{"id":13864,"date":"2018-08-24T22:17:51","date_gmt":"2018-08-24T22:17:51","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/precalcone\/?post_type=chapter&#038;p=13864"},"modified":"2020-05-21T04:53:44","modified_gmt":"2020-05-21T04:53:44","slug":"dividing-polynomials","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/chapter\/dividing-polynomials\/","title":{"raw":"Section 2.5: Dividing Polynomials","rendered":"Section 2.5: Dividing Polynomials"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use long division to divide polynomials.<\/li>\r\n \t<li>Use synthetic division to divide polynomials.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<figure id=\"Figure_03_05_001\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010740\/CNX_Precalc_Figure_03_05_0012.jpg\" alt=\"Lincoln Memorial.\" width=\"488\" height=\"286\" \/> <b>Figure 1.<\/b> Lincoln Memorial, Washington, D.C. (credit: Ron Cogswell, Flickr)[\/caption]<\/figure>\r\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\" target=\"_blank\" rel=\"noopener\">http:\/\/www.nps.gov\/linc\/historyculture\/lincoln-memorial-building-statistics.htm<\/a>. Accessed 4\/3\/2014[\/footnote]\u00a0We can easily find the volume using elementary geometry.<\/p>\r\n\r\n<div id=\"eip-435\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]\\begin{align}V&amp;=l\\cdot w\\cdot h \\\\ &amp;=61.5\\cdot 40\\cdot 30 \\\\ &amp;=73,800 \\end{align}[\/latex]<\/div>\r\n<p id=\"fs-id1165133214948\">So the volume is 73,800 cubic meters [latex]\\left(\\text{m}{^3} \\right)[\/latex].\u00a0Suppose we knew the volume, length, and width. We could divide to find the height.<\/p>\r\n\r\n<div id=\"eip-312\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]\\begin{align}h&amp;=\\frac{V}{l\\cdot w} \\\\ &amp;=\\frac{73,800}{61.5\\cdot 40} \\\\ &amp;=30 \\end{align}[\/latex]<\/div>\r\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].\u00a0The length of the solid is given by 3<em>x<\/em>;\u00a0the width is given by [latex]x - 2[\/latex].\u00a0To find the height of the solid, we can use polynomial division, which is the focus of this section.<\/p>\r\n\r\n<h2>Use long division to divide polynomials<\/h2>\r\n<p id=\"fs-id1165135191647\">We are familiar with the <strong>long division<\/strong> 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.<span id=\"fs-id1165137564295\">\r\n<img class=\" aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010740\/CNX_Precalc_Figure_03_05_0022.jpg\" alt=\"Long Division. Step 1, 5 times 3 equals 15 and 17 minus 15 equals 2. Step 2: Bring down the 8. Step 3: 9 times 3 equals 27 and 28 minus 27 equals 1. Answer: 59 with a remainder of 1 or 59 and one-third.\" width=\"487\" height=\"181\" \/><\/span><\/p>\r\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>\r\n\r\n<div id=\"eip-474\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]\\begin{align}\\text{dividend}&amp;= \\left(\\text{divisor }\\cdot \\text{ quotient}\\right)\\text{ + remainder} \\\\ 178&amp;=\\left(3\\cdot 59\\right)+1 \\\\ &amp;=177+1 \\\\ &amp;=178 \\end{align}[\/latex]<\/div>\r\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>\r\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]\u00a0by [latex]x+2[\/latex]\u00a0using the long division algorithm, it would look like this:<span id=\"eip-id1167404718588\">\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010741\/CNX_Precalc_revised_eq_12.png\" alt=\"Set up the division problem. 2x cubed divided by x is 2x squared. Multiply the sum of x and 2 by 2x squared. Subtract. Then bring down the next term. Negative 7x squared divided by x is negative 7x. Multiply the sum of x and 2 by negative 7x. Subtract, then bring down the next term. 18x divided by x is 18. Multiply the sum of x and 2 by 18. Subtract.\" width=\"574\" height=\"508\" \/><\/span><\/p>\r\n<p id=\"fs-id1165135191694\">We have found<\/p>\r\n\r\n<div id=\"eip-334\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]\\frac{2{x}^{3}-3{x}^{2}+4x+5}{x+2}=2{x}^{2}-7x+18-\\frac{31}{x+2}[\/latex]<\/div>\r\n<p id=\"fs-id1165137823279\">or<\/p>\r\n\r\n<div id=\"eip-212\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]2{x}^{3}-3{x}^{2}+4x+5=\\left(x+2\\right)\\left(2{x}^{2}-7x+18\\right)-31[\/latex]<\/div>\r\n<p id=\"fs-id1165135181270\">We can identify the <strong>dividend<\/strong>, the <strong>divisor<\/strong>, the <strong>quotient<\/strong>, and the <strong>remainder<\/strong>.<\/p>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010741\/CNX_Precalc_Figure_03_05_0032.jpg\" alt=\"The dividend is 2x cubed minus 3x squared plus 4x plus 5. The divisor is x plus 2. The quotient is 2x squared minus 7x plus 18. The remainder is negative 31.\" width=\"487\" height=\"99\" \/>\r\n<p id=\"fs-id1165135508592\">Writing the result in this manner illustrates the Division Algorithm.<\/p>\r\n\r\n<div id=\"fs-id1165135508595\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: The Division Algorithm<\/h3>\r\n<p id=\"fs-id1165137854177\">The <strong>Division Algorithm<\/strong> states that, given a polynomial dividend [latex]f\\left(x\\right)[\/latex]\u00a0and a non-zero polynomial divisor [latex]d\\left(x\\right)[\/latex]\u00a0where the degree of [latex]d\\left(x\\right)[\/latex]\u00a0is less than or equal to the degree of [latex]f\\left(x\\right)[\/latex],\u00a0there exist unique polynomials [latex]q\\left(x\\right)[\/latex]\u00a0and [latex]r\\left(x\\right)[\/latex]\u00a0such that<\/p>\r\n<p style=\"text-align: center\">[latex]f\\left(x\\right)=d\\left(x\\right)q\\left(x\\right)+r\\left(x\\right)[\/latex]<\/p>\r\n<p id=\"fs-id1165137664631\">[latex]q\\left(x\\right)[\/latex]\u00a0is the quotient and [latex]r\\left(x\\right)[\/latex]\u00a0is the remainder. The remainder is either equal to zero or has degree strictly less than [latex]d\\left(x\\right)[\/latex].<\/p>\r\n<p id=\"fs-id1165134540113\">If [latex]r\\left(x\\right)=0[\/latex],\u00a0then [latex]d\\left(x\\right)[\/latex]\u00a0divides evenly into [latex]f\\left(x\\right)[\/latex].\u00a0This means that, in this case, both [latex]d\\left(x\\right)[\/latex]\u00a0and [latex]q\\left(x\\right)[\/latex]\u00a0are factors of [latex]f\\left(x\\right)[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135638531\" class=\"note precalculus howto textbox\">\r\n<h3>How To: Given a polynomial and a binomial, use long division to divide the polynomial by the binomial.<\/h3>\r\n<ol id=\"eip-id1165134557348\">\r\n \t<li>Set up the division problem.