{"id":85,"date":"2019-08-20T17:02:14","date_gmt":"2019-08-20T21:02:14","guid":{"rendered":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/dividing-polynomials\/"},"modified":"2022-06-01T10:39:26","modified_gmt":"2022-06-01T14:39:26","slug":"dividing-polynomials","status":"publish","type":"chapter","link":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/dividing-polynomials\/","title":{"raw":"Dividing Polynomials","rendered":"Dividing Polynomials"},"content":{"raw":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\nIn this section, you will:\n<ul>\n \t<li>Use long division to divide polynomials.<\/li>\n \t<li>Use synthetic division to divide polynomials.<\/li>\n<\/ul>\n<\/div>\n<div id=\"Figure_03_05_001\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"488\" class=\"small\"]<img src=\"https:\/\/cnx.org\/resources\/975951a618866688496b5d19f096409c6269b2d1\/CNX_Precalc_Figure_03_05_001.jpg\" alt=\"Lincoln Memorial.\" width=\"488\" height=\"286\"> <strong>Figure 1. <\/strong>Lincoln Memorial, Washington, D.C. (credit: Ron Cogswell, Flickr)[\/caption]\n\n<\/div>\n<p id=\"fs-id1165135382145\">The exterior of the Lincoln Memorial in Washington, D.C., is a large rectangular solid with length 61.5 meters (m), width 40 m, and height 30 m.[footnote]National Park Service. \"Lincoln Memorial Building Statistics.\" <a href=\"http:\/\/www.nps.gov\/linc\/historyculture\/lincoln-memorial-building-statistics.htm\">http:\/\/www.nps.gov\/linc\/historyculture\/lincoln-memorial-building-statistics.htm<\/a>. Accessed 4\/3\/2014[\/footnote] We can easily find the volume using elementary geometry.<\/p>\n\n<div id=\"eip-435\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill V&amp; =&amp; l\\cdot w\\cdot h\\hfill \\\\ &amp; =&amp; 61.5\\cdot 40\\cdot 30\\hfill \\\\ &amp; =&amp; 73,800\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165133214948\">So the volume is 73,800 cubic meters[latex]\\,\\left(\\text{m}\u00b3\\right).\\,[\/latex]\nSuppose we knew the volume, length, and width. We could divide to find the height.<\/p>\n\n<div id=\"eip-312\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill h&amp; =&amp; \\frac{V}{l\\cdot w}\\hfill \\\\ &amp; =&amp; \\frac{73,800}{61.5\\cdot 40}\\hfill \\\\ &amp; =&amp; 30\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137892463\">As we can confirm from the dimensions above, the height is 30 m. We can use similar methods to find any of the missing dimensions. We can also use the same method if any, or all, of the measurements contain variable expressions. For example, suppose the volume of a rectangular solid is given by the polynomial[latex]\\,3{x}^{4}-3{x}^{3}-33{x}^{2}+54x.\\,[\/latex]\nThe length of the solid is given by[latex]\\,3x;\\,[\/latex]\nthe width is given by[latex]\\,x-2.\\,[\/latex]\nTo find the height of the solid, we can use polynomial division, which is the focus of this section.<\/p>\n\n<div id=\"fs-id1165137676949\" class=\"bc-section section\">\n<h3>Using Long Division to Divide Polynomials<\/h3>\n<p id=\"fs-id1165135191647\">We are familiar with the <span class=\"no-emphasis\">long division<\/span> algorithm for ordinary arithmetic. We begin by dividing into the digits of the dividend that have the greatest place value. We divide, multiply, subtract, include the digit in the next place value position, and repeat. For example, let\u2019s divide 178 by 3 using long division.<\/p>\n<span id=\"fs-id1165137564295\"><img class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/a3075d8a73f65e2ad04180a05fdea0723dc8cef6\/CNX_Precalc_Figure_03_05_002.jpg\" alt=\"Steps of long division for intergers.\"><\/span>\n<p id=\"fs-id1165134170235\">Another way to look at the solution is as a sum of parts. This should look familiar, since it is the same method used to check division in elementary arithmetic.<\/p>\n\n<div id=\"eip-474\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\text{dividend}&amp; =&amp; (\\text{divisor}\\cdot \\text{quotient) + remainder}\\hfill \\\\ \\hfill 178&amp; =&amp; \\left(3\\cdot 59\\right)+1\\hfill \\\\ &amp; =&amp; 177+1\\hfill \\\\ &amp; =&amp; 178\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137640958\">We call this the <strong>Division Algorithm <\/strong>and will discuss it more formally after looking at an example.<\/p>\n<p id=\"fs-id1165137933942\">Division of polynomials that contain more than one term has similarities to long division of whole numbers. We can write a polynomial dividend as the product of the divisor and the quotient added to the remainder. The terms of the polynomial division correspond to the digits (and place values) of the whole number division. This method allows us to divide two polynomials. For example, if we were to divide[latex]\\,2{x}^{3}-3{x}^{2}+4x+5\\,[\/latex]\nby[latex]\\,x+2\\,[\/latex]\nusing the long division algorithm, it would look like this:<\/p>\n<span id=\"fs-id1678300\"><img class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/6405515967a5671cb5ff219ec9a60a7a5aa0be94\/CNX_CAT_Figure_05_01_001.jpg\" alt=\"\"><\/span>\n<p id=\"fs-id1165135191694\">We have found<\/p>\n\n<div id=\"eip-334\" class=\"unnumbered aligncenter\">[latex]\\frac{2{x}^{3}-3{x}^{2}+4x+5}{x+2}=2{x}^{2}-7x+18-\\frac{31}{x+2}[\/latex]<\/div>\n<p id=\"fs-id1165137823279\">or<\/p>\n\n<div id=\"eip-212\" class=\"unnumbered aligncenter\">[latex]\\frac{2{x}^{3}-3{x}^{2}+4x+5}{x+2}=\\left(x+2\\right)\\left(2{x}^{2}-7x+18\\right)-31[\/latex]<\/div>\n<p id=\"fs-id1165135181270\">We can identify the <span class=\"no-emphasis\">dividend<\/span>, the <span class=\"no-emphasis\">divisor<\/span>, the <span class=\"no-emphasis\">quotient<\/span>, and the <span class=\"no-emphasis\">remainder<\/span>.<\/p>\n<span id=\"fs-id1165134164979\"><img class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/fdefee7ce7430fca380493b18debaba5cf907531\/CNX_Precalc_Figure_03_05_003.jpg\" alt=\"Identifying the dividend, divisor, quotient and remainder of the polynomial 2x^3-3x^2+4x+5, which is the dividend.\"><\/span>\n<p id=\"fs-id1165135508592\">Writing the result in this manner illustrates the Division Algorithm.<\/p>\n\n<div id=\"fs-id1165135508595\" class=\"textbox key-takeaways\">\n<h3>The Division Algorithm<\/h3>\n<p id=\"fs-id1165137854177\">The Division Algorithm states that, given a polynomial dividend[latex]\\,f\\left(x\\right)\\,[\/latex] and a non-zero polynomial divisor[latex]\\,d\\left(x\\right)\\,[\/latex] where the degree of[latex]\\,d\\left(x\\right)\\,[\/latex] is less than or equal to the degree of[latex]\\,f\\left(x\\right)[\/latex], there exist unique polynomials[latex]\\,q\\left(x\\right)\\,[\/latex] and[latex]\\,r\\left(x\\right)\\,[\/latex] such that [latex]f\\left(x\\right)=d\\left(x\\right)q\\left(x\\right)+r\\left(x\\right)[\/latex] [latex]q\\left(x\\right)\\,[\/latex] is the quotient and[latex]\\,r\\left(x\\right)\\,[\/latex] is the remainder. The remainder is either equal to zero or has degree strictly less than[latex]\\,d\\left(x\\right).\\,[\/latex] If[latex]\\,r\\left(x\\right)=0,\\,[\/latex] then[latex]\\,d\\left(x\\right)\\,[\/latex] divides evenly into[latex]\\,f\\left(x\\right).\\,[\/latex] This means that, in this case, both[latex]\\,d\\left(x\\right)\\,[\/latex]\nand[latex]\\,q\\left(x\\right)\\,[\/latex] are factors of[latex]\\,f\\left(x\\right).\\,[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135638531\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"eip-id1567242\"><strong>Given a polynomial and a binomial, use long division to divide the polynomial by the binomial.<\/strong><\/p>\n\n<ol id=\"eip-id1165134557348\" type=\"1\">\n \t<li>Set up the division problem.<\/li>\n \t<li>Determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor.<\/li>\n \t<li>Multiply the answer by the divisor and write it below the like terms of the dividend.<\/li>\n \t<li>Subtract the bottom <span class=\"no-emphasis\">binomial<\/span> from the top binomial.<\/li>\n \t<li>Bring down the next term of the dividend.<\/li>\n \t<li>Repeat steps 2\u20135 until reaching the last term of the dividend.<\/li>\n \t<li>If the remainder is non-zero, express as a fraction using the divisor as the denominator.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_05_01\" class=\"textbox examples\">\n<div id=\"fs-id1165137817675\">\n<div id=\"fs-id1165137817678\">\n<h3>Using Long Division to Divide a Second-Degree Polynomial<\/h3>\n<p id=\"fs-id1165137817683\">Divide[latex]\\,5{x}^{2}+3x-2\\,[\/latex] by[latex]\\,x+1.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135196885\" class=\"solution textbox shaded\">\n<div id=\"eip-id1165135533774\" class=\"unnumbered\">[reveal-answer q=\"893598\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"893598\"]\n<p id=\"fs-id1165137639118\">The quotient is[latex]\\,5x-2.\\,[\/latex] The remainder is 0. We write the result as<\/p>\n\n<div id=\"eip-id1165137701865\" class=\"unnumbered\">[latex]\\frac{5{x}^{2}+3x-2}{x+1}=5x-2[\/latex]<\/div>\n<p id=\"fs-id1165134058382\">or<\/p>\n\n<div id=\"eip-id1165135533774\" class=\"unnumbered\">[latex]5{x}^{2}+3x-2=\\left(x+1\\right)\\left(5x-2\\right)[\/latex]<\/div>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165132950610\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165135372071\">This division problem had a remainder of 0. This tells us that the dividend is divided evenly by the divisor, and that the divisor is a factor of the dividend.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_05_02\" class=\"textbox examples\">\n<div id=\"fs-id1165135372082\">\n<div id=\"fs-id1165135372084\">\n<h3>Using Long Division to Divide a Third-Degree Polynomial<\/h3>\n<p id=\"fs-id1165134352552\">Divide[latex]\\,6{x}^{3}+11{x}^{2}-31x+15\\,[\/latex] by[latex]\\,3x-2.\\,[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135503914\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135503914\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135503914\"]<img src=\"https:\/\/cnx.org\/resources\/5c1d58c86d624bbd19a2d68e7c4725b1ef41655f\/CNX_CAT_Figure_05_01_003.jpg\" alt=\"\">\n<p id=\"fs-id1165135639821\">There is a remainder of 1. We can express the result as:<\/p>\n\n<div id=\"eip-id1165134294806\" class=\"unnumbered\">[latex]\\frac{6{x}^{3}+11{x}^{2}-31x+15}{3x-2}=2{x}^{2}+5x-7+\\frac{1}{3x-2}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165135340591\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165135340597\">We can check our work by using the Division Algorithm to rewrite the solution. Then multiply.<\/p>\n\n<div id=\"eip-id1165135428302\" class=\"unnumbered\">[latex]\\left(3x-2\\right)\\left(2{x}^{2}+5x-7\\right)+1=6{x}^{3}+11{x}^{2}-31x+15[\/latex]<\/div>\n<p id=\"fs-id1165135152076\">Notice, as we write our result,<\/p>\n\n<ul id=\"fs-id1165135152079\">\n \t<li>the dividend is[latex]\\,6{x}^{3}+11{x}^{2}-31x+15\\,[\/latex]<\/li>\n \t<li>the divisor is[latex]\\,3x-2\\,[\/latex]<\/li>\n \t<li>the quotient is[latex]\\,2{x}^{2}+5x-7\\,[\/latex]<\/li>\n \t<li>the remainder is[latex]\\,1\\,[\/latex]<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134042317\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_03_05_01\">\n<div>\n<p id=\"fs-id1165135545763\">Divide[latex]\\,16{x}^{3}-12{x}^{2}+20x-3\\,[\/latex] by[latex]\\,4x+5.\\,[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135177648\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135177648\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135177648\"]\n<p id=\"fs-id1165135177649\">[latex]4{x}^{2}-8x+15-\\frac{78}{4x+5}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137932621\" class=\"bc-section section\">\n<h3>Using Synthetic Division to Divide Polynomials<\/h3>\nAs we\u2019ve seen, long division of polynomials can involve many steps and be quite cumbersome. <strong>Synthetic division<\/strong> is a shorthand method of dividing polynomials for the special case of dividing by a linear factor whose leading coefficient is 1.\n<p id=\"fs-id1165137932636\">To illustrate the process, recall the example at the beginning of the section.<\/p>\n<p id=\"fs-id1165137932639\">Divide[latex]\\,2{x}^{3}-3{x}^{2}+4x+5\\,[\/latex] by[latex]\\,x+2\\,[\/latex] using the long division algorithm.<\/p>\n<p id=\"fs-id1165135170412\">The final form of the process looked like this:<\/p>\n<span id=\"fs-id2502523\"><img class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/3572390ee6789bc61d8076e1564b505983b5eb08\/CNX_CAT_Figure_05_01_004.jpg\" alt=\"\"><\/span>\n<p id=\"fs-id1165137932377\">There is a lot of repetition in the table. If we don\u2019t write the variables but, instead, line up their coefficients in columns under the division sign and also eliminate the partial products, we already have a simpler version of the entire problem.<\/p>\n<span id=\"fs-id1165134305375\"><img class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/6e0d0074905bf0272d1d4cbe2690e6009aa46409\/CNX_Precalc_Figure_03_05_004.jpg\" alt=\"Synthetic division of the polynomial 2x^3-3x^2+4x+5 by x+2 in which it only contains the coefficients of each polynomial.\"><\/span>\n<p id=\"fs-id1165134305388\">Synthetic division carries this simplification even a few more steps. Collapse the table by moving each of the rows up to fill any vacant spots. Also, instead of dividing by 2, as we would in division of whole numbers, then multiplying and subtracting the middle product, we change the sign of the \u201cdivisor\u201d to \u20132, multiply and add. The process starts by bringing down the leading coefficient.