\\left(x+y\\right)}^{n}=\\sum _{k-0}^{n}\\left(\\begin{array}{c}n\\\\ k\\end{array}\\right){x}^{n-k}{y}^{k}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left(r+1\\right)th\\,[\/latex]term of a binomial expansion<\/td>\n<td>[latex]\\left(\\begin{array}{c}n\\\\ r\\end{array}\\right){x}^{n-r}{y}^{r}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137469809\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165135187830\">\n<li>[latex]\\left(\\begin{array}{c}n\\\\ r\\end{array}\\right)\\,[\/latex]is called a binomial coefficient and is equal to [latex]C\\left(n,r\\right).\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_11_06_01\">(Figure)<\/a>.<\/li>\n<li>The Binomial Theorem allows us to expand binomials without multiplying. See <a class=\"autogenerated-content\" href=\"#Example_11_06_02\">(Figure)<\/a>.<\/li>\n<li>We can find a given term of a binomial expansion without fully expanding the binomial. See <a class=\"autogenerated-content\" href=\"#Example_11_06_03\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165134089502\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165134089506\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165137419807\">\n<div id=\"fs-id1165135181496\">\n<p id=\"fs-id1165135181499\">What is a binomial coefficient, and how it is calculated?<\/p>\n<\/div>\n<div id=\"fs-id1165135501149\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>A binomial coefficient is an alternative way of denoting the combination [latex]\\,C\\left(n,r\\right).\\,[\/latex]It is defined as[latex]\\,\\left(\\begin{array}{c}n\\\\ r\\end{array}\\right)=\\,C\\left(n,r\\right)\\,=\\frac{n!}{r!\\left(n-r\\right)!}.[\/latex]<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137442166\">\n<div id=\"fs-id1165137442168\">\n<p id=\"fs-id1165137442170\">What role do binomial coefficients play in a binomial expansion? Are they restricted to any type of number?<\/p>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135190462\">\n<p id=\"fs-id1165135190464\">What is the Binomial Theorem and what is its use?<\/p>\n<\/div>\n<div id=\"fs-id1165137452921\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137452923\">The Binomial Theorem is defined as[latex]\\,{\\left(x+y\\right)}^{n}=\\sum _{k=0}^{n}\\left(\\begin{array}{c}n\\\\ k\\end{array}\\right){x}^{n-k}{y}^{k}\\,[\/latex]and can be used to expand any binomial.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137445986\">\n<div id=\"fs-id1165137445988\">\n<p id=\"fs-id1165137445990\">When is it an advantage to use the Binomial Theorem? Explain.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137408889\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1165135169145\">For the following exercises, evaluate the binomial coefficient.<\/p>\n<div id=\"fs-id1165135526110\">\n<div id=\"fs-id1165135526112\">\n<p id=\"fs-id1165135526114\">[latex]\\left(\\begin{array}{c}6\\\\ 2\\end{array}\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137583395\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137583397\">15<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137455051\">\n<div id=\"fs-id1165137455053\">\n<p id=\"fs-id1165137655192\">[latex]\\left(\\begin{array}{c}5\\\\ 3\\end{array}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135484521\">\n<div id=\"fs-id1165137541584\">\n<p id=\"fs-id1165137541586\">[latex]\\left(\\begin{array}{c}7\\\\ 4\\end{array}\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135484154\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135484156\">35<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137772477\">\n<div>[latex]\\left(\\begin{array}{c}9\\\\ 7\\end{array}\\right)[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165135506400\">\n<div id=\"fs-id1165135506403\">\n<p id=\"fs-id1165135570073\">[latex]\\left(\\begin{array}{c}10\\\\ 9\\end{array}\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137480607\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137480609\">10<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137656508\">\n<div