<\/li>\r\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>\r\n \t<li>Multiply the answer by the divisor and write it below the like terms of the dividend.<\/li>\r\n \t<li>Subtract the bottom <strong>binomial<\/strong> from the top binomial.<\/li>\r\n \t<li>Bring down the next term of the dividend.<\/li>\r\n \t<li>Repeat steps 2\u20135 until reaching the last term of the dividend.<\/li>\r\n \t<li>If the remainder is non-zero, express as a fraction using the divisor as the denominator.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_03_05_01\" class=\"example\">\r\n<div id=\"fs-id1165137817675\" class=\"exercise\">\r\n<div id=\"fs-id1165137817678\" class=\"problem textbox shaded\">\r\n<h3>Example 1: Using Long Division to Divide a Second-Degree Polynomial<\/h3>\r\n<p id=\"fs-id1165137817683\">Divide [latex]5{x}^{2}+3x - 2[\/latex]\u00a0by [latex]x+1[\/latex].<\/p>\r\n[reveal-answer q=\"463081\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"463081\"]\r\n<h3 id=\"eip-id1170045615164\"><span id=\"eip-id1169254772605\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010741\/CNX_Precalc_revised_eq_22.png\" alt=\"Set up the division problem. 5x squared divided by x is 5x. Multiply x plus 1 by 5x. Subtract. Bring down the next term. Negative 2x divded by x is negative 2. Multiply x + 1 by negative 2. Subtract.\" width=\"460\" height=\"312\" \/><\/span><\/h3>\r\n<p id=\"fs-id1165137639118\">The quotient is [latex]5x - 2[\/latex].\u00a0The remainder is 0. We write the result as<\/p>\r\n<p style=\"text-align: center\">[latex]\\frac{5{x}^{2}+3x - 2}{x+1}=5x - 2[\/latex]<\/p>\r\n<p id=\"fs-id1165134058382\">or<\/p>\r\n<p style=\"text-align: center\">[latex]5{x}^{2}+3x - 2=\\left(x+1\\right)\\left(5x - 2\\right)[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\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>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_03_05_02\" class=\"example\">\r\n<div id=\"fs-id1165135372082\" class=\"exercise\">\r\n<div id=\"fs-id1165135372084\" class=\"problem textbox shaded\">\r\n<h3>Example 2: Using Long Division to Divide a Third-Degree Polynomial<\/h3>\r\n<p id=\"fs-id1165134352552\">Divide [latex]6{x}^{3}+11{x}^{2}-31x+15[\/latex]\u00a0by [latex]3x - 2[\/latex].<\/p>\r\n[reveal-answer q=\"458321\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"458321\"]\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012922\/replacesquareroot.png\"><img class=\"aligncenter wp-image-11885\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012922\/replacesquareroot.png\" alt=\"6x cubed divided by 3x is 2x squared. Multiply the sum of x and 2 by 2x squared. Subtract. Bring down the next term. 15x squared divided by 3x is 5x. Multiply 3x minus 2 by 5x. Subtract. Bring down the next term. Negative 21x divided by 3x is negative 7. Multiply 3x minus 2 by negative 7. Subtract. The remainder is 1.\" width=\"874\" height=\"206\" \/><\/a>\r\n<p id=\"fs-id1165135639821\">There is a remainder of 1. We can express the result as:<\/p>\r\n\r\n<div id=\"eip-id1165134294806\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]\\frac{6{x}^{3}+11{x}^{2}-31x+15}{3x - 2}=2{x}^{2}+5x - 7+\\frac{1}{3x - 2}[\/latex]<\/div>\r\n<h4>Analysis of the Solution<\/h4>\r\n<p id=\"fs-id1165135340597\">We can check our work by using the Division Algorithm to rewrite the solution. Then multiply.<\/p>\r\n\r\n<div id=\"eip-id1165135428302\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]\\left(3x - 2\\right)\\left(2{x}^{2}+5x - 7\\right)+1=6{x}^{3}+11{x}^{2}-31x+15[\/latex]<\/div>\r\n<p id=\"fs-id1165135152076\">Notice, as we write our result,<\/p>\r\n\r\n<ul>\r\n \t<li style=\"list-style-type: none\">\r\n<ul>\r\n \t<li>the dividend is [latex]6{x}^{3}+11{x}^{2}-31x+15[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<ul id=\"fs-id1165135152079\">\r\n \t<li>the divisor is [latex]3x - 2[\/latex]<\/li>\r\n \t<li>the quotient is [latex]2{x}^{2}+5x - 7[\/latex]<\/li>\r\n \t<li>the remainder is\u00a01<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165135545763\">Divide [latex]16{x}^{3}-12{x}^{2}+20x - 3[\/latex]\u00a0by [latex]4x+5[\/latex].<\/p>\r\n[reveal-answer q=\"504428\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"504428\"]\r\n\r\n[latex]4{x}^{2}-8x+15-\\frac{78}{4x+5}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]100259[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Use synthetic division to divide polynomials<\/h2>\r\n<p id=\"fs-id1165137932627\">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>\r\n<p id=\"fs-id1165137932636\">To illustrate the process, recall the example at the beginning of the section.<\/p>\r\n<p id=\"fs-id1165137932639\">Divide [latex]2{x}^{3}-3{x}^{2}+4x+5[\/latex]\u00a0by [latex]x+2[\/latex]\u00a0using the long division algorithm.<\/p>\r\n<p id=\"fs-id1165135170412\">The final form of the process looked like this:<span id=\"eip-id1163740536072\">\r\n<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010742\/CNX_Precalc_revised_eq_42.png\" alt=\".\" width=\"292\" height=\"263\" \/><\/span><\/p>\r\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>\r\n<span id=\"fs-id1165134305375\">\r\n<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010742\/CNX_Precalc_Figure_03_05_0042.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.\" width=\"522\" height=\"118\" \/><\/span>\r\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 \"divisor\" to \u20132, multiply and add. The process starts by bringing down the leading coefficient.<\/p>\r\n<span id=\"fs-id1165137696374\"><img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010742\/CNX_Precalc_Figure_03_05_0112.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.\" width=\"553\" height=\"84\" \/><\/span>\r\n<p id=\"fs-id1165137696388\">We then multiply it by the \"divisor\" 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]2x{^2} -7x+18[\/latex]\u00a0and the remainder is \u201331.\u00a0The process will be made more clear in Example 3.<\/p>\r\n\r\n<div id=\"fs-id1165135383640\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Synthetic Division<\/h3>\r\n<p id=\"fs-id1165135383649\">Synthetic division is a shortcut that can be used when the divisor is a binomial in the form <em>x<\/em> \u2013\u00a0<em>k<\/em>.\u00a0In <strong>synthetic division<\/strong>, only the coefficients are used in the division process.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135393407\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165135393414\">How To: Given two polynomials, use synthetic division to divide.