<\/p>\n<span id=\"fs-id1165137696374\"><img class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/e37af41cac58008922e2bf6b604119e15e778d23\/CNX_Precalc_Figure_03_05_011.jpg\" alt=\"Synthetic division of the polynomial 2x^3-3x^2+4x+5 by x+2 in which it only contains the coefficients of each polynomial.\"><\/span>\n<p id=\"fs-id1165137696388\">We then multiply it by the \u201cdivisor\u201d and add, repeating this process column by column, until there are no entries left. The bottom row represents the coefficients of the quotient; the last entry of the bottom row is the remainder. In this case, the quotient is[latex]\\,2{x}^{2}\u20137x+18\\,[\/latex]and the remainder is[latex]\\,\u201331.\\,[\/latex] The process will be made more clear in <a class=\"autogenerated-content\" href=\"#Example_03_05_03\">(Figure)<\/a>.<\/p>\n\n<div id=\"fs-id1165135383640\" class=\"textbox key-takeaways\">\n<h3>Synthetic Division<\/h3>\n<p id=\"fs-id1165135383649\">Synthetic division is a shortcut that can be used when the divisor is a binomial in the form[latex]\\,x-k\\,[\/latex]where[latex]\\,k\\,[\/latex]is a real number.<\/p>\nIn synthetic division, only the coefficients are used in the division process.\n\n<\/div>\n<div id=\"fs-id1165135393407\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135393414\"><strong>Given two polynomials, use synthetic division to divide.<\/strong><\/p>\n\n<ol id=\"fs-id1165135393418\" type=\"1\">\n \t<li>Write[latex]\\,k\\,[\/latex] for the divisor.<\/li>\n \t<li>Write the coefficients of the dividend.<\/li>\n \t<li>Bring the lead coefficient down.<\/li>\n \t<li>Multiply the lead coefficient by[latex]\\,k.\\,[\/latex] Write the product in the next column.<\/li>\n \t<li>Add the terms of the second column.<\/li>\n \t<li>Multiply the result by[latex]\\,k.\\,[\/latex] Write the product in the next column.<\/li>\n \t<li>Repeat steps 5 and 6 for the remaining columns.<\/li>\n \t<li>Use the bottom numbers to write the quotient. The number in the last column is the remainder and has degree 0, the next number from the right has degree 1, the next number from the right has degree 2, and so on.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_05_03\" class=\"textbox examples\">\n<div id=\"fs-id1165135383099\">\n<div id=\"fs-id1165135383101\">\n<h3>Using Synthetic Division to Divide a Second-Degree Polynomial<\/h3>\n<p id=\"fs-id1165135383107\">Use synthetic division to divide[latex]\\,5{x}^{2}-3x-36\\,[\/latex]\nby[latex]\\,x-3.\\,[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135177606\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135177606\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135177606\"]\n<p id=\"fs-id1165135177608\">Begin by setting up the synthetic division. Write[latex]\\,k\\,[\/latex] and the coefficients.<\/p>\n<span id=\"fs-id1165135177629\"><img src=\"https:\/\/cnx.org\/resources\/9f9b7c3f2afaf1c98578471f8ec6fd5c3b34749f\/CNX_Precalc_Figure_03_05_005.jpg\" alt=\"A collapsed version of the previous synthetic division.\"><\/span>\n<p id=\"fs-id1165135439942\">Bring down the lead coefficient. Multiply the lead coefficient by[latex]\\,k.\\,[\/latex]<\/p>\n<span id=\"fs-id1165135439966\"><img src=\"https:\/\/cnx.org\/resources\/8b638e9a3fafd6b1e6496583a605b4a0b4858d0e\/CNX_Precalc_Figure_03_05_006.jpg\" alt=\"The set-up of the synthetic division for the polynomial 5x^2-3x-36 by x-3, which renders {5, -3, -36} by 3.\"><\/span>\n<p id=\"fs-id1165135179942\">Continue by adding the numbers in the second column. Multiply the resulting number by[latex]\\,k.\\,[\/latex] Write the result in the next column. Then add the numbers in the third column.<\/p>\n<span id=\"fs-id1165135179966\"><img src=\"https:\/\/cnx.org\/resources\/a77e7bc9a6d09d92cfe1e1f485fc8887a5ba4e6f\/CNX_Precalc_Figure_03_05_007.jpg\" alt=\"Multiplied by the lead coefficient, 5, in the second column, and the lead coefficient is brought down to the second row. \"><\/span>\n\nThe result is[latex]\\,5x+12.\\,[\/latex] The remainder is 0. So[latex]\\,x-3\\,[\/latex] is a factor of the original polynomial.[\/hidden-answer]\n\n<\/div>\n<div id=\"fs-id1165135463242\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165135463247\">Just as with long division, we can check our work by multiplying the quotient by the divisor and adding the remainder.<\/p>\n<p id=\"fs-id1165135463251\">[latex]\\left(x-3\\right)\\left(5x+12\\right)+0=5{x}^{2}-3x-36[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_05_04\" class=\"textbox examples\">\n<div id=\"fs-id1165135549012\">\n<div id=\"fs-id1165135549014\">\n<h3>Using Synthetic Division to Divide a Third-Degree Polynomial<\/h3>\n<p id=\"fs-id1165135549019\">Use synthetic division to divide[latex]\\,4{x}^{3}+10{x}^{2}-6x-20\\,[\/latex] by[latex]\\,x+2.\\,[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135173365\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135173365\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135173365\"]\n<p id=\"fs-id1165135173367\">The binomial divisor is[latex]\\,x+2\\,[\/latex] so[latex]\\,k=-2.\\,[\/latex]\nAdd each column, multiply the result by \u20132, and repeat until the last column is reached.<\/p>\n<span id=\"fs-id1165134176031\"><img src=\"https:\/\/cnx.org\/resources\/c129e0af0a8095a5958635373b6337b02d8d41a4\/CNX_Precalc_Figure_03_05_008.jpg\" alt=\"Synthetic division of 4x^3+10x^2-6x-20 divided by x+2.\"><\/span>\n<p id=\"fs-id1165134433356\">The result is[latex]\\,4{x}^{2}+2x-10.\\,[\/latex] The remainder is 0. Thus,[latex]\\,x+2\\,[\/latex] is a factor of[latex]\\,4{x}^{3}+10{x}^{2}-6x-20.\\,[\/latex][\/hidden-answer]<\/p>\n\n<\/div>\n<div id=\"fs-id1165133061700\">\n<h4>Analysis<\/h4>\nThe graph of the polynomial function[latex]\\,f\\left(x\\right)=4{x}^{3}+10{x}^{2}-6x-20\\,[\/latex] in <a class=\"autogenerated-content\" href=\"#Figure_03_05_009\">(Figure)<\/a> shows a zero at[latex]\\,x=k=-2.\\,[\/latex] This confirms that[latex]\\,x+2\\,[\/latex] is a factor of[latex]\\,4{x}^{3}+10{x}^{2}-6x-20.\\,[\/latex]\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/cnx.org\/resources\/4e9f36eda9dde4f8ff108084685bf7661b4038c8\/CNX_Precalc_Figure_03_05_009.jpg\" alt=\"Synthetic division of 4x^3+10x^2-6x-20 divided by x+2.\" width=\"487\" height=\"742\"> <strong>Figure 2.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_05_05\" class=\"textbox examples\">\n<div id=\"fs-id1165133260470\">\n<div id=\"fs-id1165133260472\">\n<h3>Using Synthetic Division to Divide a Fourth-Degree Polynomial<\/h3>\n<p id=\"fs-id1165135481144\">Use synthetic division to divide[latex]\\,-9{x}^{4}+10{x}^{3}+7{x}^{2}-6\\,[\/latex] by[latex]\\,x-1.\\,[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135571792\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135571792\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135571792\"]\n<p id=\"fs-id1165135571794\">Notice there is no <em>x<\/em>-term. We will use a zero as the coefficient for that term.<\/p>\n\n<div><\/div>\n<span id=\"Figure_05_04_001\"><img src=\"https:\/\/cnx.org\/resources\/21c33f8114c651702043fec1fafe25f629656a9a\/CNX_CAT_Figure_05_04_001.jpg\" alt=\"..\"><\/span>\n\nThe result is[latex]\\,-9{x}^{3}+{x}^{2}+8x+8+\\frac{2}{x-1}.[\/latex]\n<div>\n\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134037571\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_03_05_02\">\n<div id=\"fs-id1165134037583\">\n<p id=\"fs-id1165134037584\">Use synthetic division to divide[latex]\\,3{x}^{4}+18{x}^{3}-3x+40\\,[\/latex]\nby[latex]\\,x+7.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165133365553\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133365553\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133365553\"]\n<p id=\"fs-id1165133365554\">[latex]3{x}^{3}-3{x}^{2}+21x-150+\\frac{1,090}{x+7}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135403412\" class=\"bc-section section\">\n<h3>Using Polynomial Division to Solve Application Problems<\/h3>\n<p id=\"fs-id1165135403417\">Polynomial division can be used to solve a variety of application problems involving expressions for area and volume. We looked at an application at the beginning of this section. Now we will solve that problem in the following example.<\/p>\n\n<div id=\"Example_03_05_06\" class=\"textbox examples\">\n<div id=\"fs-id1165135403427\">\n<div id=\"fs-id1165135403429\">\n<h3>Using Polynomial Division in an Application Problem<\/h3>\n<p id=\"fs-id1165135403434\">The volume of a rectangular solid is given by the polynomial[latex]\\,3{x}^{4}-3{x}^{3}-33{x}^{2}+54x.\\,[\/latex] The length of the solid is given by[latex]\\,3x\\,[\/latex] and the width is given by[latex]\\,x-2.\\,[\/latex]\nFind the height,[latex]\\,h,[\/latex]of the solid.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135685835\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135685835\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135685835\"]\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/cnx.org\/resources\/fc1a8c7d3dd929f331a9e0b414d36568e7f722dd\/CNX_Precalc_Figure_03_05_010.jpg\" alt=\"Graph of f(x)=4x^3+10x^2-6x-20 with a close up on x+2.\" width=\"487\" height=\"140\"> <strong>Figure 3.<\/strong>[\/caption]\n<p id=\"fs-id1165135685837\">There are a few ways to approach this problem. We need to divide the expression for the volume of the solid by the expressions for the length and width. Let us create a sketch as in <a class=\"autogenerated-content\" href=\"#Figure_03_05_010\">(Figure)<\/a>.<\/p>\n<p id=\"fs-id1165137843229\">We can now write an equation by substituting the known values into the formula for the volume of a rectangular solid.<\/p>\n\n<div id=\"eip-id1165135439925\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill V&amp; =&amp; l\\cdot w\\cdot h\\hfill \\\\ \\hfill 3{x}^{4}-3{x}^{3}-33{x}^{2}+54x&amp; =&amp; 3x\\cdot \\left(x-2\\right)\\cdot h\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135457104\">To solve for[latex]\\,h,\\,[\/latex]first divide both sides by[latex]\\,3x.[\/latex]<\/p>\n\n<div id=\"eip-id1165135438421\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill \\frac{3x\\cdot \\left(x-2\\right)\\cdot h}{3x}&amp; =&amp; \\frac{3{x}^{4}-3{x}^{3}-33{x}^{2}+54x}{3x}\\hfill \\\\ \\hfill \\left(x-2\\right)h&amp; =&amp; {x}^{3}-{x}^{2}-11x+18\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135528878\">Now solve for[latex]\\,h\\,[\/latex] using synthetic division.<\/p>\n\n<div id=\"eip-id1165134103025\" class=\"unnumbered\">[latex]h=\\frac{{x}^{3}-{x}^{2}-11x+18}{x-2}[\/latex]<\/div>\n<span id=\"fs-id1349001\"><img src=\"https:\/\/cnx.org\/resources\/ba678b7589f59945aa041cce2af5b441e7d759c5\/CNX_CAT_Figure_05_01_005.jpg\" alt=\"\"><\/span>\n<p id=\"fs-id1165134152722\">The quotient is[latex]\\,{x}^{2}+x-9\\,[\/latex] and the remainder is 0. The height of the solid is[latex]\\,{x}^{2}+x-9.[\/latex]<span id=\"fs-id1165137843218\"><\/span>[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135694534\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_03_05_03\">\n<div id=\"fs-id1165135694546\">\n<p id=\"fs-id1165135694547\">The area of a rectangle is given by[latex]\\,3{x}^{3}+14{x}^{2}-23x+6.\\,[\/latex] The width of the rectangle is given by[latex]\\,x+6.\\,[\/latex]\nFind an expression for the length of the rectangle.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135407022\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135407022\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135407022\"]\n<p id=\"fs-id1165135407023\">[latex]3{x}^{2}-4x+1[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135571867\" class=\"precalculus media\">\n<p id=\"fs-id1165135571875\">Access these online resources for additional instruction and practice with polynomial division.<\/p>\n\n<ul id=\"fs-id1165135571879\">\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/dividetribild\">Dividing a Trinomial by a Binomial Using Long Division<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/dividepolybild\">Dividing a Polynomial by a Binomial Using Long Division <\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/dividepolybisd2\">Ex 2: Dividing a Polynomial by a Binomial Using Synthetic Division<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/dividepolybisd4\">Ex 4: Dividing a Polynomial by a Binomial Using Synthetic Division<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135487276\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"eip-id1165133432926\" summary=\"..\">\n<tbody>\n<tr>\n<td>Division Algorithm<\/td>\n<td>[latex]f\\left(x\\right)=d\\left(x\\right)q\\left(x\\right)+r\\left(x\\right)\\text{ where }q\\left(x\\right)\\ne 0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165135531548\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165135531552\">\n \t<li>Polynomial long division can be used to divide a polynomial by any polynomial with equal or lower degree. See <a class=\"autogenerated-content\" href=\"#Example_03_05_01\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_03_05_02\">(Figure)<\/a><strong>.<\/strong><\/li>\n \t<li>The Division Algorithm tells us that a polynomial dividend can be written as the product of the divisor and the quotient added to the remainder.