id=\"fs-id1165137656510\">\n<p id=\"fs-id1165137530730\">[latex]\\left(\\begin{array}{c}25\\\\ 11\\end{array}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135241276\">\n<div id=\"fs-id1165135424707\">\n<p id=\"fs-id1165135424709\">[latex]\\left(\\begin{array}{c}17\\\\ 6\\end{array}\\right)[\/latex]<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>12,376<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135193638\">\n<div id=\"fs-id1165135193640\">\n<p id=\"fs-id1165137409164\">[latex]\\left(\\begin{array}{c}200\\\\ 199\\end{array}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137807329\">For the following exercises, use the Binomial Theorem to expand each binomial.<\/p>\n<div id=\"fs-id1165137442493\">\n<div id=\"fs-id1165137442495\">\n<p id=\"fs-id1165137724327\">[latex]{\\left(4a-b\\right)}^{3}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137831242\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>[latex]64{a}^{3}-48{a}^{2}b+12a{b}^{2}-{b}^{3}[\/latex]<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137548190\">\n<div id=\"fs-id1165137548192\">\n<p id=\"fs-id1165137548194\">[latex]{\\left(5a+2\\right)}^{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137542019\">\n<div>[latex]{\\left(3a+2b\\right)}^{3}[\/latex]<\/div>\n<div id=\"fs-id1165135394319\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135394321\">[latex]27{a}^{3}+54{a}^{2}b+36a{b}^{2}+8{b}^{3}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137414737\">\n<div id=\"fs-id1165137414739\">\n<p id=\"fs-id1165137647996\">[latex]{\\left(2x+3y\\right)}^{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1165137896227\">[latex]{\\left(4x+2y\\right)}^{5}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135516855\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135516857\">[latex]1024{x}^{5}+2560{x}^{4}y+2560{x}^{3}{y}^{2}+1280{x}^{2}{y}^{3}+320x{y}^{4}+32{y}^{5}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137678200\">\n<div id=\"fs-id1165137678202\">\n<p id=\"fs-id1165137678204\">[latex]{\\left(3x-2y\\right)}^{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137667609\">\n<div id=\"fs-id1165137667611\">\n<p id=\"fs-id1165137667613\">[latex]{\\left(4x-3y\\right)}^{5}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137837121\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137642794\">[latex]1024{x}^{5}-3840{x}^{4}y+5760{x}^{3}{y}^{2}-4320{x}^{2}{y}^{3}+1620x{y}^{4}-243{y}^{5}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137418542\">\n<div id=\"fs-id1165137418544\">\n<p id=\"fs-id1165137418546\">[latex]{\\left(\\frac{1}{x}+3y\\right)}^{5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135526126\">\n<div id=\"fs-id1165135526128\">\n<p id=\"fs-id1165135526130\">[latex]{\\left({x}^{-1}+2{y}^{-1}\\right)}^{4}[\/latex]<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137805817\">[latex]\\frac{1}{{x}^{4}}+\\frac{8}{{x}^{3}y}+\\frac{24}{{x}^{2}{y}^{2}}+\\frac{32}{x{y}^{3}}+\\frac{16}{{y}^{4}}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137828074\">\n<p id=\"fs-id1165137828076\">[latex]{\\left(\\sqrt{x}-\\sqrt{y}\\right)}^{5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135149859\">For the following exercises, use the Binomial Theorem to write the first three terms of each binomial.