<\/h3>\r\n<ol id=\"fs-id1165135393418\">\r\n \t<li>Write <em>k<\/em>\u00a0for the divisor.<\/li>\r\n \t<li>Write the coefficients of the dividend.<\/li>\r\n \t<li>Bring the lead coefficient down.<\/li>\r\n \t<li>Multiply the lead coefficient by <em>k<\/em>.\u00a0Write the product in the next column.<\/li>\r\n \t<li>Add the terms of the second column.<\/li>\r\n \t<li>Multiply the result by <em>k<\/em>.\u00a0Write the product in the next column.<\/li>\r\n \t<li>Repeat steps 5 and 6 for the remaining columns.<\/li>\r\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>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_03_05_03\" class=\"example\">\r\n<div id=\"fs-id1165135383099\" class=\"exercise\">\r\n<div id=\"fs-id1165135383101\" class=\"problem textbox shaded\">\r\n<h3>Example 3: Using Synthetic Division to Divide a Second-Degree Polynomial<\/h3>\r\n<p id=\"fs-id1165135383107\">Use synthetic division to divide [latex]5{x}^{2}-3x - 36[\/latex]\u00a0by [latex]x - 3[\/latex].<\/p>\r\n[reveal-answer q=\"125978\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"125978\"]\r\n<p id=\"fs-id1165135177608\">Begin by setting up the synthetic division. Write <em>k<\/em>\u00a0and the coefficients.<\/p>\r\n<span id=\"fs-id1165135177629\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010743\/CNX_Precalc_Figure_03_05_0052.jpg\" alt=\"A collapsed version of the previous synthetic division.\" \/><\/span>\r\n<p id=\"fs-id1165135439942\">Bring down the lead coefficient. Multiply the lead coefficient by <em>k<\/em>.<\/p>\r\n<span id=\"fs-id1165135439966\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010743\/CNX_Precalc_Figure_03_05_0062.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>\r\n<p id=\"fs-id1165135179942\">Continue by adding the numbers in the second column. Multiply the resulting number by <em>k<\/em>.\u00a0Write the result in the next column. Then add the numbers in the third column.<\/p>\r\n<span id=\"fs-id1165135179966\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010743\/CNX_Precalc_Figure_03_05_0072.jpg\" alt=\"Multiplied by the lead coefficient, 5, in the second column, and the lead coefficient is brought down to the second row. \" \/><\/span>\r\n<p id=\"fs-id1165135628639\">The result is [latex]5x+12[\/latex].\u00a0The remainder is 0. So [latex]x - 3[\/latex]\u00a0is a factor of the original polynomial.<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\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>\r\n<p id=\"fs-id1165135463251\" style=\"text-align: center\">[latex]\\left(x - 3\\right)\\left(5x+12\\right)+0=5{x}^{2}-3x - 36[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_03_05_04\" class=\"example\">\r\n<div id=\"fs-id1165135549012\" class=\"exercise\">\r\n<div id=\"fs-id1165135549014\" class=\"problem textbox shaded\">\r\n<h3>Example 4: Using Synthetic Division to Divide a Third-Degree Polynomial<\/h3>\r\n<p id=\"fs-id1165135549019\">Use synthetic division to divide [latex]4{x}^{3}+10{x}^{2}-6x - 20[\/latex]\u00a0by [latex]x+2[\/latex].<\/p>\r\n[reveal-answer q=\"388484\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"388484\"]\r\n<p id=\"fs-id1165135173367\">The binomial divisor is [latex]x+2[\/latex]\u00a0so [latex]k=-2[\/latex].\u00a0Add each column, multiply the result by \u20132, and repeat until the last column is reached.<span id=\"fs-id1165134176031\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010743\/CNX_Precalc_Figure_03_05_0082.jpg\" alt=\"Synthetic division of 4x^3+10x^2-6x-20 divided by x+2.\" \/><\/span><\/p>\r\n<p id=\"fs-id1165134433356\">The result is [latex]4{x}^{2}+2x - 10[\/latex].\u00a0The remainder is 0. Thus, [latex]x+2[\/latex]\u00a0is a factor of [latex]4{x}^{3}+10{x}^{2}-6x - 20[\/latex].<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\nThe graph of the polynomial function [latex]f\\left(x\\right)=4{x}^{3}+10{x}^{2}-6x - 20[\/latex]\u00a0in Figure 2\u00a0shows a zero at [latex]x=k=-2[\/latex].\u00a0This confirms that [latex]x+2[\/latex]\u00a0is a factor of [latex]4{x}^{3}+10{x}^{2}-6x - 20[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010744\/CNX_Precalc_Figure_03_05_0092.jpg\" alt=\"Synthetic division of 4x^3+10x^2-6x-20 divided by x+2.\" width=\"487\" height=\"742\" \/> <b>Figure 2<\/b>[\/caption]\r\n\r\n[\/hidden-answer]<b><\/b>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_03_05_05\" class=\"example\">\r\n<div id=\"fs-id1165133260470\" class=\"exercise\">\r\n<div id=\"fs-id1165133260472\" class=\"problem textbox shaded\">\r\n<h3>Example 5: Using Synthetic Division to Divide a Fourth-Degree Polynomial<\/h3>\r\n<p id=\"fs-id1165135481144\">Use synthetic division to divide [latex]-9{x}^{4}+10{x}^{3}+7{x}^{2}-6[\/latex]\u00a0by [latex]x - 1[\/latex].<\/p>\r\n[reveal-answer q=\"565402\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"565402\"]\r\n<p id=\"fs-id1165135571794\">Notice there is no <em>x<\/em>-term. We will use a zero as the coefficient for that term.<span id=\"eip-id6273758\">\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010744\/CNX_Precalc_revised_eq_52.png\" alt=\"Synthetic Division of -9x^4+10x^3+7x^2-6 by x-1\" width=\"230\" height=\"300\" \/><\/span><\/p>\r\n<p id=\"fs-id1165135341342\">The result is [latex]-9{x}^{3}+{x}^{2}+8x+8+\\frac{2}{x - 1}[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165134037584\">Use synthetic division to divide [latex]3{x}^{4}+18{x}^{3}-3x+40[\/latex]\u00a0by [latex]x+7[\/latex].<\/p>\r\n[reveal-answer q=\"620042\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"620042\"]\r\n\r\n[latex]3{x}^{3}-3{x}^{2}+21x - 150+\\frac{1,090}{x+7}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]126107[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Use polynomial division to solve application problems<\/h2>\r\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>\r\n\r\n<div id=\"Example_03_05_06\" class=\"example\">\r\n<div id=\"fs-id1165135403427\" class=\"exercise\">\r\n<div id=\"fs-id1165135403429\" class=\"problem textbox shaded\">\r\n<h3>Example 6: Using Polynomial Division in an Application Problem<\/h3>\r\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].\u00a0The length of the solid is given by 3<em>x<\/em>\u00a0and the width is given by <em>x<\/em>\u00a0\u2013 2.\u00a0Find the height of the solid.<\/p>\r\n[reveal-answer q=\"423911\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"423911\"]\r\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.