<\/li>\n \t<li>Synthetic division is a shortcut that can be used to divide a polynomial by a binomial in the form[latex]\\,x-k.\\,[\/latex]\nSee <a class=\"autogenerated-content\" href=\"#Example_03_05_03\">(Figure)<\/a><strong>, <\/strong><a class=\"autogenerated-content\" href=\"#Example_03_05_04\">(Figure)<\/a><strong>, <\/strong>and <a class=\"autogenerated-content\" href=\"#Example_03_05_05\">(Figure)<\/a><strong>.<\/strong><\/li>\n \t<li>Polynomial division can be used to solve application problems, including area and volume. See <a class=\"autogenerated-content\" href=\"#Example_03_05_06\">(Figure)<\/a><strong>.<\/strong><\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165135255120\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165135255124\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165135255129\">\n<div id=\"fs-id1165135255131\">\n<p id=\"fs-id1165135443966\">If division of a polynomial by a binomial results in a remainder of zero, what can be conclude?<\/p>\n\n<\/div>\n<div id=\"fs-id1165135443970\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135443970\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135443970\"]\n<p id=\"fs-id1165135443971\">The binomial is a factor of the polynomial.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135443975\">\n<div id=\"fs-id1165135443976\">\n<p id=\"fs-id1165135443977\">If a polynomial of degree[latex]\\,n\\,[\/latex] is divided by a binomial of degree 1, what is the degree of the quotient?<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135443995\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1165135444000\">For the following exercises, use long division to divide. Specify the quotient and the remainder.<\/p>\n\n<div id=\"fs-id1165135349098\">\n<div id=\"fs-id1165135349099\">\n<p id=\"fs-id1165135349100\">[latex]\\left({x}^{2}+5x-1\\right)\u00f7\\left(x-1\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137849077\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137849077\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137849077\"]\n<p id=\"fs-id1165137849078\">[latex]x+6+\\frac{5}{x-1}\\text{,}\\,\\text{quotient:}\\,x+6\\text{,}\\,\\text{remainder:}\\,\\text{5}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137932684\">\n<div>\n<p id=\"fs-id1165137932686\">[latex]\\left(2{x}^{2}-9x-5\\right)\u00f7\\left(x-5\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135643143\">\n<div id=\"fs-id1165135643144\">\n<p id=\"fs-id1165135643146\">[latex]\\left(3{x}^{2}+23x+14\\right)\u00f7\\left(x+7\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134149896\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134149896\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134149896\"]\n<p id=\"fs-id1165134149897\">[latex]3x+2\\text{,}\\,\\text{quotient: }3x+2\\text{,}\\,\\text{remainder: 0}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135453058\">\n<div id=\"fs-id1165135453059\">\n<p id=\"fs-id1165135453060\">[latex]\\left(4{x}^{2}-10x+6\\right)\u00f7\\left(4x+2\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137833881\">\n<div id=\"fs-id1165137833882\">\n<p id=\"fs-id1165137833883\">[latex]\\left(6{x}^{2}-25x-25\\right)\u00f7\\left(6x+5\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135532496\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135532496\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135532496\"]\n<p id=\"fs-id1165135532497\">[latex]x-5\\text{,}\\,\\text{quotient:}\\,x-5\\text{,}\\,\\text{remainder:}\\,\\text{0}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135363204\">\n<div id=\"fs-id1165135363205\">\n<p id=\"fs-id1165135363206\">[latex]\\left(-{x}^{2}-1\\right)\u00f7\\left(x+1\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134472270\">\n<div id=\"fs-id1165134472271\">\n<p id=\"fs-id1165134472272\">[latex]\\left(2{x}^{2}-3x+2\\right)\u00f7\\left(x+2\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134234239\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134234239\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134234239\"]\n<p id=\"fs-id1165135694449\">[latex]2x-7+\\frac{16}{x+2}\\text{,}\\,\\text{quotient:}\\text{\u200b}\\,2x-7\\text{,}\\,\\text{remainder:}\\,\\text{16}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134352532\">\n<div id=\"fs-id1165134352533\">\n<p id=\"fs-id1165134352534\">[latex]\\left({x}^{3}-126\\right)\u00f7\\left(x-5\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135678617\">\n<div id=\"fs-id1165135678618\">\n<p id=\"fs-id1165135678619\">[latex]\\left(3{x}^{2}-5x+4\\right)\u00f7\\left(3x+1\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165132962052\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165132962052\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165132962052\"]\n<p id=\"fs-id1165132962053\">[latex]x-2+\\frac{6}{3x+1}\\text{,}\\,\\text{quotient:}\\,x-2\\text{,}\\,\\text{remainder:}\\,\\text{6}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165133093355\">\n<div id=\"fs-id1165133093356\">\n<p id=\"fs-id1165133093357\">[latex]\\left({x}^{3}-3{x}^{2}+5x-6\\right)\u00f7\\left(x-2\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135435668\">[latex]\\left(2{x}^{3}+3{x}^{2}-4x+15\\right)\u00f7\\left(x+3\\right)[\/latex]<\/div>\n<div id=\"fs-id1165135632104\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135632104\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135632104\"]\n<p id=\"fs-id1165135632105\">[latex]2{x}^{2}-3x+5\\text{,}\\,\\text{quotient:}\\,2{x}^{2}-3x+5\\text{,}\\,\\text{remainder:}\\,\\text{0}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135321927\">For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.)<\/p>\n\n<div id=\"fs-id1165135321931\">\n<div id=\"fs-id1165135321932\">\n<p id=\"fs-id1165135321933\">[latex]\\left(3{x}^{3}-2{x}^{2}+x-4\\right)\u00f7\\left(x+3\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134474179\">\n<div id=\"fs-id1165134474180\">\n<p id=\"fs-id1165134474181\">[latex]\\left(2{x}^{3}-6{x}^{2}-7x+6\\right)\u00f7\\left(x-4\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135298453\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135298453\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135298453\"]\n<p id=\"fs-id1165135298454\">[latex]2{x}^{2}+2x+1+\\frac{10}{x-4}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135199519\">\n<div id=\"fs-id1165137680594\">\n<p id=\"fs-id1165137680595\">[latex]\\left(6{x}^{3}-10{x}^{2}-7x-15\\right)\u00f7\\left(x+1\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165133233060\">\n<div id=\"fs-id1165133233062\">\n<p id=\"fs-id1165133233063\">[latex]\\left(4{x}^{3}-12{x}^{2}-5x-1\\right)\u00f7\\left(2x+1\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135486057\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135486057\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135486057\"]\n<p id=\"fs-id1165135486058\">[latex]2{x}^{2}-7x+1-\\frac{2}{2x+1}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165133311045\">\n<div id=\"fs-id1165133311046\">\n<p id=\"fs-id1165133311047\">[latex]\\left(9{x}^{3}-9{x}^{2}+18x+5\\right)\u00f7\\left(3x-1\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135560631\">\n<p id=\"fs-id1165135560632\">[latex]\\left(3{x}^{3}-2{x}^{2}+x-4\\right)\u00f7\\left(x+3\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135517139\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135517139\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135517139\"]\n<p id=\"fs-id1165135517140\">[latex]3{x}^{2}-11x+34-\\frac{106}{x+3}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137642766\">\n<div id=\"fs-id1165137642767\">\n<p id=\"fs-id1165137642768\">[latex]\\left(-6{x}^{3}+{x}^{2}-4\\right)\u00f7\\left(2x-3\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135530633\">\n<div id=\"fs-id1165135530634\">\n<p id=\"fs-id1165135530635\">[latex]\\left(2{x}^{3}+7{x}^{2}-13x-3\\right)\u00f7\\left(2x-3\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134173709\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134173709\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134173709\"]\n<p id=\"fs-id1165134173710\">[latex]{x}^{2}+5x+1[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134173742\">\n<div id=\"fs-id1165134173743\">\n<p id=\"fs-id1165134173744\">[latex]\\left(3{x}^{3}-5{x}^{2}+2x+3\\right)\u00f7\\left(x+2\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135443762\">\n<div id=\"fs-id1165135443764\">\n<p id=\"fs-id1165135443765\">[latex]\\left(4{x}^{3}-5{x}^{2}+13\\right)\u00f7\\left(x+4\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137851483\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137851483\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137851483\"]\n<p id=\"fs-id1165137851484\">[latex]4{x}^{2}-21x+84-\\frac{323}{x+4}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135403522\">\n<div id=\"fs-id1165135403523\">\n<p id=\"fs-id1165135403524\">[latex]\\left({x}^{3}-3x+2\\right)\u00f7\\left(x+2\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134061948\">\n<div id=\"fs-id1165134061949\">\n<p id=\"fs-id1165134061950\">[latex]\\left({x}^{3}-21{x}^{2}+147x-343\\right)\u00f7\\left(x-7\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134156036\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134156036\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134156036\"]\n<p id=\"fs-id1165134156037\">[latex]{x}^{2}-14x+49[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134156069\">\n<div id=\"fs-id1165134156070\">\n<p id=\"fs-id1165134156071\">[latex]\\left({x}^{3}-15{x}^{2}+75x-125\\right)\u00f7\\left(x-5\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135344106\">\n<div id=\"fs-id1165135344107\">\n<p id=\"fs-id1165135344108\">[latex]\\left(9{x}^{3}-x+2\\right)\u00f7\\left(3x-1\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134118337\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134118337\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134118337\"]\n<p id=\"fs-id1165134118338\">[latex]3{x}^{2}+x+\\frac{2}{3x-1}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135485710\">\n<div id=\"fs-id1165135485711\">\n<p id=\"fs-id1165135485712\">[latex]\\left(6{x}^{3}-{x}^{2}+5x+2\\right)\u00f7\\left(3x+1\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135485136\">\n<div id=\"fs-id1165135485137\">\n<p id=\"fs-id1165135485138\">[latex]\\left({x}^{4}+{x}^{3}-3{x}^{2}-2x+1\\right)\u00f7\\left(x+1\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134357257\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134357257\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134357257\"]\n<p id=\"fs-id1165134357258\">[latex]{x}^{3}-3x+1[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134357289\">\n<div id=\"fs-id1165134357290\">\n<p id=\"fs-id1165134357291\">[latex]\\left({x}^{4}-3{x}^{2}+1\\right)\u00f7\\left(x-1\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135390850\">\n<div id=\"fs-id1165135390852\">\n<p id=\"fs-id1165135390853\">[latex]\\left({x}^{4}+2{x}^{3}-3{x}^{2}+2x+6\\right)\u00f7\\left(x+3\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135575976\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135575976\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135575976\"]\n<p id=\"fs-id1165135575977\">[latex]{x}^{3}-{x}^{2}+2[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137850245\">\n<div id=\"fs-id1165137850246\">[latex]\\left({x}^{4}-10{x}^{3}+37{x}^{2}-60x+36\\right)\u00f7\\left(x-2\\right)[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137705401\">\n<div id=\"fs-id1165137705402\">\n<p id=\"fs-id1165137705403\">[latex]\\left({x}^{4}-8{x}^{3}+24{x}^{2}-32x+16\\right)\u00f7\\left(x-2\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134261862\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134261862\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134261862\"]\n<p id=\"fs-id1165134261863\">[latex]{x}^{3}-6{x}^{2}+12x-8[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134389913\">\n<div id=\"fs-id1165134389914\">\n<p id=\"fs-id1165134389915\">[latex]\\left({x}^{4}+5{x}^{3}-3{x}^{2}-13x+10\\right)\u00f7\\left(x+5\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134484904\">\n<div id=\"fs-id1165134484905\">\n<p id=\"fs-id1165134484906\">[latex]\\left({x}^{4}-12{x}^{3}+54{x}^{2}-108x+81\\right)\u00f7\\left(x-3\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135486004\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135486004\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135486004\"]\n<p id=\"fs-id1165135486005\">[latex]{x}^{3}-9{x}^{2}+27x-27[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137696095\">\n<div id=\"fs-id1165137696096\">\n<p id=\"fs-id1165137696097\">[latex]\\left(4{x}^{4}-2{x}^{3}-4x+2\\right)\u00f7\\left(2x-1\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135575253\">\n<div id=\"fs-id1165135575254\">\n<p id=\"fs-id1165135575255\">[latex]\\left(4{x}^{4}+2{x}^{3}-4{x}^{2}+2x+2\\right)\u00f7\\left(2x+1\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135403348\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135403348\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135403348\"]\n<p id=\"fs-id1165135403350\">[latex]2{x}^{3}-2x+2[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135353027\">For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization.<\/p>\n\n<div id=\"fs-id1165135353032\">\n<div id=\"fs-id1165135353033\">\n<p id=\"fs-id1165135353034\">[latex]x-2,\\,4{x}^{3}-3{x}^{2}-8x+4[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134031382\">\n<div id=\"fs-id1165134031383\">\n<p id=\"fs-id1165134031384\">[latex]x-2,\\,3{x}^{4}-6{x}^{3}-5x+10[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134431791\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134431791\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134431791\"]\n<p id=\"fs-id1165134431792\">Yes[latex]\\,\\left(x-2\\right)\\left(3{x}^{3}-5\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135240972\">\n<div id=\"fs-id1165135240974\">\n<p id=\"fs-id1165135240975\">[latex]x+3,\\,-4{x}^{3}+5{x}^{2}+8[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135394052\">\n<div id=\"fs-id1165135394053\">\n<p id=\"fs-id1165135394054\">[latex]x-2,\\,4{x}^{4}-15{x}^{2}-4[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135329837\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135329837\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135329837\"]\n<p id=\"fs-id1165135329838\">Yes[latex]\\,\\left(x-2\\right)\\left(4{x}^{3}+8{x}^{2}+x+2\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137851216\">\n<div id=\"fs-id1165135397213\">\n<p id=\"fs-id1165135397214\">[latex]x-\\frac{1}{2},\\,2{x}^{4}-{x}^{3}+2x-1[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135547391\">\n<div id=\"fs-id1165135547392\">\n<p id=\"fs-id1165135547394\">[latex]x+\\frac{1}{3},\\,3{x}^{4}+{x}^{3}-3x+1[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135481294\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135481294\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135481294\"]\n<p id=\"fs-id1165135481295\">No<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135481299\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1165135481304\">For the following exercises, use the graph of the third-degree polynomial and one factor to write the factored form of the polynomial suggested by the graph. The leading coefficient is one.<\/p>\n\n<div id=\"fs-id1165135436617\">\n<div id=\"fs-id1165135436618\">\n\nFactor is[latex]\\,{x}^{2}-x+3[\/latex]\n\n<span id=\"fs-id1165135436654\"><img src=\"https:\/\/cnx.org\/resources\/c6d107e4540de20a05d0ec88e4a90387178d41b2\/CNX_PreCalc_Figure_03_05_201.jpg\" alt=\"Graph of a polynomial that has a x-intercept at -1.\"><\/span>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135423510\">\n<div id=\"fs-id1165135423511\">\n<p id=\"fs-id1165135423512\">Factor is[latex]\\,\\left({x}^{2}+2x+4\\right)[\/latex]<\/p>\n<span id=\"fs-id1165134197955\"><img src=\"https:\/\/cnx.org\/resources\/7fb421522dcd4d4d0f220fdd9ceb28d54774ffcc\/CNX_PreCalc_Figure_03_05_202.jpg\" alt=\"Graph of a polynomial that has a x-intercept at 1.\"><\/span>\n\n<\/div>\n<div id=\"fs-id1165134197969\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134197969\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134197969\"]\n<p id=\"fs-id1165134197970\">[latex]\\left(x-1\\right)\\left({x}^{2}+2x+4\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135412915\">\n<div id=\"fs-id1165135412916\">\n<p id=\"fs-id1165135412917\">Factor is[latex]\\,{x}^{2}+2x+5[\/latex]<\/p>\n<span id=\"fs-id1165135176349\"><img src=\"https:\/\/cnx.org\/resources\/a592de91e6672cd41eef495d53d6002f85a9c36a\/CNX_PreCalc_Figure_03_05_203.jpg\" alt=\"Graph of a polynomial that has a x-intercept at 2.\"><\/span>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135176362\">\n<div id=\"fs-id1165135176363\">\n<p id=\"fs-id1165135176364\">Factor is[latex]\\,{x}^{2}+x+1[\/latex]<\/p>\n<span id=\"fs-id1165134164936\"><img src=\"https:\/\/cnx.org\/resources\/0de0bff90e32c4df60909455bf7cbd34294d97fa\/CNX_PreCalc_Figure_03_05_204.jpg\" alt=\"Graph of a polynomial that has a x-intercept at 5.\"><\/span>\n\n<\/div>\n<div id=\"fs-id1165134164949\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134164949\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134164949\"]\n<p id=\"fs-id1165134164950\">[latex]\\left(x-5\\right)\\left({x}^{2}+x+1\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134089490\">\n<div id=\"fs-id1165134089491\">\n<p id=\"fs-id1165134089492\">Factor is[latex]{x}^{2}+2x+2[\/latex]<\/p>\n<span id=\"fs-id1165137888963\"><img src=\"https:\/\/cnx.org\/resources\/028b1bf1cdc846cc20ae1ee2d89d755f2999dc89\/CNX_PreCalc_Figure_03_05_205.jpg\" alt=\"Graph of a polynomial that has a x-intercept at -3.\"><\/span>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137888975\">For the following exercises, use synthetic division to find the quotient and remainder.<\/p>\n\n<div id=\"fs-id1165137888978\">\n<div id=\"fs-id1165137888979\">\n<p id=\"fs-id1165137888980\">[latex]\\frac{4{x}^{3}-33}{x-2}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134061088\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134061088\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134061088\"]\n<p id=\"fs-id1165134061090\">[latex]\\text{Quotient:}\\,4{x}^{2}+8x+16\\text{,}\\,\\text{remainder:}\\,-1[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135209652\">\n<div id=\"fs-id1165135209653\">\n<p id=\"fs-id1165135209654\">[latex]\\frac{2{x}^{3}+25}{x+3}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135443854\">\n<div id=\"fs-id1165135443855\">\n<p id=\"fs-id1165135443856\">[latex]\\frac{3{x}^{3}+2x-5}{x-1}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137646978\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137646978\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137646978\"]\n<p id=\"fs-id1165137646979\">[latex]\\text{Quotient:}\\,3{x}^{2}+3x+5\\text{,}\\,\\text{remainder:}\\,0[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135154363\">\n<div id=\"fs-id1165135154364\">\n<p id=\"fs-id1165135154366\">[latex]\\frac{-4{x}^{3}-{x}^{2}-12}{x+4}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135155437\">\n<div id=\"fs-id1165135155438\">\n<p id=\"fs-id1165135155439\">[latex]\\frac{{x}^{4}-22}{x+2}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134118486\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134118486\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134118486\"]\n<p id=\"fs-id1165134118487\">[latex]\\text{Quotient:}\\,{x}^{3}-2{x}^{2}+4x-8\\text{,}\\,\\text{remainder:}\\,-6[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135501101\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<p id=\"fs-id1165135501106\">For the following exercises, use a calculator with CAS to answer the questions.<\/p>\n\n<div id=\"fs-id1165137889809\">\n<div id=\"fs-id1165137889810\">\n<p id=\"fs-id1165137889811\">Consider[latex]\\,\\frac{{x}^{k}-1}{x-1}\\,[\/latex]with[latex]\\,k=1, 2, 3.\\,[\/latex]What do you expect the result to be if[latex]\\,k=4?[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135534969\">\n<div id=\"fs-id1165135534970\">\n<p id=\"fs-id1165135534971\">Consider[latex]\\,\\frac{{x}^{k}+1}{x+1}\\,[\/latex]for[latex]\\,k=1, 3, 5.\\,[\/latex]What do you expect the result to be if[latex]\\,k=7?[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134148458\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134148458\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134148458\"]\n<p id=\"fs-id1165134148459\">[latex]{x}^{6}-{x}^{5}+{x}^{4}-{x}^{3}+{x}^{2}-x+1[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135241323\">\n<div id=\"fs-id1165135241324\">\n<p id=\"fs-id1165135241326\">Consider[latex]\\,\\frac{{x}^{4}-{k}^{4}}{x-k}\\,[\/latex]for[latex]\\,k=1, 2, 3.\\,[\/latex]What do you expect the result to be if[latex]\\,k=4?[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165132939184\">\n<div id=\"fs-id1165132939185\">\n<p id=\"fs-id1165132939186\">Consider[latex]\\,\\frac{{x}^{k}}{x+1}\\,[\/latex]with[latex]\\,k=1, 2, 3.\\,[\/latex]What do you expect the result to be if[latex]\\,k=4?[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134153037\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134153037\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134153037\"]\n<p id=\"fs-id1165134153038\">[latex]{x}^{3}-{x}^{2}+x-1+\\frac{1}{x+1}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134261659\">\n<div id=\"fs-id1165134261660\">\n<p id=\"fs-id1165134261661\">Consider[latex]\\,\\frac{{x}^{k}}{x-1}\\,[\/latex]with[latex]\\,k=1, 2, 3.\\,[\/latex]What do you expect the result to be if[latex]\\,k=4?[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135191971\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<p id=\"fs-id1165135191976\">For the following exercises, use synthetic division to determine the quotient involving a complex number.<\/p>\n\n<div id=\"fs-id1165135191981\">\n<div id=\"fs-id1165135191982\">\n<p id=\"fs-id1165135191983\">[latex]\\frac{x+1}{x-i}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135259591\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135259591\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135259591\"]\n<p id=\"fs-id1165135259592\">[latex]1+\\frac{1+i}{x-i}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135593134\">\n<p id=\"fs-id1165135593135\">[latex]\\frac{{x}^{2}+1}{x-i}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135593178\">\n<div id=\"fs-id1165135593179\">\n<p id=\"fs-id1165135593180\">[latex]\\frac{x+1}{x+i}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137834324\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137834324\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137834324\"]\n<p id=\"fs-id1165137834325\">[latex]1+\\frac{1-i}{x+i}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137777664\">\n<div id=\"fs-id1165137777665\">\n<p id=\"fs-id1165137777666\">[latex]\\frac{{x}^{2}+1}{x+i}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134248021\">\n<div id=\"fs-id1165134248022\">\n<p id=\"fs-id1165134248023\">[latex]\\frac{{x}^{3}+1}{x-i}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134248065\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134248065\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134248065\"]\n<p id=\"fs-id1165134248066\">[latex]{x}^{2}-ix-1+\\frac{1-i}{x-i}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135180069\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<p id=\"fs-id1165135180074\">For the following exercises, use the given length and area of a rectangle to express the width algebraically.<\/p>\n\n<div id=\"fs-id1165135180078\">\n<div id=\"fs-id1165135440138\">\n<p id=\"fs-id1165135440139\">Length is[latex]\\,x+5,\\,[\/latex]area is[latex]\\,2{x}^{2}+9x-5.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135440195\">\n<div id=\"fs-id1165135440196\">\n<p id=\"fs-id1165135440197\">Length is[latex]\\,2x\\text{ }+\\text{ }5,\\,[\/latex]area is[latex]\\,4{x}^{3}+10{x}^{2}+6x+15[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135264690\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135264690\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135264690\"]\n<p id=\"fs-id1165135264691\">[latex]2{x}^{2}+3[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135264719\">\n<div id=\"fs-id1165135264720\">\n<p id=\"fs-id1165135264721\">Length is[latex]\\,3x\u20134,\\,[\/latex]area is[latex]\\,6{x}^{4}-8{x}^{3}+9{x}^{2}-9x-4[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135339554\">For the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically.<\/p>\n\n<div id=\"fs-id1165135339559\">\n<div id=\"fs-id1165135339560\">\n<p id=\"fs-id1165135339561\">Volume is[latex]\\,12{x}^{3}+20{x}^{2}-21x-36,\\,[\/latex]length is[latex]\\,2x+3,\\,[\/latex]width is[latex]\\,3x-4.[\/latex]<\/p>\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165135317480\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135317480\"]\n<p id=\"fs-id1165135317480\">[latex]2x+3[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134071557\">\n<div id=\"fs-id1165134071558\">\n<p id=\"fs-id1165134071559\">Volume is[latex]\\,18{x}^{3}-21{x}^{2}-40x+48,\\,[\/latex]length is[latex]\\,3x\u20134,\\,[\/latex] width is[latex]\\,3x\u20134.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165132920308\">\n<div id=\"fs-id1165132920309\">\n<p id=\"fs-id1165132920310\">Volume is[latex]\\,10{x}^{3}+27{x}^{2}+2x-24,\\,[\/latex]length is[latex]\\,5x\u20134,\\,[\/latex] width is[latex]\\,2x+3.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135564161\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135564161\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135564161\"]\n<p id=\"fs-id1165135564162\">[latex]x+2[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135564178\">\n<div id=\"fs-id1165135564179\">\n<p id=\"fs-id1165135564180\">Volume is[latex]\\,10{x}^{3}+30{x}^{2}-8x-24,\\,[\/latex]length is[latex]\\,2,\\,[\/latex]width is[latex]\\,x+3.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135514530\">For the following exercises, use the given volume and radius of a cylinder to express the height of the cylinder algebraically.<\/p>\n\n<div id=\"fs-id1165135514534\">\n<div id=\"fs-id1165135514535\">\n<p id=\"fs-id1165135514536\">Volume is[latex]\\,\\pi \\left(25{x}^{3}-65{x}^{2}-29x-3\\right),\\,[\/latex]radius is[latex]\\,5x+1.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134031308\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134031308\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134031308\"]\n<p id=\"fs-id1165134031310\">[latex]x-3[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165134031329\">\n\nVolume is[latex]\\,\\pi \\left(4{x}^{3}+12{x}^{2}-15x-50\\right),\\,[\/latex]radius is[latex]\\,2x+5.