<\/p>\n<div>\n<div>\n<p id=\"fs-id1165137565759\">[latex]{\\left(a+b\\right)}^{17}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137527392\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137535275\">[latex]{a}^{17}+17{a}^{16}b+136{a}^{15}{b}^{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135182971\">\n<div id=\"fs-id1165137871899\">\n<p id=\"fs-id1165137871901\">[latex]{\\left(x-1\\right)}^{18}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137736266\">\n<div id=\"fs-id1165137736268\">\n<p id=\"fs-id1165137409564\">[latex]{\\left(a-2b\\right)}^{15}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135188234\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>[latex]{a}^{15}-30{a}^{14}b+420{a}^{13}{b}^{2}[\/latex]<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137543302\">\n<div id=\"fs-id1165137543304\">\n<p id=\"fs-id1165137543306\">[latex]{\\left(x-2y\\right)}^{8}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137635106\">\n<div>\n<p id=\"fs-id1165137874644\">[latex]{\\left(3a+b\\right)}^{20}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137659105\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137659107\">[latex]3,486,784,401{a}^{20}+23,245,229,340{a}^{19}b+73,609,892,910{a}^{18}{b}^{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137646617\">\n<div id=\"fs-id1165137526699\">\n<p id=\"fs-id1165137526701\">[latex]{\\left(2a+4b\\right)}^{7}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137732098\">\n<div id=\"fs-id1165137573195\">\n<p id=\"fs-id1165137573197\">[latex]{\\left({x}^{3}-\\sqrt{y}\\right)}^{8}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137433769\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137414388\">[latex]{x}^{24}-8{x}^{21}\\sqrt{y}+28{x}^{18}y[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165134032296\">For the following exercises, find the indicated term of each binomial without fully expanding the binomial.<\/p>\n<div id=\"fs-id1165135344957\">\n<div id=\"fs-id1165135344959\">\n<p id=\"fs-id1165135344961\">The fourth term of[latex]\\,{\\left(2x-3y\\right)}^{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137696650\">\n<div id=\"fs-id1165137696653\">\n<p id=\"fs-id1165135193960\">The fourth term of[latex]\\,{\\left(3x-2y\\right)}^{5}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137827785\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137827787\">[latex]-720{x}^{2}{y}^{3}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137898985\">\n<div id=\"fs-id1165137898987\">\n<p id=\"fs-id1165137898989\">The third term of[latex]\\,{\\left(6x-3y\\right)}^{7}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135195054\">\n<div id=\"fs-id1165135195056\">\n<p id=\"fs-id1165137737799\">The eighth term of[latex]\\,{\\left(7+5y\\right)}^{14}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137549805\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137659226\">[latex]220,812,466,875,000{y}^{7}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137466522\">\n<div id=\"fs-id1165137466524\">\n<p id=\"fs-id1165137466526\">The seventh term of[latex]\\,{\\left(a+b\\right)}^{11}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135173932\">\n<div id=\"fs-id1165135173934\">\n<p id=\"fs-id1165135173936\">The fifth term of[latex]\\,{\\left(x-y\\right)}^{7}[\/latex]<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137673496\">[latex]35{x}^{3}{y}^{4}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137768225\">\n<p id=\"fs-id1165137768227\">The tenth term of[latex]\\,{\\left(x-1\\right)}^{12}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135541625\">\n<div id=\"fs-id1165135541627\">\n<p id=\"fs-id1165135541629\">The ninth term of[latex]\\,{\\left(a-3{b}^{2}\\right)}^{11}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137423625\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>[latex]1,082,565{a}^{3}{b}^{16}[\/latex]<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137723425\">\n<div id=\"fs-id1165137536035\">\n<p id=\"fs-id1165137536037\">The fourth term of[latex]\\,{\\left({x}^{3}-\\frac{1}{2}\\right)}^{10}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135152237\">\n<div id=\"fs-id1165135152240\">\n<p id=\"fs-id1165137445389\">The eighth term of[latex]\\,{\\left(\\frac{y}{2}+\\frac{2}{x}\\right)}^{9}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137692780\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137692782\">[latex]\\frac{1152{y}^{2}}{{x}^{7}}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137653220\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1165137635092\">For the following exercises, use the Binomial Theorem to expand the binomial [latex]f\\left(x\\right)={\\left(x+3\\right)}^{4}.