<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010744\/CNX_Precalc_Figure_03_05_0102.jpg\" alt=\"Graph of f(x)=4x^3+10x^2-6x-20 with a close up on x+2.\" width=\"487\" height=\"140\" \/> <b>Figure 3<\/b>[\/caption]\r\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>\r\n<p style=\"text-align: center\">[latex]V=l\\cdot w\\cdot h[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]3{x}^{4}-3{x}^{3}-33{x}^{2}+54x=3x\\cdot \\left(x - 2\\right)\\cdot h[\/latex]<\/p>\r\n<p id=\"fs-id1165135457104\">To solve for <em>h<\/em>, first divide both sides by 3<em>x<\/em>.<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{gathered}\\frac{3x\\cdot \\left(x - 2\\right)\\cdot h}{3x}=\\frac{3{x}^{4}-3{x}^{3}-33{x}^{2}+54x}{3x} \\\\[1 mm] \\left(x - 2\\right)h={x}^{3}-{x}^{2}-11x+18\\end{gathered}[\/latex]<\/p>\r\n<p id=\"fs-id1165135528878\">Now solve for <em>h<\/em>\u00a0using synthetic division.<\/p>\r\n<p style=\"text-align: center\">[latex]h=\\frac{{x}^{3}-{x}^{2}-11x+18}{x - 2}[\/latex]<\/p>\r\n\r\n<div class=\"equation unnumbered\" style=\"text-align: center\"><\/div>\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/09\/03012930\/Screen-Shot-2015-09-11-at-2.58.28-PM.png\"><img class=\"aligncenter wp-image-13106\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/09\/03012930\/Screen-Shot-2015-09-11-at-2.58.28-PM.png\" alt=\"Synthetic division with 2 as the divisor and {1, -1, -11, 18} as the quotient. The result is {1, 1, -9, 0}\" width=\"247\" height=\"142\" \/><\/a>\r\n<p id=\"fs-id1165134152722\">The quotient is [latex]{x}^{2}+x - 9[\/latex]\u00a0and the remainder is 0. The height of the solid is [latex]{x}^{2}+x - 9[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165135694547\">The area of a rectangle is given by [latex]3{x}^{3}+14{x}^{2}-23x+6[\/latex].\u00a0The width of the rectangle is given by <em>x\u00a0<\/em>+ 6.\u00a0Find an expression for the length of the rectangle.<\/p>\r\n[reveal-answer q=\"145892\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"145892\"]\r\n\r\n[latex]3{x}^{2}-4x+1[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<section id=\"fs-id1165135487276\" class=\"key-equations\">\r\n<table id=\"eip-id1165133432926\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td>Division Algorithm<\/td>\r\n<td>[latex]f\\left(x\\right)=d\\left(x\\right)q\\left(x\\right)+r\\left(x\\right)[\/latex] where [latex]q\\left(x\\right)\\ne 0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section><section id=\"fs-id1165135531548\" class=\"key-concepts\">\r\n<h2>Key Concepts<\/h2>\r\n<ul id=\"fs-id1165135531552\">\r\n \t<li>Polynomial long division can be used to divide a polynomial by any polynomial with equal or lower degree.<\/li>\r\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>\r\n \t<li>Synthetic division is a shortcut that can be used to divide a polynomial by a binomial in the form <em>x \u2013\u00a0k<\/em>.<\/li>\r\n \t<li>Polynomial division can be used to solve application problems, including area and volume.<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165135471190\" class=\"definition\">\r\n \t<dt><strong>Division Algorithm<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135471195\">given a polynomial dividend [latex]f\\left(x\\right)[\/latex]\u00a0and a non-zero polynomial divisor [latex]d\\left(x\\right)[\/latex]\u00a0where the degree of [latex]d\\left(x\\right)[\/latex]\u00a0is less than or equal to the degree of [latex]f\\left(x\\right)[\/latex],\u00a0there exist unique polynomials [latex]q\\left(x\\right)[\/latex]\u00a0and [latex]r\\left(x\\right)[\/latex]\u00a0such that [latex]f\\left(x\\right)=d\\left(x\\right)q\\left(x\\right)+r\\left(x\\right)[\/latex]\u00a0where [latex]q\\left(x\\right)[\/latex]\u00a0is the quotient and [latex]r\\left(x\\right)[\/latex]\u00a0is the remainder. The remainder is either equal to zero or has degree strictly less than [latex]d\\left(x\\right)[\/latex].<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165134486770\" class=\"definition\">\r\n \t<dt><strong>synthetic division<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165134486776\">a shortcut method that can be used to divide a polynomial by a binomial of the form <em>x<\/em> \u2013<em> k<\/em><\/dd>\r\n<\/dl>\r\n<\/section>&nbsp;\r\n<h2 style=\"text-align: center\">Section 2.5 Homework Exercises<\/h2>\r\n1. If division of a polynomial by a binomial results in a remainder of zero, what can be conclude?\r\n\r\n2.\u00a0If a polynomial of degree <em>n<\/em>\u00a0is divided by a binomial of degree 1, what is the degree of the quotient?\r\n\r\nFor the following exercises, use long division to divide. Specify the quotient and the remainder.\r\n\r\n3. [latex]\\left({x}^{2}+5x - 1\\right)\\div \\left(x - 1\\right)[\/latex]\r\n\r\n4.\u00a0[latex]\\left(2{x}^{2}-9x - 5\\right)\\div \\left(x - 5\\right)[\/latex]\r\n\r\n5. [latex]\\left(3{x}^{2}+23x+14\\right)\\div \\left(x+7\\right)[\/latex]\r\n\r\n6.\u00a0[latex]\\left(4{x}^{2}-10x+6\\right)\\div \\left(4x+2\\right)[\/latex]\r\n\r\n7. [latex]\\left(6{x}^{2}-25x - 25\\right)\\div \\left(6x+5\\right)[\/latex]\r\n\r\n8.\u00a0[latex]\\left(-{x}^{2}-1\\right)\\div \\left(x+1\\right)[\/latex]\r\n\r\n9. [latex]\\left(2{x}^{2}-3x+2\\right)\\div \\left(x+2\\right)[\/latex]\r\n\r\n10.\u00a0[latex]\\left({x}^{3}-126\\right)\\div \\left(x - 5\\right)[\/latex]\r\n\r\n11. [latex]\\left(3{x}^{2}-5x+4\\right)\\div \\left(3x+1\\right)[\/latex]\r\n\r\n12.\u00a0[latex]\\left({x}^{3}-3{x}^{2}+5x - 6\\right)\\div \\left(x - 2\\right)[\/latex]\r\n\r\n13. [latex]\\left(2{x}^{3}+3{x}^{2}-4x+15\\right)\\div \\left(x+3\\right)[\/latex]\r\n\r\nFor the following exercises, use synthetic division to find the quotient.\r\n\r\n14. [latex]\\left(3{x}^{3}-2{x}^{2}+x - 4\\right)\\div \\left(x+3\\right)[\/latex]\r\n\r\n15. [latex]\\left(2{x}^{3}-6{x}^{2}-7x+6\\right)\\div \\left(x - 4\\right)[\/latex]\r\n\r\n16.\u00a0[latex]\\left(6{x}^{3}-10{x}^{2}-7x - 15\\right)\\div \\left(x+1\\right)[\/latex]\r\n\r\n17. [latex]\\left(4{x}^{3}-12{x}^{2}-5x - 1\\right)\\div \\left(2x+1\\right)[\/latex]\r\n\r\n18.\u00a0[latex]\\left(9{x}^{3}-9{x}^{2}+18x+5\\right)\\div \\left(3x - 1\\right)[\/latex]\r\n\r\n19. [latex]\\left(3{x}^{3}-2{x}^{2}+x - 4\\right)\\div \\left(x+3\\right)[\/latex]\r\n\r\n20.\u00a0[latex]\\left(-6{x}^{3}+{x}^{2}-4\\right)\\div \\left(2x - 3\\right)[\/latex]\r\n\r\n21. [latex]\\left(2{x}^{3}+7{x}^{2}-13x - 3\\right)\\div \\left(2x - 3\\right)[\/latex]\r\n\r\n22.\u00a0[latex]\\left(3{x}^{3}-5{x}^{2}+2x+3\\right)\\div \\left(x+2\\right)[\/latex]\r\n\r\n23. [latex]\\left(4{x}^{3}-5{x}^{2}+13\\right)\\div \\left(x+4\\right)[\/latex]\r\n\r\n24.