[\/latex]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134258601\">\n<div id=\"fs-id1165134258602\">\n<p id=\"fs-id1165134258603\">Volume is[latex]\\,\\pi \\left(3{x}^{4}+24{x}^{3}+46{x}^{2}-16x-32\\right),\\,[\/latex]radius is[latex]\\,x+4.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137527638\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137527638\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137527638\"]\n<p id=\"fs-id1165137527639\">[latex]3{x}^{2}-2[\/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-id1165135471190\">\n \t<dt>Division Algorithm<\/dt>\n \t<dd id=\"fs-id1165135471195\">given a polynomial dividend[latex]\\,f\\left(x\\right)\\,[\/latex] and a non-zero polynomial divisor[latex]\\,d\\left(x\\right)\\,[\/latex] where the degree of[latex]\\,d\\left(x\\right)\\,[\/latex] is less than or equal to the degree of[latex]\\,f\\left(x\\right)[\/latex], there exist unique polynomials[latex]\\,q\\left(x\\right)\\,[\/latex] and[latex]\\,r\\left(x\\right)\\,[\/latex] such that[latex]\\,f\\left(x\\right)=d\\left(x\\right)q\\left(x\\right)+r\\left(x\\right)\\,[\/latex] where[latex]\\,q\\left(x\\right)\\,[\/latex] is the quotient and[latex]\\,r\\left(x\\right)\\,[\/latex] is the remainder. The remainder is either equal to zero or has degree strictly less than[latex]\\,d\\left(x\\right).\\,[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134486770\">\n \t<dt>synthetic division<\/dt>\n \t<dd id=\"fs-id1165134486776\">a shortcut method that can be used to divide a polynomial by a binomial of the form[latex]\\,x-k\\,[\/latex]<\/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>Use long division to divide polynomials.<\/li>\n<li>Use synthetic division to divide polynomials.<\/li>\n<\/ul>\n<\/div>\n<div id=\"Figure_03_05_001\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 488px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/975951a618866688496b5d19f096409c6269b2d1\/CNX_Precalc_Figure_03_05_001.jpg\" alt=\"Lincoln Memorial.\" width=\"488\" height=\"286\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 1. <\/strong>Lincoln Memorial, Washington, D.C. (credit: Ron Cogswell, Flickr)<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165135382145\">The exterior of the Lincoln Memorial in Washington, D.C., is a large rectangular solid with length 61.5 meters (m), width 40 m, and height 30 m.<a class=\"footnote\" title=\"National Park Service. &quot;Lincoln Memorial Building Statistics.&quot; http:\/\/www.nps.gov\/linc\/historyculture\/lincoln-memorial-building-statistics.htm. Accessed 4\/3\/2014\" id=\"return-footnote-85-1\" href=\"#footnote-85-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a> We can easily find the volume using elementary geometry.<\/p>\n<div id=\"eip-435\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill V& =& l\\cdot w\\cdot h\\hfill \\\\ & =& 61.5\\cdot 40\\cdot 30\\hfill \\\\ & =& 73,800\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165133214948\">So the volume is 73,800 cubic meters[latex]\\,\\left(\\text{m}\u00b3\\right).\\,[\/latex]<br \/>\nSuppose we knew the volume, length, and width. We could divide to find the height.<\/p>\n<div id=\"eip-312\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill h& =& \\frac{V}{l\\cdot w}\\hfill \\\\ & =& \\frac{73,800}{61.5\\cdot 40}\\hfill \\\\ & =& 30\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137892463\">As we can confirm from the dimensions above, the height is 30 m. We can use similar methods to find any of the missing dimensions. We can also use the same method if any, or all, of the measurements contain variable expressions. For example, suppose the volume of a rectangular solid is given by the polynomial[latex]\\,3{x}^{4}-3{x}^{3}-33{x}^{2}+54x.\\,[\/latex]<br \/>\nThe length of the solid is given by[latex]\\,3x;\\,[\/latex]<br \/>\nthe width is given by[latex]\\,x-2.\\,[\/latex]<br \/>\nTo find the height of the solid, we can use polynomial division, which is the focus of this section.<\/p>\n<div id=\"fs-id1165137676949\" class=\"bc-section section\">\n<h3>Using Long Division to Divide Polynomials<\/h3>\n<p id=\"fs-id1165135191647\">We are familiar with the <span class=\"no-emphasis\">long division<\/span> algorithm for ordinary arithmetic. We begin by dividing into the digits of the dividend that have the greatest place value. We divide, multiply, subtract, include the digit in the next place value position, and repeat. For example, let\u2019s divide 178 by 3 using long division.<\/p>\n<p><span id=\"fs-id1165137564295\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/a3075d8a73f65e2ad04180a05fdea0723dc8cef6\/CNX_Precalc_Figure_03_05_002.jpg\" alt=\"Steps of long division for intergers.\" \/><\/span><\/p>\n<p id=\"fs-id1165134170235\">Another way to look at the solution is as a sum of parts. This should look familiar, since it is the same method used to check division in elementary arithmetic.<\/p>\n<div id=\"eip-474\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\text{dividend}& =& (\\text{divisor}\\cdot \\text{quotient) + remainder}\\hfill \\\\ \\hfill 178& =& \\left(3\\cdot 59\\right)+1\\hfill \\\\ & =& 177+1\\hfill \\\\ & =& 178\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137640958\">We call this the <strong>Division Algorithm <\/strong>and will discuss it more formally after looking at an example.<\/p>\n<p id=\"fs-id1165137933942\">Division of polynomials that contain more than one term has similarities to long division of whole numbers. We can write a polynomial dividend as the product of the divisor and the quotient added to the remainder. The terms of the polynomial division correspond to the digits (and place values) of the whole number division. This method allows us to divide two polynomials. For example, if we were to divide[latex]\\,2{x}^{3}-3{x}^{2}+4x+5\\,[\/latex]<br \/>\nby[latex]\\,x+2\\,[\/latex]<br \/>\nusing the long division algorithm, it would look like this:<\/p>\n<p><span id=\"fs-id1678300\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/6405515967a5671cb5ff219ec9a60a7a5aa0be94\/CNX_CAT_Figure_05_01_001.jpg\" alt=\"\" \/><\/span><\/p>\n<p id=\"fs-id1165135191694\">We have found<\/p>\n<div id=\"eip-334\" class=\"unnumbered aligncenter\">[latex]\\frac{2{x}^{3}-3{x}^{2}+4x+5}{x+2}=2{x}^{2}-7x+18-\\frac{31}{x+2}[\/latex]<\/div>\n<p id=\"fs-id1165137823279\">or<\/p>\n<div id=\"eip-212\" class=\"unnumbered aligncenter\">[latex]\\frac{2{x}^{3}-3{x}^{2}+4x+5}{x+2}=\\left(x+2\\right)\\left(2{x}^{2}-7x+18\\right)-31[\/latex]<\/div>\n<p id=\"fs-id1165135181270\">We can identify the <span class=\"no-emphasis\">dividend<\/span>, the <span class=\"no-emphasis\">divisor<\/span>, the <span class=\"no-emphasis\">quotient<\/span>, and the <span class=\"no-emphasis\">remainder<\/span>.<\/p>\n<p><span id=\"fs-id1165134164979\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/fdefee7ce7430fca380493b18debaba5cf907531\/CNX_Precalc_Figure_03_05_003.jpg\" alt=\"Identifying the dividend, divisor, quotient and remainder of the polynomial 2x^3-3x^2+4x+5, which is the dividend.\" \/><\/span><\/p>\n<p id=\"fs-id1165135508592\">Writing the result in this manner illustrates the Division Algorithm.<\/p>\n<div id=\"fs-id1165135508595\" class=\"textbox key-takeaways\">\n<h3>The Division Algorithm<\/h3>\n<p id=\"fs-id1165137854177\">The Division Algorithm states that, given a polynomial dividend[latex]\\,f\\left(x\\right)\\,[\/latex] and a non-zero polynomial divisor[latex]\\,d\\left(x\\right)\\,[\/latex] where the degree of[latex]\\,d\\left(x\\right)\\,[\/latex] is less than or equal to the degree of[latex]\\,f\\left(x\\right)[\/latex], there exist unique polynomials[latex]\\,q\\left(x\\right)\\,[\/latex] and[latex]\\,r\\left(x\\right)\\,[\/latex] such that [latex]f\\left(x\\right)=d\\left(x\\right)q\\left(x\\right)+r\\left(x\\right)[\/latex] [latex]q\\left(x\\right)\\,[\/latex] is the quotient and[latex]\\,r\\left(x\\right)\\,[\/latex] is the remainder. The remainder is either equal to zero or has degree strictly less than[latex]\\,d\\left(x\\right).\\,[\/latex] If[latex]\\,r\\left(x\\right)=0,\\,[\/latex] then[latex]\\,d\\left(x\\right)\\,[\/latex] divides evenly into[latex]\\,f\\left(x\\right).\\,[\/latex] This means that, in this case, both[latex]\\,d\\left(x\\right)\\,[\/latex]<br \/>\nand[latex]\\,q\\left(x\\right)\\,[\/latex] are factors of[latex]\\,f\\left(x\\right).\\,[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135638531\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"eip-id1567242\"><strong>Given a polynomial and a binomial, use long division to divide the polynomial by the binomial.<\/strong><\/p>\n<ol id=\"eip-id1165134557348\" type=\"1\">\n<li>Set up the division problem.<\/li>\n<li>Determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor.<\/li>\n<li>Multiply the answer by the divisor and write it below the like terms of the dividend.<\/li>\n<li>Subtract the bottom <span class=\"no-emphasis\">binomial<\/span> from the top binomial.<\/li>\n<li>Bring down the next term of the dividend.<\/li>\n<li>Repeat steps 2\u20135 until reaching the last term of the dividend.<\/li>\n<li>If the remainder is non-zero, express as a fraction using the divisor as the denominator.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_05_01\" class=\"textbox examples\">\n<div id=\"fs-id1165137817675\">\n<div id=\"fs-id1165137817678\">\n<h3>Using Long Division to Divide a Second-Degree Polynomial<\/h3>\n<p id=\"fs-id1165137817683\">Divide[latex]\\,5{x}^{2}+3x-2\\,[\/latex] by[latex]\\,x+1.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135196885\" class=\"solution textbox shaded\">\n<div id=\"eip-id1165135533774\" class=\"unnumbered\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137639118\">The quotient is[latex]\\,5x-2.\\,[\/latex] The remainder is 0. We write the result as<\/p>\n<div id=\"eip-id1165137701865\" class=\"unnumbered\">[latex]\\frac{5{x}^{2}+3x-2}{x+1}=5x-2[\/latex]<\/div>\n<p id=\"fs-id1165134058382\">or<\/p>\n<div id=\"eip-id1165135533774\" class=\"unnumbered\">[latex]5{x}^{2}+3x-2=\\left(x+1\\right)\\left(5x-2\\right)[\/latex]<\/div>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165132950610\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165135372071\">This division problem had a remainder of 0. This tells us that the dividend is divided evenly by the divisor, and that the divisor is a factor of the dividend.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_05_02\" class=\"textbox examples\">\n<div id=\"fs-id1165135372082\">\n<div id=\"fs-id1165135372084\">\n<h3>Using Long Division to Divide a Third-Degree Polynomial<\/h3>\n<p id=\"fs-id1165134352552\">Divide[latex]\\,6{x}^{3}+11{x}^{2}-31x+15\\,[\/latex] by[latex]\\,3x-2.\\,[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135503914\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/5c1d58c86d624bbd19a2d68e7c4725b1ef41655f\/CNX_CAT_Figure_05_01_003.jpg\" alt=\"\" \/><\/p>\n<p id=\"fs-id1165135639821\">There is a remainder of 1. We can express the result as:<\/p>\n<div id=\"eip-id1165134294806\" class=\"unnumbered\">[latex]\\frac{6{x}^{3}+11{x}^{2}-31x+15}{3x-2}=2{x}^{2}+5x-7+\\frac{1}{3x-2}[\/latex]<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135340591\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165135340597\">We can check our work by using the Division Algorithm to rewrite the solution. Then multiply.<\/p>\n<div id=\"eip-id1165135428302\" class=\"unnumbered\">[latex]\\left(3x-2\\right)\\left(2{x}^{2}+5x-7\\right)+1=6{x}^{3}+11{x}^{2}-31x+15[\/latex]<\/div>\n<p id=\"fs-id1165135152076\">Notice, as we write our result,<\/p>\n<ul id=\"fs-id1165135152079\">\n<li>the dividend is[latex]\\,6{x}^{3}+11{x}^{2}-31x+15\\,[\/latex]<\/li>\n<li>the divisor is[latex]\\,3x-2\\,[\/latex]<\/li>\n<li>the quotient is[latex]\\,2{x}^{2}+5x-7\\,[\/latex]<\/li>\n<li>the remainder is[latex]\\,1\\,[\/latex]<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134042317\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_03_05_01\">\n<div>\n<p id=\"fs-id1165135545763\">Divide[latex]\\,16{x}^{3}-12{x}^{2}+20x-3\\,[\/latex] by[latex]\\,4x+5.\\,[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135177648\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135177649\">[latex]4{x}^{2}-8x+15-\\frac{78}{4x+5}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137932621\" class=\"bc-section section\">\n<h3>Using Synthetic Division to Divide Polynomials<\/h3>\n<p>As we\u2019ve seen, long division of polynomials can involve many steps and be quite cumbersome. <strong>Synthetic division<\/strong> is a shorthand method of dividing polynomials for the special case of dividing by a linear factor whose leading coefficient is 1.<\/p>\n<p id=\"fs-id1165137932636\">To illustrate the process, recall the example at the beginning of the section.<\/p>\n<p id=\"fs-id1165137932639\">Divide[latex]\\,2{x}^{3}-3{x}^{2}+4x+5\\,[\/latex] by[latex]\\,x+2\\,[\/latex] using the long division algorithm.