[\/latex] Then find and graph each indicated sum on one set of axes.<\/p>\n<div id=\"fs-id1165137407882\">\n<div id=\"fs-id1165137407884\">\n<p id=\"fs-id1165137693683\">Find and graph[latex]\\,{f}_{1}\\left(x\\right),\\,[\/latex]such that[latex]\\,{f}_{1}\\left(x\\right)\\,[\/latex]is the first term of the expansion.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137925296\">\n<div id=\"fs-id1165137925298\">\n<p id=\"fs-id1165137925300\">Find and graph[latex]\\,{f}_{2}\\left(x\\right),\\,[\/latex]such that[latex]\\,{f}_{2}\\left(x\\right)\\,[\/latex]is the sum of the first two terms of the expansion.<\/p>\n<\/div>\n<div id=\"fs-id1165137575802\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137575804\">[latex]{f}_{2}\\left(x\\right)={x}^{4}+12{x}^{3}[\/latex]<\/p>\n<p><span id=\"fs-id1165137726516\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19155129\/CNX_Precalc_Figure_11_06_202.jpg\" alt=\"Graph of the function f_2.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137812692\">\n<div id=\"fs-id1165137812694\">\n<p id=\"fs-id1165137812696\">Find and graph[latex]\\,{f}_{3}\\left(x\\right),\\,[\/latex]such that[latex]\\,{f}_{3}\\left(x\\right)\\,[\/latex] is the sum of the first three terms of the expansion.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137723651\">\n<div id=\"fs-id1165137723653\">\n<p id=\"fs-id1165137723655\">Find and graph[latex]\\,{f}_{4}\\left(x\\right),\\,[\/latex]such that[latex]\\,{f}_{4}\\left(x\\right)\\,[\/latex]is the sum of the first four terms of the expansion.<\/p>\n<\/div>\n<div id=\"fs-id1165135408467\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135408469\">[latex]{f}_{4}\\left(x\\right)={x}^{4}+12{x}^{3}+54{x}^{2}+108x[\/latex]<\/p>\n<p><span id=\"fs-id1165137482837\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19155139\/CNX_Precalc_Figure_11_06_204.jpg\" alt=\"Graph of the function f_4.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137557834\">\n<div id=\"fs-id1165137557836\">\n<p id=\"fs-id1165137557838\">Find and graph[latex]\\,{f}_{5}\\left(x\\right),\\,[\/latex]such that[latex]\\,{f}_{5}\\left(x\\right)\\,[\/latex]is the sum of the first five terms of the expansion.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137836668\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<div id=\"fs-id1165137836673\">\n<div id=\"fs-id1165137836675\">\n<p id=\"fs-id1165137932593\">In the expansion of[latex]\\,{\\left(5x+3y\\right)}^{n},\\,[\/latex]each term has the form[latex]\\,\\left(\\begin{array}{c}n\\\\ k\\end{array}\\right){a}^{n\u2013k}{b}^{k}, \\text{where} k\\,[\/latex]successively takes on the value[latex]\\,0,1,2,\\,...,\\,n.[\/latex]If[latex]\\,\\left(\\begin{array}{c}n\\\\ k\\end{array}\\right)=\\left(\\begin{array}{c}7\\\\ 2\\end{array}\\right),\\,[\/latex]what is the corresponding term?<\/p>\n<\/div>\n<div id=\"fs-id1165137871611\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137476761\">[latex]590,625{x}^{5}{y}^{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137441704\">\n<div id=\"fs-id1165137441706\">\n<p>In the expansion of[latex]\\,{\\left(a+b\\right)}^{n},\\,[\/latex]the coefficient of[latex]\\,{a}^{n-k}{b}^{k}\\,[\/latex]is the same as the coefficient of which other term?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137651016\">\n<div id=\"fs-id1165137651019\">\n<p id=\"fs-id1165135209329\">Consider the expansion of[latex]\\,{\\left(x+b\\right)}^{40}.\\,[\/latex]What is the exponent of [latex]b[\/latex] in the [latex]k\\text{th}[\/latex] term?<\/p>\n<\/div>\n<div id=\"fs-id1165137731710\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137731712\">[latex]k-1[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137755502\">\n<div id=\"fs-id1165137755504\">\n<p>Find[latex]\\,\\left(\\begin{array}{c}n\\\\ k-1\\end{array}\\right)+\\left(\\begin{array}{c}n\\\\ k\\end{array}\\right)\\,[\/latex]and write the answer as a binomial coefficient in the form[latex]\\,\\left(\\begin{array}{c}n\\\\ k\\end{array}\\right).