\u00a0[latex]\\left({x}^{3}-3x+2\\right)\\div \\left(x+2\\right)[\/latex]\r\n\r\n25. [latex]\\left({x}^{3}-21{x}^{2}+147x - 343\\right)\\div \\left(x - 7\\right)[\/latex]\r\n\r\n26.\u00a0[latex]\\left({x}^{3}-15{x}^{2}+75x - 125\\right)\\div \\left(x - 5\\right)[\/latex]\r\n\r\n27. [latex]\\left(9{x}^{3}-x+2\\right)\\div \\left(3x - 1\\right)[\/latex]\r\n\r\n28.\u00a0[latex]\\left(6{x}^{3}-{x}^{2}+5x+2\\right)\\div \\left(3x+1\\right)[\/latex]\r\n\r\n29. [latex]\\left({x}^{4}+{x}^{3}-3{x}^{2}-2x+1\\right)\\div \\left(x+1\\right)[\/latex]\r\n\r\n30.\u00a0[latex]\\left({x}^{4}-3{x}^{2}+1\\right)\\div \\left(x - 1\\right)[\/latex]\r\n\r\n31. [latex]\\left({x}^{4}+2{x}^{3}-3{x}^{2}+2x+6\\right)\\div \\left(x+3\\right)[\/latex]\r\n\r\n32.\u00a0[latex]\\left({x}^{4}-10{x}^{3}+37{x}^{2}-60x+36\\right)\\div \\left(x - 2\\right)[\/latex]\r\n\r\n33. [latex]\\left({x}^{4}-8{x}^{3}+24{x}^{2}-32x+16\\right)\\div \\left(x - 2\\right)[\/latex]\r\n\r\n34.\u00a0[latex]\\left({x}^{4}+5{x}^{3}-3{x}^{2}-13x+10\\right)\\div \\left(x+5\\right)[\/latex]\r\n\r\n35. [latex]\\left({x}^{4}-12{x}^{3}+54{x}^{2}-108x+81\\right)\\div \\left(x - 3\\right)[\/latex]\r\n\r\n36.\u00a0[latex]\\left(4{x}^{4}-2{x}^{3}-4x+2\\right)\\div \\left(2x - 1\\right)[\/latex]\r\n\r\n37. [latex]\\left(4{x}^{4}+2{x}^{3}-4{x}^{2}+2x+2\\right)\\div \\left(2x+1\\right)[\/latex]\r\n\r\nFor 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.\r\n\r\n38. Factor is [latex]{x}^{2}-x+3[\/latex]\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010745\/CNX_PreCalc_Figure_03_05_2012.jpg\" alt=\"Graph of a polynomial that has a x-intercept at -1.\" \/>\r\n\r\n39. Factor is [latex]\\left({x}^{2}+2x+4\\right)[\/latex]\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010745\/CNX_PreCalc_Figure_03_05_2022.jpg\" alt=\"Graph of a polynomial that has a x-intercept at 1.\" \/>\r\n\r\n40. Factor is [latex]{x}^{2}+2x+5[\/latex]\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010745\/CNX_PreCalc_Figure_03_05_2032.jpg\" alt=\"Graph of a polynomial that has a x-intercept at 2.\" \/>\r\n\r\n41. Factor is [latex]{x}^{2}+x+1[\/latex]\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010745\/CNX_PreCalc_Figure_03_05_2042.jpg\" alt=\"Graph of a polynomial that has a x-intercept at 5.\" \/>\r\n\r\n42.\u00a0Factor is [latex]{x}^{2}+2x+2[\/latex]\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010746\/CNX_PreCalc_Figure_03_05_2052.jpg\" alt=\"Graph of a polynomial that has a x-intercept at -3.\" \/>\r\n\r\nFor the following exercises, use synthetic division to find the quotient and remainder.\r\n\r\n43. [latex]\\frac{4{x}^{3}-33}{x - 2}[\/latex]\r\n\r\n44.\u00a0[latex]\\frac{2{x}^{3}+25}{x+3}[\/latex]\r\n\r\n45. [latex]\\frac{3{x}^{3}+2x - 5}{x - 1}[\/latex]\r\n\r\n46.\u00a0[latex]\\frac{-4{x}^{3}-{x}^{2}-12}{x+4}[\/latex]\r\n\r\n47. [latex]\\frac{{x}^{4}-22}{x+2}[\/latex]\r\n\r\nFor the following exercises, use a calculator with CAS to answer the questions.\r\n\r\n48. 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 <em>k<\/em> = 4?\r\n\r\n49. 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\u00a0<em>k<\/em> = 7?\r\n\r\n50.\u00a0Consider [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\u00a0<em>k<\/em> = 4?\r\n\r\n51. Consider [latex]\\frac{{x}^{k}}{x+1}[\/latex] with [latex]k=1, 2, 3[\/latex]. What do you expect the result to be if\u00a0<em>k<\/em> = 4?\r\n\r\n52.\u00a0Consider [latex]\\frac{{x}^{k}}{x - 1}[\/latex] with [latex]k=1, 2, 3[\/latex]. What do you expect the result to be if\u00a0<em>k<\/em> = 4?\r\n\r\nFor the following exercises, use synthetic division to determine the quotient involving a complex number.\r\n\r\n53. [latex]\\frac{x+1}{x-i}[\/latex]\r\n\r\n54.\u00a0[latex]\\frac{{x}^{2}+1}{x-i}[\/latex]\r\n\r\n55. [latex]\\frac{x+1}{x+i}[\/latex]\r\n\r\n56.\u00a0[latex]\\frac{{x}^{2}+1}{x+i}[\/latex]\r\n\r\n57. [latex]\\frac{{x}^{3}+1}{x-i}[\/latex]\r\n\r\nFor the following exercises, use the given length and area of a rectangle to express the width algebraically.\r\n\r\n58. Length is [latex]x+5[\/latex], area is [latex]2{x}^{2}+9x - 5[\/latex].\r\n\r\n59. Length is [latex]2x\\text{ }+\\text{ }5[\/latex], area is [latex]4{x}^{3}+10{x}^{2}+6x+15[\/latex]\r\n\r\n60.\u00a0Length is [latex]3x - 4[\/latex], area is [latex]6{x}^{4}-8{x}^{3}+9{x}^{2}-9x - 4[\/latex]\r\n\r\nFor the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically.\r\n\r\n61. Volume is [latex]12{x}^{3}+20{x}^{2}-21x - 36[\/latex], length is [latex]2x+3[\/latex], width is [latex]3x - 4[\/latex].\r\n\r\n62.\u00a0Volume is [latex]18{x}^{3}-21{x}^{2}-40x+48[\/latex], length is [latex]3x - 4[\/latex],\u00a0width is [latex]3x - 4[\/latex].\r\n\r\n63. Volume is [latex]10{x}^{3}+27{x}^{2}+2x - 24[\/latex], length is [latex]5x - 4[\/latex],\u00a0width is [latex]2x+3[\/latex].\r\n\r\n64.\u00a0Volume is [latex]10{x}^{3}+30{x}^{2}-8x - 24[\/latex], length is 2, width is [latex]x+3[\/latex].\r\n\r\nFor the following exercises, use the given volume and radius of a cylinder to express the height of the cylinder algebraically.\r\n\r\n65. Volume is [latex]\\pi \\left(25{x}^{3}-65{x}^{2}-29x - 3\\right)[\/latex], radius is [latex]5x+1[\/latex].\r\n\r\n66.\u00a0Volume is [latex]\\pi \\left(4{x}^{3}+12{x}^{2}-15x - 50\\right)[\/latex], radius is [latex]2x+5[\/latex].\r\n\r\n67. Volume is [latex]\\pi \\left(3{x}^{4}+24{x}^{3}+46{x}^{2}-16x - 32\\right)[\/latex], radius is [latex]x+4[\/latex].","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\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<figure id=\"Figure_03_05_001\">\n<div style=\"width: 498px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010740\/CNX_Precalc_Figure_03_05_0012.jpg\" alt=\"Lincoln Memorial.\" width=\"488\" height=\"286\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 1.<\/b> Lincoln Memorial, Washington, D.C. (credit: Ron Cogswell, Flickr)<\/p>\n<\/div>\n<\/figure>\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-13864-1\" href=\"#footnote-13864-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a>\u00a0We can easily find the volume using elementary geometry.<\/p>\n<div id=\"eip-435\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]\\begin{align}V&=l\\cdot w\\cdot h \\\\ &=61.5\\cdot 40\\cdot 30 \\\\ &=73,800 \\end{align}[\/latex]<\/div>\n<p id=\"fs-id1165133214948\">So the volume is 73,800 cubic meters [latex]\\left(\\text{m}{^3} \\right)[\/latex].\u00a0Suppose we knew the volume, length, and width. We could divide to find the height.