<\/p>\n<p id=\"fs-id1165135170412\">The final form of the process looked like this:<\/p>\n<p><span id=\"fs-id2502523\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/3572390ee6789bc61d8076e1564b505983b5eb08\/CNX_CAT_Figure_05_01_004.jpg\" alt=\"\" \/><\/span><\/p>\n<p id=\"fs-id1165137932377\">There is a lot of repetition in the table. If we don\u2019t write the variables but, instead, line up their coefficients in columns under the division sign and also eliminate the partial products, we already have a simpler version of the entire problem.<\/p>\n<p><span id=\"fs-id1165134305375\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/6e0d0074905bf0272d1d4cbe2690e6009aa46409\/CNX_Precalc_Figure_03_05_004.jpg\" alt=\"Synthetic division of the polynomial 2x^3-3x^2+4x+5 by x+2 in which it only contains the coefficients of each polynomial.\" \/><\/span><\/p>\n<p id=\"fs-id1165134305388\">Synthetic division carries this simplification even a few more steps. Collapse the table by moving each of the rows up to fill any vacant spots. Also, instead of dividing by 2, as we would in division of whole numbers, then multiplying and subtracting the middle product, we change the sign of the \u201cdivisor\u201d to \u20132, multiply and add. The process starts by bringing down the leading coefficient.<\/p>\n<p><span id=\"fs-id1165137696374\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/e37af41cac58008922e2bf6b604119e15e778d23\/CNX_Precalc_Figure_03_05_011.jpg\" alt=\"Synthetic division of the polynomial 2x^3-3x^2+4x+5 by x+2 in which it only contains the coefficients of each polynomial.\" \/><\/span><\/p>\n<p id=\"fs-id1165137696388\">We then multiply it by the \u201cdivisor\u201d and add, repeating this process column by column, until there are no entries left. The bottom row represents the coefficients of the quotient; the last entry of the bottom row is the remainder. In this case, the quotient is[latex]\\,2{x}^{2}\u20137x+18\\,[\/latex]and the remainder is[latex]\\,\u201331.\\,[\/latex] The process will be made more clear in <a class=\"autogenerated-content\" href=\"#Example_03_05_03\">(Figure)<\/a>.<\/p>\n<div id=\"fs-id1165135383640\" class=\"textbox key-takeaways\">\n<h3>Synthetic Division<\/h3>\n<p id=\"fs-id1165135383649\">Synthetic division is a shortcut that can be used when the divisor is a binomial in the form[latex]\\,x-k\\,[\/latex]where[latex]\\,k\\,[\/latex]is a real number.<\/p>\n<p>In synthetic division, only the coefficients are used in the division process.<\/p>\n<\/div>\n<div id=\"fs-id1165135393407\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135393414\"><strong>Given two polynomials, use synthetic division to divide.<\/strong><\/p>\n<ol id=\"fs-id1165135393418\" type=\"1\">\n<li>Write[latex]\\,k\\,[\/latex] for the divisor.<\/li>\n<li>Write the coefficients of the dividend.<\/li>\n<li>Bring the lead coefficient down.<\/li>\n<li>Multiply the lead coefficient by[latex]\\,k.\\,[\/latex] Write the product in the next column.<\/li>\n<li>Add the terms of the second column.<\/li>\n<li>Multiply the result by[latex]\\,k.\\,[\/latex] Write the product in the next column.<\/li>\n<li>Repeat steps 5 and 6 for the remaining columns.<\/li>\n<li>Use the bottom numbers to write the quotient. The number in the last column is the remainder and has degree 0, the next number from the right has degree 1, the next number from the right has degree 2, and so on.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_05_03\" class=\"textbox examples\">\n<div id=\"fs-id1165135383099\">\n<div id=\"fs-id1165135383101\">\n<h3>Using Synthetic Division to Divide a Second-Degree Polynomial<\/h3>\n<p id=\"fs-id1165135383107\">Use synthetic division to divide[latex]\\,5{x}^{2}-3x-36\\,[\/latex]<br \/>\nby[latex]\\,x-3.\\,[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135177606\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135177608\">Begin by setting up the synthetic division. Write[latex]\\,k\\,[\/latex] and the coefficients.<\/p>\n<p><span id=\"fs-id1165135177629\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/9f9b7c3f2afaf1c98578471f8ec6fd5c3b34749f\/CNX_Precalc_Figure_03_05_005.jpg\" alt=\"A collapsed version of the previous synthetic division.\" \/><\/span><\/p>\n<p id=\"fs-id1165135439942\">Bring down the lead coefficient. Multiply the lead coefficient by[latex]\\,k.\\,[\/latex]<\/p>\n<p><span id=\"fs-id1165135439966\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/8b638e9a3fafd6b1e6496583a605b4a0b4858d0e\/CNX_Precalc_Figure_03_05_006.jpg\" alt=\"The set-up of the synthetic division for the polynomial 5x^2-3x-36 by x-3, which renders {5, -3, -36} by 3.\" \/><\/span><\/p>\n<p id=\"fs-id1165135179942\">Continue by adding the numbers in the second column. Multiply the resulting number by[latex]\\,k.\\,[\/latex] Write the result in the next column. Then add the numbers in the third column.<\/p>\n<p><span id=\"fs-id1165135179966\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/a77e7bc9a6d09d92cfe1e1f485fc8887a5ba4e6f\/CNX_Precalc_Figure_03_05_007.jpg\" alt=\"Multiplied by the lead coefficient, 5, in the second column, and the lead coefficient is brought down to the second row.\" \/><\/span><\/p>\n<p>The result is[latex]\\,5x+12.\\,[\/latex] The remainder is 0. So[latex]\\,x-3\\,[\/latex] is a factor of the original polynomial.<\/details>\n<\/div>\n<div id=\"fs-id1165135463242\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165135463247\">Just as with long division, we can check our work by multiplying the quotient by the divisor and adding the remainder.<\/p>\n<p id=\"fs-id1165135463251\">[latex]\\left(x-3\\right)\\left(5x+12\\right)+0=5{x}^{2}-3x-36[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_05_04\" class=\"textbox examples\">\n<div id=\"fs-id1165135549012\">\n<div id=\"fs-id1165135549014\">\n<h3>Using Synthetic Division to Divide a Third-Degree Polynomial<\/h3>\n<p id=\"fs-id1165135549019\">Use synthetic division to divide[latex]\\,4{x}^{3}+10{x}^{2}-6x-20\\,[\/latex] by[latex]\\,x+2.\\,[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135173365\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135173367\">The binomial divisor is[latex]\\,x+2\\,[\/latex] so[latex]\\,k=-2.\\,[\/latex]<br \/>\nAdd each column, multiply the result by \u20132, and repeat until the last column is reached.<\/p>\n<p><span id=\"fs-id1165134176031\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/c129e0af0a8095a5958635373b6337b02d8d41a4\/CNX_Precalc_Figure_03_05_008.jpg\" alt=\"Synthetic division of 4x^3+10x^2-6x-20 divided by x+2.\" \/><\/span><\/p>\n<p id=\"fs-id1165134433356\">The result is[latex]\\,4{x}^{2}+2x-10.\\,[\/latex] The remainder is 0. Thus,[latex]\\,x+2\\,[\/latex] is a factor of[latex]\\,4{x}^{3}+10{x}^{2}-6x-20.\\,[\/latex]<\/details>\n<\/p>\n<\/div>\n<div id=\"fs-id1165133061700\">\n<h4>Analysis<\/h4>\n<p>The graph of the polynomial function[latex]\\,f\\left(x\\right)=4{x}^{3}+10{x}^{2}-6x-20\\,[\/latex] in <a class=\"autogenerated-content\" href=\"#Figure_03_05_009\">(Figure)<\/a> shows a zero at[latex]\\,x=k=-2.\\,[\/latex] This confirms that[latex]\\,x+2\\,[\/latex] is a factor of[latex]\\,4{x}^{3}+10{x}^{2}-6x-20.\\,[\/latex]<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/4e9f36eda9dde4f8ff108084685bf7661b4038c8\/CNX_Precalc_Figure_03_05_009.jpg\" alt=\"Synthetic division of 4x^3+10x^2-6x-20 divided by x+2.\" width=\"487\" height=\"742\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 2.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_05_05\" class=\"textbox examples\">\n<div id=\"fs-id1165133260470\">\n<div id=\"fs-id1165133260472\">\n<h3>Using Synthetic Division to Divide a Fourth-Degree Polynomial<\/h3>\n<p id=\"fs-id1165135481144\">Use synthetic division to divide[latex]\\,-9{x}^{4}+10{x}^{3}+7{x}^{2}-6\\,[\/latex] by[latex]\\,x-1.\\,[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135571792\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135571794\">Notice there is no <em>x<\/em>-term. We will use a zero as the coefficient for that term.<\/p>\n<div><\/div>\n<p><span id=\"Figure_05_04_001\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/21c33f8114c651702043fec1fafe25f629656a9a\/CNX_CAT_Figure_05_04_001.jpg\" alt=\"..\" \/><\/span><\/p>\n<p>The result is[latex]\\,-9{x}^{3}+{x}^{2}+8x+8+\\frac{2}{x-1}.[\/latex]<\/p>\n<div>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134037571\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_03_05_02\">\n<div id=\"fs-id1165134037583\">\n<p id=\"fs-id1165134037584\">Use synthetic division to divide[latex]\\,3{x}^{4}+18{x}^{3}-3x+40\\,[\/latex]<br \/>\nby[latex]\\,x+7.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165133365553\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165133365554\">[latex]3{x}^{3}-3{x}^{2}+21x-150+\\frac{1,090}{x+7}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135403412\" class=\"bc-section section\">\n<h3>Using Polynomial Division to Solve Application Problems<\/h3>\n<p id=\"fs-id1165135403417\">Polynomial division can be used to solve a variety of application problems involving expressions for area and volume. We looked at an application at the beginning of this section. Now we will solve that problem in the following example.<\/p>\n<div id=\"Example_03_05_06\" class=\"textbox examples\">\n<div id=\"fs-id1165135403427\">\n<div id=\"fs-id1165135403429\">\n<h3>Using Polynomial Division in an Application Problem<\/h3>\n<p id=\"fs-id1165135403434\">The volume of a rectangular solid is given by the polynomial[latex]\\,3{x}^{4}-3{x}^{3}-33{x}^{2}+54x.\\,[\/latex] The length of the solid is given by[latex]\\,3x\\,[\/latex] and the width is given by[latex]\\,x-2.\\,[\/latex]<br \/>\nFind the height,[latex]\\,h,[\/latex]of the solid.<\/p>\n<\/div>\n<div id=\"fs-id1165135685835\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/fc1a8c7d3dd929f331a9e0b414d36568e7f722dd\/CNX_Precalc_Figure_03_05_010.jpg\" alt=\"Graph of f(x)=4x^3+10x^2-6x-20 with a close up on x+2.\" width=\"487\" height=\"140\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 3.<\/strong><\/figcaption><\/figure>\n<p id=\"fs-id1165135685837\">There are a few ways to approach this problem. We need to divide the expression for the volume of the solid by the expressions for the length and width. Let us create a sketch as in <a class=\"autogenerated-content\" href=\"#Figure_03_05_010\">(Figure)<\/a>.<\/p>\n<p id=\"fs-id1165137843229\">We can now write an equation by substituting the known values into the formula for the volume of a rectangular solid.<\/p>\n<div id=\"eip-id1165135439925\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill V& =& l\\cdot w\\cdot h\\hfill \\\\ \\hfill 3{x}^{4}-3{x}^{3}-33{x}^{2}+54x& =& 3x\\cdot \\left(x-2\\right)\\cdot h\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135457104\">To solve for[latex]\\,h,\\,[\/latex]first divide both sides by[latex]\\,3x.[\/latex]<\/p>\n<div id=\"eip-id1165135438421\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill \\frac{3x\\cdot \\left(x-2\\right)\\cdot h}{3x}& =& \\frac{3{x}^{4}-3{x}^{3}-33{x}^{2}+54x}{3x}\\hfill \\\\ \\hfill \\left(x-2\\right)h& =& {x}^{3}-{x}^{2}-11x+18\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135528878\">Now solve for[latex]\\,h\\,[\/latex] using synthetic division.<\/p>\n<div id=\"eip-id1165134103025\" class=\"unnumbered\">[latex]h=\\frac{{x}^{3}-{x}^{2}-11x+18}{x-2}[\/latex]<\/div>\n<p><span id=\"fs-id1349001\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/ba678b7589f59945aa041cce2af5b441e7d759c5\/CNX_CAT_Figure_05_01_005.jpg\" alt=\"\" \/><\/span><\/p>\n<p id=\"fs-id1165134152722\">The quotient is[latex]\\,{x}^{2}+x-9\\,[\/latex] and the remainder is 0. The height of the solid is[latex]\\,{x}^{2}+x-9.[\/latex]<span id=\"fs-id1165137843218\"><\/span><\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135694534\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_03_05_03\">\n<div id=\"fs-id1165135694546\">\n<p id=\"fs-id1165135694547\">The area of a rectangle is given by[latex]\\,3{x}^{3}+14{x}^{2}-23x+6.\\,[\/latex] The width of the rectangle is given by[latex]\\,x+6.\\,[\/latex]<br \/>\nFind an expression for the length of the rectangle.<\/p>\n<\/div>\n<div id=\"fs-id1165135407022\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135407023\">[latex]3{x}^{2}-4x+1[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135571867\" class=\"precalculus media\">\n<p id=\"fs-id1165135571875\">Access these online resources for additional instruction and practice with polynomial division.<\/p>\n<ul id=\"fs-id1165135571879\">\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/dividetribild\">Dividing a Trinomial by a Binomial Using Long Division<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/dividepolybild\">Dividing a Polynomial by a Binomial Using Long Division <\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/dividepolybisd2\">Ex 2: Dividing a Polynomial by a Binomial Using Synthetic Division<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/dividepolybisd4\">Ex 4: Dividing a Polynomial by a Binomial Using Synthetic Division<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135487276\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"eip-id1165133432926\" summary=\"..\">\n<tbody>\n<tr>\n<td>Division Algorithm<\/td>\n<td>[latex]f\\left(x\\right)=d\\left(x\\right)q\\left(x\\right)+r\\left(x\\right)\\text{ where }q\\left(x\\right)\\ne 0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165135531548\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165135531552\">\n<li>Polynomial long division can be used to divide a polynomial by any polynomial with equal or lower degree. See <a class=\"autogenerated-content\" href=\"#Example_03_05_01\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_03_05_02\">(Figure)<\/a><strong>.