\\,[\/latex]Prove it. <em>Hint:<\/em> Use the fact that, for any integer[latex]\\,p,\\,[\/latex]such that[latex]\\,p\\ge 1,\\,p!=p\\left(p-1\\right)!\\text{.}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137692126\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137692128\">[latex]\\left(\\begin{array}{c}n\\\\ k-1\\end{array}\\right)+\\left(\\begin{array}{l}n\\\\ k\\end{array}\\right)=\\left(\\begin{array}{c}n+1\\\\ k\\end{array}\\right);\\,[\/latex]Proof:<\/p>\n<p id=\"fs-id1165137759673\">[latex]\\begin{array}{}\\\\ \\\\ \\\\ \\,\\,\\,\\,\\,\\left(\\begin{array}{c}n\\\\ k-1\\end{array}\\right)+\\left(\\begin{array}{l}n\\\\ k\\end{array}\\right)\\\\ =\\frac{n!}{k!\\left(n-k\\right)!}+\\frac{n!}{\\left(k-1\\right)!\\left(n-\\left(k-1\\right)\\right)!}\\\\ =\\frac{n!}{k!\\left(n-k\\right)!}+\\frac{n!}{\\left(k-1\\right)!\\left(n-k+1\\right)!}\\\\ =\\frac{\\left(n-k+1\\right)n!}{\\left(n-k+1\\right)k!\\left(n-k\\right)!}+\\frac{kn!}{k\\left(k-1\\right)!\\left(n-k+1\\right)!}\\\\ =\\frac{\\left(n-k+1\\right)n!+kn!}{k!\\left(n-k+1\\right)!}\\\\ =\\frac{\\left(n+1\\right)n!}{k!\\left(\\left(n+1\\right)-k\\right)!}\\\\ =\\frac{\\left(n+1\\right)!}{k!\\left(\\left(n+1\\right)-k\\right)!}\\\\ =\\left(\\begin{array}{c}n+1\\\\ k\\end{array}\\right)\\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137757975\">\n<div id=\"fs-id1165137757977\">\n<p>Which expression cannot be expanded using the Binomial Theorem? Explain.<\/p>\n<ul id=\"eip-id1165137732312\">\n<li>[latex]\\left({x}^{2}-2x+1\\right)[\/latex]<\/li>\n<li>[latex]{\\left(\\sqrt{a}+4\\sqrt{a}-5\\right)}^{8}[\/latex]<\/li>\n<li>[latex]{\\left({x}^{3}+2{y}^{2}-z\\right)}^{5}[\/latex]<\/li>\n<li>[latex]{\\left(3{x}^{2}-\\sqrt{2{y}^{3}}\\right)}^{12}[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165135176542\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135176544\">The expression[latex]\\,{\\left({x}^{3}+2{y}^{2}-z\\right)}^{5}\\,[\/latex]cannot be expanded using the Binomial Theorem because it cannot be rewritten as a binomial.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165137673583\">\n<dt>binomial coefficient<\/dt>\n<dd id=\"fs-id1165137673588\">the number of ways to choose<em> r<\/em> objects from <em>n<\/em> objects where order does not matter; equivalent to[latex]\\,C\\left(n,r\\right),\\,[\/latex]denoted[latex]\\,\\left(\\begin{array}{c}n\\\\ r\\end{array}\\right)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137812031\">\n<dt>binomial expansion<\/dt>\n<dd id=\"fs-id1165137812163\">the result of expanding[latex]\\,{\\left(x+y\\right)}^{n}\\,[\/latex]by multiplying<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135161212\">\n<dt>Binomial Theorem<\/dt>\n<dd id=\"fs-id1165135161217\">a formula that can be used to expand any binomial<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":291,"menu_order":7,"template":"","meta":{"pb_show_title":null,"pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-211","chapter","type-chapter","status-publish","hentry"],"part":198,"_links":{"self":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/211","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\/211\/revisions"}],"predecessor-version":[{"id":212,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/211\/revisions\/212"}],"part":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/parts\/198"}],"metadata":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/211\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/media?parent=211"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapter-type?post=211"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/contributor?post=211"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/license?post=211"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}