<\/p>\n<div id=\"eip-312\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]\\begin{align}h&=\\frac{V}{l\\cdot w} \\\\ &=\\frac{73,800}{61.5\\cdot 40} \\\\ &=30 \\end{align}[\/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].\u00a0The length of the solid is given by 3<em>x<\/em>;\u00a0the width is given by [latex]x - 2[\/latex].\u00a0To find the height of the solid, we can use polynomial division, which is the focus of this section.<\/p>\n<h2>Use long division to divide polynomials<\/h2>\n<p id=\"fs-id1165135191647\">We are familiar with the <strong>long division<\/strong> 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.<span id=\"fs-id1165137564295\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010740\/CNX_Precalc_Figure_03_05_0022.jpg\" alt=\"Long Division. Step 1, 5 times 3 equals 15 and 17 minus 15 equals 2. Step 2: Bring down the 8. Step 3: 9 times 3 equals 27 and 28 minus 27 equals 1. Answer: 59 with a remainder of 1 or 59 and one-third.\" width=\"487\" height=\"181\" \/><\/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=\"equation unnumbered\" style=\"text-align: center\">[latex]\\begin{align}\\text{dividend}&= \\left(\\text{divisor }\\cdot \\text{ quotient}\\right)\\text{ + remainder} \\\\ 178&=\\left(3\\cdot 59\\right)+1 \\\\ &=177+1 \\\\ &=178 \\end{align}[\/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]\u00a0by [latex]x+2[\/latex]\u00a0using the long division algorithm, it would look like this:<span id=\"eip-id1167404718588\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010741\/CNX_Precalc_revised_eq_12.png\" alt=\"Set up the division problem. 2x cubed divided by x is 2x squared. Multiply the sum of x and 2 by 2x squared. Subtract. Then bring down the next term. Negative 7x squared divided by x is negative 7x. Multiply the sum of x and 2 by negative 7x. Subtract, then bring down the next term. 18x divided by x is 18. Multiply the sum of x and 2 by 18. Subtract.\" width=\"574\" height=\"508\" \/><\/span><\/p>\n<p id=\"fs-id1165135191694\">We have found<\/p>\n<div id=\"eip-334\" class=\"equation unnumbered\" style=\"text-align: center\">[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=\"equation unnumbered\" style=\"text-align: center\">[latex]2{x}^{3}-3{x}^{2}+4x+5=\\left(x+2\\right)\\left(2{x}^{2}-7x+18\\right)-31[\/latex]<\/div>\n<p id=\"fs-id1165135181270\">We can identify the <strong>dividend<\/strong>, the <strong>divisor<\/strong>, the <strong>quotient<\/strong>, and the <strong>remainder<\/strong>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010741\/CNX_Precalc_Figure_03_05_0032.jpg\" alt=\"The dividend is 2x cubed minus 3x squared plus 4x plus 5. The divisor is x plus 2. The quotient is 2x squared minus 7x plus 18. The remainder is negative 31.\" width=\"487\" height=\"99\" \/><\/p>\n<p id=\"fs-id1165135508592\">Writing the result in this manner illustrates the Division Algorithm.<\/p>\n<div id=\"fs-id1165135508595\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: The Division Algorithm<\/h3>\n<p id=\"fs-id1165137854177\">The <strong>Division Algorithm<\/strong> states that, given a polynomial dividend [latex]f\\left(x\\right)[\/latex]\u00a0and a non-zero polynomial divisor [latex]d\\left(x\\right)[\/latex]\u00a0where the degree of [latex]d\\left(x\\right)[\/latex]\u00a0is less than or equal to the degree of [latex]f\\left(x\\right)[\/latex],\u00a0there exist unique polynomials [latex]q\\left(x\\right)[\/latex]\u00a0and [latex]r\\left(x\\right)[\/latex]\u00a0such that<\/p>\n<p style=\"text-align: center\">[latex]f\\left(x\\right)=d\\left(x\\right)q\\left(x\\right)+r\\left(x\\right)[\/latex]<\/p>\n<p id=\"fs-id1165137664631\">[latex]q\\left(x\\right)[\/latex]\u00a0is the quotient and [latex]r\\left(x\\right)[\/latex]\u00a0is the remainder. The remainder is either equal to zero or has degree strictly less than [latex]d\\left(x\\right)[\/latex].<\/p>\n<p id=\"fs-id1165134540113\">If [latex]r\\left(x\\right)=0[\/latex],\u00a0then [latex]d\\left(x\\right)[\/latex]\u00a0divides evenly into [latex]f\\left(x\\right)[\/latex].\u00a0This means that, in this case, both [latex]d\\left(x\\right)[\/latex]\u00a0and [latex]q\\left(x\\right)[\/latex]\u00a0are factors of [latex]f\\left(x\\right)[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1165135638531\" class=\"note precalculus howto textbox\">\n<h3>How To: Given a polynomial and a binomial, use long division to divide the polynomial by the binomial.<\/h3>\n<ol id=\"eip-id1165134557348\">\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 <strong>binomial<\/strong> 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=\"example\">\n<div id=\"fs-id1165137817675\" class=\"exercise\">\n<div id=\"fs-id1165137817678\" class=\"problem textbox shaded\">\n<h3>Example 1: Using Long Division to Divide a Second-Degree Polynomial<\/h3>\n<p id=\"fs-id1165137817683\">Divide [latex]5{x}^{2}+3x - 2[\/latex]\u00a0by [latex]x+1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q463081\">Show Solution<\/span><\/p>\n<div id=\"q463081\" class=\"hidden-answer\" style=\"display: none\">\n<h3 id=\"eip-id1170045615164\"><span id=\"eip-id1169254772605\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010741\/CNX_Precalc_revised_eq_22.png\" alt=\"Set up the division problem. 5x squared divided by x is 5x. Multiply x plus 1 by 5x. Subtract. Bring down the next term. Negative 2x divded by x is negative 2. Multiply x + 1 by negative 2. Subtract.\" width=\"460\" height=\"312\" \/><\/span><\/h3>\n<p id=\"fs-id1165137639118\">The quotient is [latex]5x - 2[\/latex].\u00a0The remainder is 0. We write the result as<\/p>\n<p style=\"text-align: center\">[latex]\\frac{5{x}^{2}+3x - 2}{x+1}=5x - 2[\/latex]<\/p>\n<p id=\"fs-id1165134058382\">or<\/p>\n<p style=\"text-align: center\">[latex]5{x}^{2}+3x - 2=\\left(x+1\\right)\\left(5x - 2\\right)[\/latex]<\/p>\n<h4>Analysis of the Solution<\/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>\n<\/div>\n<div id=\"Example_03_05_02\" class=\"example\">\n<div id=\"fs-id1165135372082\" class=\"exercise\">\n<div id=\"fs-id1165135372084\" class=\"problem textbox shaded\">\n<h3>Example 2: 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]\u00a0by [latex]3x - 2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q458321\">Show Solution<\/span><\/p>\n<div id=\"q458321\" class=\"hidden-answer\" style=\"display: none\">\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012922\/replacesquareroot.