<\/strong><\/li>\n<li>The Division Algorithm tells us that a polynomial dividend can be written as the product of the divisor and the quotient added to the remainder.<\/li>\n<li>Synthetic division is a shortcut that can be used to divide a polynomial by a binomial in the form[latex]\\,x-k.\\,[\/latex]<br \/>\nSee <a class=\"autogenerated-content\" href=\"#Example_03_05_03\">(Figure)<\/a><strong>, <\/strong><a class=\"autogenerated-content\" href=\"#Example_03_05_04\">(Figure)<\/a><strong>, <\/strong>and <a class=\"autogenerated-content\" href=\"#Example_03_05_05\">(Figure)<\/a><strong>.<\/strong><\/li>\n<li>Polynomial division can be used to solve application problems, including area and volume. See <a class=\"autogenerated-content\" href=\"#Example_03_05_06\">(Figure)<\/a><strong>.<\/strong><\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165135255120\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165135255124\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165135255129\">\n<div id=\"fs-id1165135255131\">\n<p id=\"fs-id1165135443966\">If division of a polynomial by a binomial results in a remainder of zero, what can be conclude?<\/p>\n<\/div>\n<div id=\"fs-id1165135443970\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135443971\">The binomial is a factor of the polynomial.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135443975\">\n<div id=\"fs-id1165135443976\">\n<p id=\"fs-id1165135443977\">If a polynomial of degree[latex]\\,n\\,[\/latex] is divided by a binomial of degree 1, what is the degree of the quotient?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135443995\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1165135444000\">For the following exercises, use long division to divide. Specify the quotient and the remainder.<\/p>\n<div id=\"fs-id1165135349098\">\n<div id=\"fs-id1165135349099\">\n<p id=\"fs-id1165135349100\">[latex]\\left({x}^{2}+5x-1\\right)\u00f7\\left(x-1\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137849077\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137849078\">[latex]x+6+\\frac{5}{x-1}\\text{,}\\,\\text{quotient:}\\,x+6\\text{,}\\,\\text{remainder:}\\,\\text{5}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137932684\">\n<div>\n<p id=\"fs-id1165137932686\">[latex]\\left(2{x}^{2}-9x-5\\right)\u00f7\\left(x-5\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135643143\">\n<div id=\"fs-id1165135643144\">\n<p id=\"fs-id1165135643146\">[latex]\\left(3{x}^{2}+23x+14\\right)\u00f7\\left(x+7\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134149896\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134149897\">[latex]3x+2\\text{,}\\,\\text{quotient: }3x+2\\text{,}\\,\\text{remainder: 0}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135453058\">\n<div id=\"fs-id1165135453059\">\n<p id=\"fs-id1165135453060\">[latex]\\left(4{x}^{2}-10x+6\\right)\u00f7\\left(4x+2\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137833881\">\n<div id=\"fs-id1165137833882\">\n<p id=\"fs-id1165137833883\">[latex]\\left(6{x}^{2}-25x-25\\right)\u00f7\\left(6x+5\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135532496\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135532497\">[latex]x-5\\text{,}\\,\\text{quotient:}\\,x-5\\text{,}\\,\\text{remainder:}\\,\\text{0}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135363204\">\n<div id=\"fs-id1165135363205\">\n<p id=\"fs-id1165135363206\">[latex]\\left(-{x}^{2}-1\\right)\u00f7\\left(x+1\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134472270\">\n<div id=\"fs-id1165134472271\">\n<p id=\"fs-id1165134472272\">[latex]\\left(2{x}^{2}-3x+2\\right)\u00f7\\left(x+2\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134234239\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135694449\">[latex]2x-7+\\frac{16}{x+2}\\text{,}\\,\\text{quotient:}\\text{\u200b}\\,2x-7\\text{,}\\,\\text{remainder:}\\,\\text{16}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134352532\">\n<div id=\"fs-id1165134352533\">\n<p id=\"fs-id1165134352534\">[latex]\\left({x}^{3}-126\\right)\u00f7\\left(x-5\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135678617\">\n<div id=\"fs-id1165135678618\">\n<p id=\"fs-id1165135678619\">[latex]\\left(3{x}^{2}-5x+4\\right)\u00f7\\left(3x+1\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165132962052\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165132962053\">[latex]x-2+\\frac{6}{3x+1}\\text{,}\\,\\text{quotient:}\\,x-2\\text{,}\\,\\text{remainder:}\\,\\text{6}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133093355\">\n<div id=\"fs-id1165133093356\">\n<p id=\"fs-id1165133093357\">[latex]\\left({x}^{3}-3{x}^{2}+5x-6\\right)\u00f7\\left(x-2\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135435668\">[latex]\\left(2{x}^{3}+3{x}^{2}-4x+15\\right)\u00f7\\left(x+3\\right)[\/latex]<\/div>\n<div id=\"fs-id1165135632104\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135632105\">[latex]2{x}^{2}-3x+5\\text{,}\\,\\text{quotient:}\\,2{x}^{2}-3x+5\\text{,}\\,\\text{remainder:}\\,\\text{0}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135321927\">For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.)<\/p>\n<div id=\"fs-id1165135321931\">\n<div id=\"fs-id1165135321932\">\n<p id=\"fs-id1165135321933\">[latex]\\left(3{x}^{3}-2{x}^{2}+x-4\\right)\u00f7\\left(x+3\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134474179\">\n<div id=\"fs-id1165134474180\">\n<p id=\"fs-id1165134474181\">[latex]\\left(2{x}^{3}-6{x}^{2}-7x+6\\right)\u00f7\\left(x-4\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135298453\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135298454\">[latex]2{x}^{2}+2x+1+\\frac{10}{x-4}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135199519\">\n<div id=\"fs-id1165137680594\">\n<p id=\"fs-id1165137680595\">[latex]\\left(6{x}^{3}-10{x}^{2}-7x-15\\right)\u00f7\\left(x+1\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133233060\">\n<div id=\"fs-id1165133233062\">\n<p id=\"fs-id1165133233063\">[latex]\\left(4{x}^{3}-12{x}^{2}-5x-1\\right)\u00f7\\left(2x+1\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135486057\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135486058\">[latex]2{x}^{2}-7x+1-\\frac{2}{2x+1}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133311045\">\n<div id=\"fs-id1165133311046\">\n<p id=\"fs-id1165133311047\">[latex]\\left(9{x}^{3}-9{x}^{2}+18x+5\\right)\u00f7\\left(3x-1\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135560631\">\n<p id=\"fs-id1165135560632\">[latex]\\left(3{x}^{3}-2{x}^{2}+x-4\\right)\u00f7\\left(x+3\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135517139\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135517140\">[latex]3{x}^{2}-11x+34-\\frac{106}{x+3}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137642766\">\n<div id=\"fs-id1165137642767\">\n<p id=\"fs-id1165137642768\">[latex]\\left(-6{x}^{3}+{x}^{2}-4\\right)\u00f7\\left(2x-3\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135530633\">\n<div id=\"fs-id1165135530634\">\n<p id=\"fs-id1165135530635\">[latex]\\left(2{x}^{3}+7{x}^{2}-13x-3\\right)\u00f7\\left(2x-3\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134173709\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134173710\">[latex]{x}^{2}+5x+1[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134173742\">\n<div id=\"fs-id1165134173743\">\n<p id=\"fs-id1165134173744\">[latex]\\left(3{x}^{3}-5{x}^{2}+2x+3\\right)\u00f7\\left(x+2\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135443762\">\n<div id=\"fs-id1165135443764\">\n<p id=\"fs-id1165135443765\">[latex]\\left(4{x}^{3}-5{x}^{2}+13\\right)\u00f7\\left(x+4\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137851483\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137851484\">[latex]4{x}^{2}-21x+84-\\frac{323}{x+4}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135403522\">\n<div id=\"fs-id1165135403523\">\n<p id=\"fs-id1165135403524\">[latex]\\left({x}^{3}-3x+2\\right)\u00f7\\left(x+2\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134061948\">\n<div id=\"fs-id1165134061949\">\n<p id=\"fs-id1165134061950\">[latex]\\left({x}^{3}-21{x}^{2}+147x-343\\right)\u00f7\\left(x-7\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134156036\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134156037\">[latex]{x}^{2}-14x+49[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134156069\">\n<div id=\"fs-id1165134156070\">\n<p id=\"fs-id1165134156071\">[latex]\\left({x}^{3}-15{x}^{2}+75x-125\\right)\u00f7\\left(x-5\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135344106\">\n<div id=\"fs-id1165135344107\">\n<p id=\"fs-id1165135344108\">[latex]\\left(9{x}^{3}-x+2\\right)\u00f7\\left(3x-1\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134118337\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134118338\">[latex]3{x}^{2}+x+\\frac{2}{3x-1}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135485710\">\n<div id=\"fs-id1165135485711\">\n<p id=\"fs-id1165135485712\">[latex]\\left(6{x}^{3}-{x}^{2}+5x+2\\right)\u00f7\\left(3x+1\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135485136\">\n<div id=\"fs-id1165135485137\">\n<p id=\"fs-id1165135485138\">[latex]\\left({x}^{4}+{x}^{3}-3{x}^{2}-2x+1\\right)\u00f7\\left(x+1\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134357257\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134357258\">[latex]{x}^{3}-3x+1[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134357289\">\n<div id=\"fs-id1165134357290\">\n<p id=\"fs-id1165134357291\">[latex]\\left({x}^{4}-3{x}^{2}+1\\right)\u00f7\\left(x-1\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135390850\">\n<div id=\"fs-id1165135390852\">\n<p id=\"fs-id1165135390853\">[latex]\\left({x}^{4}+2{x}^{3}-3{x}^{2}+2x+6\\right)\u00f7\\left(x+3\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135575976\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135575977\">[latex]{x}^{3}-{x}^{2}+2[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137850245\">\n<div id=\"fs-id1165137850246\">[latex]\\left({x}^{4}-10{x}^{3}+37{x}^{2}-60x+36\\right)\u00f7\\left(x-2\\right)[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137705401\">\n<div id=\"fs-id1165137705402\">\n<p id=\"fs-id1165137705403\">[latex]\\left({x}^{4}-8{x}^{3}+24{x}^{2}-32x+16\\right)\u00f7\\left(x-2\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134261862\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134261863\">[latex]{x}^{3}-6{x}^{2}+12x-8[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134389913\">\n<div id=\"fs-id1165134389914\">\n<p id=\"fs-id1165134389915\">[latex]\\left({x}^{4}+5{x}^{3}-3{x}^{2}-13x+10\\right)\u00f7\\left(x+5\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134484904\">\n<div id=\"fs-id1165134484905\">\n<p id=\"fs-id1165134484906\">[latex]\\left({x}^{4}-12{x}^{3}+54{x}^{2}-108x+81\\right)\u00f7\\left(x-3\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135486004\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135486005\">[latex]{x}^{3}-9{x}^{2}+27x-27[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137696095\">\n<div id=\"fs-id1165137696096\">\n<p id=\"fs-id1165137696097\">[latex]\\left(4{x}^{4}-2{x}^{3}-4x+2\\right)\u00f7\\left(2x-1\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135575253\">\n<div id=\"fs-id1165135575254\">\n<p id=\"fs-id1165135575255\">[latex]\\left(4{x}^{4}+2{x}^{3}-4{x}^{2}+2x+2\\right)\u00f7\\left(2x+1\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135403348\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135403350\">[latex]2{x}^{3}-2x+2[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135353027\">For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization.<\/p>\n<div id=\"fs-id1165135353032\">\n<div id=\"fs-id1165135353033\">\n<p id=\"fs-id1165135353034\">[latex]x-2,\\,4{x}^{3}-3{x}^{2}-8x+4[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134031382\">\n<div id=\"fs-id1165134031383\">\n<p id=\"fs-id1165134031384\">[latex]x-2,\\,3{x}^{4}-6{x}^{3}-5x+10[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134431791\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134431792\">Yes[latex]\\,\\left(x-2\\right)\\left(3{x}^{3}-5\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135240972\">\n<div id=\"fs-id1165135240974\">\n<p id=\"fs-id1165135240975\">[latex]x+3,\\,-4{x}^{3}+5{x}^{2}+8[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135394052\">\n<div id=\"fs-id1165135394053\">\n<p id=\"fs-id1165135394054\">[latex]x-2,\\,4{x}^{4}-15{x}^{2}-4[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135329837\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135329838\">Yes[latex]\\,\\left(x-2\\right)\\left(4{x}^{3}+8{x}^{2}+x+2\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137851216\">\n<div id=\"fs-id1165135397213\">\n<p id=\"fs-id1165135397214\">[latex]x-\\frac{1}{2},\\,2{x}^{4}-{x}^{3}+2x-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135547391\">\n<div id=\"fs-id1165135547392\">\n<p id=\"fs-id1165135547394\">[latex]x+\\frac{1}{3},\\,3{x}^{4}+{x}^{3}-3x+1[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135481294\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135481295\">No<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135481299\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1165135481304\">For the following exercises, use the graph of the third-degree polynomial and one factor to write the factored form of the polynomial suggested by the graph. The leading coefficient is one.