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-11885\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/08\/03012922\/replacesquareroot.png\" alt=\"6x cubed divided by 3x is 2x squared. Multiply the sum of x and 2 by 2x squared. Subtract. Bring down the next term. 15x squared divided by 3x is 5x. Multiply 3x minus 2 by 5x. Subtract. Bring down the next term. Negative 21x divided by 3x is negative 7. Multiply 3x minus 2 by negative 7. Subtract. The remainder is 1.\" width=\"874\" height=\"206\" \/><\/a><\/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=\"equation unnumbered\" style=\"text-align: center\">[latex]\\frac{6{x}^{3}+11{x}^{2}-31x+15}{3x - 2}=2{x}^{2}+5x - 7+\\frac{1}{3x - 2}[\/latex]<\/div>\n<h4>Analysis of the Solution<\/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=\"equation unnumbered\" style=\"text-align: center\">[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>\n<li style=\"list-style-type: none\">\n<ul>\n<li>the dividend is [latex]6{x}^{3}+11{x}^{2}-31x+15[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<ul id=\"fs-id1165135152079\">\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\u00a01<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165135545763\">Divide [latex]16{x}^{3}-12{x}^{2}+20x - 3[\/latex]\u00a0by [latex]4x+5[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q504428\">Show Solution<\/span><\/p>\n<div id=\"q504428\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]4{x}^{2}-8x+15-\\frac{78}{4x+5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm100259\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=100259&theme=oea&iframe_resize_id=ohm100259\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Use synthetic division to divide polynomials<\/h2>\n<p id=\"fs-id1165137932627\">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]\u00a0by [latex]x+2[\/latex]\u00a0using the long division algorithm.<\/p>\n<p id=\"fs-id1165135170412\">The final form of the process looked like this:<span id=\"eip-id1163740536072\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010742\/CNX_Precalc_revised_eq_42.png\" alt=\".\" width=\"292\" height=\"263\" \/><\/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\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010742\/CNX_Precalc_Figure_03_05_0042.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.\" width=\"522\" height=\"118\" \/><\/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 &#8220;divisor&#8221; to \u20132, multiply and add. The process starts by bringing down the leading coefficient.<\/p>\n<p><span id=\"fs-id1165137696374\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010742\/CNX_Precalc_Figure_03_05_0112.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.\" width=\"553\" height=\"84\" \/><\/span><\/p>\n<p id=\"fs-id1165137696388\">We then multiply it by the &#8220;divisor&#8221; 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]2x{^2} -7x+18[\/latex]\u00a0and the remainder is \u201331.\u00a0The process will be made more clear in Example 3.<\/p>\n<div id=\"fs-id1165135383640\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: 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 <em>x<\/em> \u2013\u00a0<em>k<\/em>.\u00a0In <strong>synthetic division<\/strong>, only the coefficients are used in the division process.<\/p>\n<\/div>\n<div id=\"fs-id1165135393407\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165135393414\">How To: Given two polynomials, use synthetic division to divide.<\/h3>\n<ol id=\"fs-id1165135393418\">\n<li>Write <em>k<\/em>\u00a0for 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 <em>k<\/em>.\u00a0Write the product in the next column.<\/li>\n<li>Add the terms of the second column.<\/li>\n<li>Multiply the result by <em>k<\/em>.\u00a0Write 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=\"example\">\n<div id=\"fs-id1165135383099\" class=\"exercise\">\n<div id=\"fs-id1165135383101\" class=\"problem textbox shaded\">\n<h3>Example 3: 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]\u00a0by [latex]x - 3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q125978\">Show Solution<\/span><\/p>\n<div id=\"q125978\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135177608\">Begin by setting up the synthetic division. Write <em>k<\/em>\u00a0and the coefficients.<\/p>\n<p><span id=\"fs-id1165135177629\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010743\/CNX_Precalc_Figure_03_05_0052.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 <em>k<\/em>.<\/p>\n<p><span id=\"fs-id1165135439966\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010743\/CNX_Precalc_Figure_03_05_0062.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 <em>k<\/em>.\u00a0Write the result in the next column. Then add the numbers in the third column.<\/p>\n<p><span id=\"fs-id1165135179966\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010743\/CNX_Precalc_Figure_03_05_0072.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 id=\"fs-id1165135628639\">The result is [latex]5x+12[\/latex].\u00a0The remainder is 0. So [latex]x - 3[\/latex]\u00a0is a factor of the original polynomial.<\/p>\n<h4>Analysis of the Solution<\/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\" style=\"text-align: center\">[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>\n<\/div>\n<div id=\"Example_03_05_04\" class=\"example\">\n<div id=\"fs-id1165135549012\" class=\"exercise\">\n<div id=\"fs-id1165135549014\" class=\"problem textbox shaded\">\n<h3>Example 4: 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]\u00a0by [latex]x+2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q388484\">Show Solution<\/span><\/p>\n<div id=\"q388484\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135173367\">The binomial divisor is [latex]x+2[\/latex]\u00a0so [latex]k=-2[\/latex].\u00a0Add each column, multiply the result by \u20132, and repeat until the last column is reached.<span id=\"fs-id1165134176031\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010743\/CNX_Precalc_Figure_03_05_0082.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].\u00a0The remainder is 0. Thus, [latex]x+2[\/latex]\u00a0is a factor of [latex]4{x}^{3}+10{x}^{2}-6x - 20[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>The graph of the polynomial function [latex]f\\left(x\\right)=4{x}^{3}+10{x}^{2}-6x - 20[\/latex]\u00a0in Figure 2\u00a0shows a zero at [latex]x=k=-2[\/latex].\u00a0This confirms that [latex]x+2[\/latex]\u00a0is a factor of [latex]4{x}^{3}+10{x}^{2}-6x - 20[\/latex].