<\/p>\n<div id=\"fs-id1165135436617\">\n<div id=\"fs-id1165135436618\">\n<p>Factor is[latex]\\,{x}^{2}-x+3[\/latex]<\/p>\n<p><span id=\"fs-id1165135436654\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/c6d107e4540de20a05d0ec88e4a90387178d41b2\/CNX_PreCalc_Figure_03_05_201.jpg\" alt=\"Graph of a polynomial that has a x-intercept at -1.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135423510\">\n<div id=\"fs-id1165135423511\">\n<p id=\"fs-id1165135423512\">Factor is[latex]\\,\\left({x}^{2}+2x+4\\right)[\/latex]<\/p>\n<p><span id=\"fs-id1165134197955\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/7fb421522dcd4d4d0f220fdd9ceb28d54774ffcc\/CNX_PreCalc_Figure_03_05_202.jpg\" alt=\"Graph of a polynomial that has a x-intercept at 1.\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1165134197969\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134197970\">[latex]\\left(x-1\\right)\\left({x}^{2}+2x+4\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135412915\">\n<div id=\"fs-id1165135412916\">\n<p id=\"fs-id1165135412917\">Factor is[latex]\\,{x}^{2}+2x+5[\/latex]<\/p>\n<p><span id=\"fs-id1165135176349\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/a592de91e6672cd41eef495d53d6002f85a9c36a\/CNX_PreCalc_Figure_03_05_203.jpg\" alt=\"Graph of a polynomial that has a x-intercept at 2.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135176362\">\n<div id=\"fs-id1165135176363\">\n<p id=\"fs-id1165135176364\">Factor is[latex]\\,{x}^{2}+x+1[\/latex]<\/p>\n<p><span id=\"fs-id1165134164936\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/0de0bff90e32c4df60909455bf7cbd34294d97fa\/CNX_PreCalc_Figure_03_05_204.jpg\" alt=\"Graph of a polynomial that has a x-intercept at 5.\" \/><\/span><\/p>\n<\/div>\n<div id=\"fs-id1165134164949\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134164950\">[latex]\\left(x-5\\right)\\left({x}^{2}+x+1\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134089490\">\n<div id=\"fs-id1165134089491\">\n<p id=\"fs-id1165134089492\">Factor is[latex]{x}^{2}+2x+2[\/latex]<\/p>\n<p><span id=\"fs-id1165137888963\"><img decoding=\"async\" src=\"https:\/\/cnx.org\/resources\/028b1bf1cdc846cc20ae1ee2d89d755f2999dc89\/CNX_PreCalc_Figure_03_05_205.jpg\" alt=\"Graph of a polynomial that has a x-intercept at -3.\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137888975\">For the following exercises, use synthetic division to find the quotient and remainder.<\/p>\n<div id=\"fs-id1165137888978\">\n<div id=\"fs-id1165137888979\">\n<p id=\"fs-id1165137888980\">[latex]\\frac{4{x}^{3}-33}{x-2}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134061088\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134061090\">[latex]\\text{Quotient:}\\,4{x}^{2}+8x+16\\text{,}\\,\\text{remainder:}\\,-1[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135209652\">\n<div id=\"fs-id1165135209653\">\n<p id=\"fs-id1165135209654\">[latex]\\frac{2{x}^{3}+25}{x+3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135443854\">\n<div id=\"fs-id1165135443855\">\n<p id=\"fs-id1165135443856\">[latex]\\frac{3{x}^{3}+2x-5}{x-1}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137646978\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137646979\">[latex]\\text{Quotient:}\\,3{x}^{2}+3x+5\\text{,}\\,\\text{remainder:}\\,0[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135154363\">\n<div id=\"fs-id1165135154364\">\n<p id=\"fs-id1165135154366\">[latex]\\frac{-4{x}^{3}-{x}^{2}-12}{x+4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135155437\">\n<div id=\"fs-id1165135155438\">\n<p id=\"fs-id1165135155439\">[latex]\\frac{{x}^{4}-22}{x+2}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134118486\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134118487\">[latex]\\text{Quotient:}\\,{x}^{3}-2{x}^{2}+4x-8\\text{,}\\,\\text{remainder:}\\,-6[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135501101\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<p id=\"fs-id1165135501106\">For the following exercises, use a calculator with CAS to answer the questions.<\/p>\n<div id=\"fs-id1165137889809\">\n<div id=\"fs-id1165137889810\">\n<p id=\"fs-id1165137889811\">Consider[latex]\\,\\frac{{x}^{k}-1}{x-1}\\,[\/latex]with[latex]\\,k=1, 2, 3.\\,[\/latex]What do you expect the result to be if[latex]\\,k=4?[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135534969\">\n<div id=\"fs-id1165135534970\">\n<p id=\"fs-id1165135534971\">Consider[latex]\\,\\frac{{x}^{k}+1}{x+1}\\,[\/latex]for[latex]\\,k=1, 3, 5.\\,[\/latex]What do you expect the result to be if[latex]\\,k=7?[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134148458\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134148459\">[latex]{x}^{6}-{x}^{5}+{x}^{4}-{x}^{3}+{x}^{2}-x+1[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135241323\">\n<div id=\"fs-id1165135241324\">\n<p id=\"fs-id1165135241326\">Consider[latex]\\,\\frac{{x}^{4}-{k}^{4}}{x-k}\\,[\/latex]for[latex]\\,k=1, 2, 3.\\,[\/latex]What do you expect the result to be if[latex]\\,k=4?[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165132939184\">\n<div id=\"fs-id1165132939185\">\n<p id=\"fs-id1165132939186\">Consider[latex]\\,\\frac{{x}^{k}}{x+1}\\,[\/latex]with[latex]\\,k=1, 2, 3.\\,[\/latex]What do you expect the result to be if[latex]\\,k=4?[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134153037\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134153038\">[latex]{x}^{3}-{x}^{2}+x-1+\\frac{1}{x+1}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134261659\">\n<div id=\"fs-id1165134261660\">\n<p id=\"fs-id1165134261661\">Consider[latex]\\,\\frac{{x}^{k}}{x-1}\\,[\/latex]with[latex]\\,k=1, 2, 3.\\,[\/latex]What do you expect the result to be if[latex]\\,k=4?[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135191971\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<p id=\"fs-id1165135191976\">For the following exercises, use synthetic division to determine the quotient involving a complex number.<\/p>\n<div id=\"fs-id1165135191981\">\n<div id=\"fs-id1165135191982\">\n<p id=\"fs-id1165135191983\">[latex]\\frac{x+1}{x-i}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135259591\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135259592\">[latex]1+\\frac{1+i}{x-i}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135593134\">\n<p id=\"fs-id1165135593135\">[latex]\\frac{{x}^{2}+1}{x-i}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135593178\">\n<div id=\"fs-id1165135593179\">\n<p id=\"fs-id1165135593180\">[latex]\\frac{x+1}{x+i}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137834324\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137834325\">[latex]1+\\frac{1-i}{x+i}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137777664\">\n<div id=\"fs-id1165137777665\">\n<p id=\"fs-id1165137777666\">[latex]\\frac{{x}^{2}+1}{x+i}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134248021\">\n<div id=\"fs-id1165134248022\">\n<p id=\"fs-id1165134248023\">[latex]\\frac{{x}^{3}+1}{x-i}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134248065\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134248066\">[latex]{x}^{2}-ix-1+\\frac{1-i}{x-i}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135180069\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<p id=\"fs-id1165135180074\">For the following exercises, use the given length and area of a rectangle to express the width algebraically.<\/p>\n<div id=\"fs-id1165135180078\">\n<div id=\"fs-id1165135440138\">\n<p id=\"fs-id1165135440139\">Length is[latex]\\,x+5,\\,[\/latex]area is[latex]\\,2{x}^{2}+9x-5.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135440195\">\n<div id=\"fs-id1165135440196\">\n<p id=\"fs-id1165135440197\">Length is[latex]\\,2x\\text{ }+\\text{ }5,\\,[\/latex]area is[latex]\\,4{x}^{3}+10{x}^{2}+6x+15[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135264690\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135264691\">[latex]2{x}^{2}+3[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135264719\">\n<div id=\"fs-id1165135264720\">\n<p id=\"fs-id1165135264721\">Length is[latex]\\,3x\u20134,\\,[\/latex]area is[latex]\\,6{x}^{4}-8{x}^{3}+9{x}^{2}-9x-4[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135339554\">For the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically.<\/p>\n<div id=\"fs-id1165135339559\">\n<div id=\"fs-id1165135339560\">\n<p id=\"fs-id1165135339561\">Volume is[latex]\\,12{x}^{3}+20{x}^{2}-21x-36,\\,[\/latex]length is[latex]\\,2x+3,\\,[\/latex]width is[latex]\\,3x-4.[\/latex]<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135317480\">[latex]2x+3[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134071557\">\n<div id=\"fs-id1165134071558\">\n<p id=\"fs-id1165134071559\">Volume is[latex]\\,18{x}^{3}-21{x}^{2}-40x+48,\\,[\/latex]length is[latex]\\,3x\u20134,\\,[\/latex] width is[latex]\\,3x\u20134.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165132920308\">\n<div id=\"fs-id1165132920309\">\n<p id=\"fs-id1165132920310\">Volume is[latex]\\,10{x}^{3}+27{x}^{2}+2x-24,\\,[\/latex]length is[latex]\\,5x\u20134,\\,[\/latex] width is[latex]\\,2x+3.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135564161\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135564162\">[latex]x+2[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135564178\">\n<div id=\"fs-id1165135564179\">\n<p id=\"fs-id1165135564180\">Volume is[latex]\\,10{x}^{3}+30{x}^{2}-8x-24,\\,[\/latex]length is[latex]\\,2,\\,[\/latex]width is[latex]\\,x+3.[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135514530\">For the following exercises, use the given volume and radius of a cylinder to express the height of the cylinder algebraically.<\/p>\n<div id=\"fs-id1165135514534\">\n<div id=\"fs-id1165135514535\">\n<p id=\"fs-id1165135514536\">Volume is[latex]\\,\\pi \\left(25{x}^{3}-65{x}^{2}-29x-3\\right),\\,[\/latex]radius is[latex]\\,5x+1.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134031308\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134031310\">[latex]x-3[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165134031329\">\n<p>Volume is[latex]\\,\\pi \\left(4{x}^{3}+12{x}^{2}-15x-50\\right),\\,[\/latex]radius is[latex]\\,2x+5.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134258601\">\n<div id=\"fs-id1165134258602\">\n<p id=\"fs-id1165134258603\">Volume is[latex]\\,\\pi \\left(3{x}^{4}+24{x}^{3}+46{x}^{2}-16x-32\\right),\\,[\/latex]radius is[latex]\\,x+4.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137527638\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137527639\">[latex]3{x}^{2}-2[\/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-id1165135471190\">\n<dt>Division Algorithm<\/dt>\n<dd id=\"fs-id1165135471195\">given a polynomial dividend[latex]\\,f\\left(x\\right)\\,[\/latex] and a non-zero polynomial divisor[latex]\\,d\\left(x\\right)\\,[\/latex] where the degree of[latex]\\,d\\left(x\\right)\\,[\/latex] is less than or equal to the degree of[latex]\\,f\\left(x\\right)[\/latex], there exist unique polynomials[latex]\\,q\\left(x\\right)\\,[\/latex] and[latex]\\,r\\left(x\\right)\\,[\/latex] such that[latex]\\,f\\left(x\\right)=d\\left(x\\right)q\\left(x\\right)+r\\left(x\\right)\\,[\/latex] where[latex]\\,q\\left(x\\right)\\,[\/latex] is the quotient and[latex]\\,r\\left(x\\right)\\,[\/latex] is the remainder. The remainder is either equal to zero or has degree strictly less than[latex]\\,d\\left(x\\right).\\,[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134486770\">\n<dt>synthetic division<\/dt>\n<dd id=\"fs-id1165134486776\">a shortcut method that can be used to divide a polynomial by a binomial of the form[latex]\\,x-k\\,[\/latex]<\/dd>\n<\/dl>\n<\/div>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-85-1\">National Park Service. \"Lincoln Memorial Building Statistics.\" <a href=\"http:\/\/www.nps.gov\/linc\/historyculture\/lincoln-memorial-building-statistics.htm\">http:\/\/www.nps.gov\/linc\/historyculture\/lincoln-memorial-building-statistics.htm<\/a>. Accessed 4\/3\/2014 <a href=\"#return-footnote-85-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","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-85","chapter","type-chapter","status-publish","hentry"],"part":76,"_links":{"self":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/85","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\/85\/revisions"}],"predecessor-version":[{"id":86,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/85\/revisions\/86"}],"part":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/parts\/76"}],"metadata":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/85\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/media?parent=85"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapter-type?post=85"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/contributor?post=85"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/license?post=85"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}