<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010744\/CNX_Precalc_Figure_03_05_0092.jpg\" alt=\"Synthetic division of 4x^3+10x^2-6x-20 divided by x+2.\" width=\"487\" height=\"742\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 2<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_05_05\" class=\"example\">\n<div id=\"fs-id1165133260470\" class=\"exercise\">\n<div id=\"fs-id1165133260472\" class=\"problem textbox shaded\">\n<h3>Example 5: 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]\u00a0by [latex]x - 1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q565402\">Show Solution<\/span><\/p>\n<div id=\"q565402\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135571794\">Notice there is no <em>x<\/em>-term. We will use a zero as the coefficient for that term.<span id=\"eip-id6273758\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010744\/CNX_Precalc_revised_eq_52.png\" alt=\"Synthetic Division of -9x^4+10x^3+7x^2-6 by x-1\" width=\"230\" height=\"300\" \/><\/span><\/p>\n<p id=\"fs-id1165135341342\">The result is [latex]-9{x}^{3}+{x}^{2}+8x+8+\\frac{2}{x - 1}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165134037584\">Use synthetic division to divide [latex]3{x}^{4}+18{x}^{3}-3x+40[\/latex]\u00a0by [latex]x+7[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q620042\">Show Solution<\/span><\/p>\n<div id=\"q620042\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]3{x}^{3}-3{x}^{2}+21x - 150+\\frac{1,090}{x+7}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm126107\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=126107&theme=oea&iframe_resize_id=ohm126107\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Use polynomial division to solve application problems<\/h2>\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=\"example\">\n<div id=\"fs-id1165135403427\" class=\"exercise\">\n<div id=\"fs-id1165135403429\" class=\"problem textbox shaded\">\n<h3>Example 6: 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].\u00a0The length of the solid is given by 3<em>x<\/em>\u00a0and the width is given by <em>x<\/em>\u00a0\u2013 2.\u00a0Find the height of the solid.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q423911\">Show Solution<\/span><\/p>\n<div id=\"q423911\" class=\"hidden-answer\" style=\"display: none\">\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.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010744\/CNX_Precalc_Figure_03_05_0102.jpg\" alt=\"Graph of f(x)=4x^3+10x^2-6x-20 with a close up on x+2.\" width=\"487\" height=\"140\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 3<\/b><\/p>\n<\/div>\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<p style=\"text-align: center\">[latex]V=l\\cdot w\\cdot h[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]3{x}^{4}-3{x}^{3}-33{x}^{2}+54x=3x\\cdot \\left(x - 2\\right)\\cdot h[\/latex]<\/p>\n<p id=\"fs-id1165135457104\">To solve for <em>h<\/em>, first divide both sides by 3<em>x<\/em>.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{gathered}\\frac{3x\\cdot \\left(x - 2\\right)\\cdot h}{3x}=\\frac{3{x}^{4}-3{x}^{3}-33{x}^{2}+54x}{3x} \\\\[1 mm] \\left(x - 2\\right)h={x}^{3}-{x}^{2}-11x+18\\end{gathered}[\/latex]<\/p>\n<p id=\"fs-id1165135528878\">Now solve for <em>h<\/em>\u00a0using synthetic division.<\/p>\n<p style=\"text-align: center\">[latex]h=\\frac{{x}^{3}-{x}^{2}-11x+18}{x - 2}[\/latex]<\/p>\n<div class=\"equation unnumbered\" style=\"text-align: center\"><\/div>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/09\/03012930\/Screen-Shot-2015-09-11-at-2.58.28-PM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-13106\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/09\/03012930\/Screen-Shot-2015-09-11-at-2.58.28-PM.png\" alt=\"Synthetic division with 2 as the divisor and {1, -1, -11, 18} as the quotient. The result is {1, 1, -9, 0}\" width=\"247\" height=\"142\" \/><\/a><\/p>\n<p id=\"fs-id1165134152722\">The quotient is [latex]{x}^{2}+x - 9[\/latex]\u00a0and the remainder is 0. The height of the solid is [latex]{x}^{2}+x - 9[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165135694547\">The area of a rectangle is given by [latex]3{x}^{3}+14{x}^{2}-23x+6[\/latex].\u00a0The width of the rectangle is given by <em>x\u00a0<\/em>+ 6.\u00a0Find an expression for the length of the rectangle.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q145892\">Show Solution<\/span><\/p>\n<div id=\"q145892\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]3{x}^{2}-4x+1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Key Equations<\/h2>\n<section id=\"fs-id1165135487276\" class=\"key-equations\">\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)[\/latex] where [latex]q\\left(x\\right)\\ne 0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<section id=\"fs-id1165135531548\" class=\"key-concepts\">\n<h2>Key Concepts<\/h2>\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.<\/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 <em>x \u2013\u00a0k<\/em>.<\/li>\n<li>Polynomial division can be used to solve application problems, including area and volume.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165135471190\" class=\"definition\">\n<dt><strong>Division Algorithm<\/strong><\/dt>\n<dd id=\"fs-id1165135471195\">given a polynomial dividend [latex]f\\left(x\\right)[\/latex]\u00a0and a non-zero polynomial divisor [latex]d\\left(x\\right)[\/latex]\u00a0where the degree of [latex]d\\left(x\\right)[\/latex]\u00a0is less than or equal to the degree of [latex]f\\left(x\\right)[\/latex],\u00a0there exist unique polynomials [latex]q\\left(x\\right)[\/latex]\u00a0and [latex]r\\left(x\\right)[\/latex]\u00a0such that [latex]f\\left(x\\right)=d\\left(x\\right)q\\left(x\\right)+r\\left(x\\right)[\/latex]\u00a0where [latex]q\\left(x\\right)[\/latex]\u00a0is the quotient and [latex]r\\left(x\\right)[\/latex]\u00a0is 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\" class=\"definition\">\n<dt><strong>synthetic division<\/strong><\/dt>\n<dd id=\"fs-id1165134486776\">a shortcut method that can be used to divide a polynomial by a binomial of the form <em>x<\/em> \u2013<em> k<\/em><\/dd>\n<\/dl>\n<\/section>\n<p>&nbsp;<\/p>\n<h2 style=\"text-align: center\">Section 2.5 Homework Exercises<\/h2>\n<p>1. If division of a polynomial by a binomial results in a remainder of zero, what can be conclude?<\/p>\n<p>2.\u00a0If a polynomial of degree <em>n<\/em>\u00a0is divided by a binomial of degree 1, what is the degree of the quotient?<\/p>\n<p>For the following exercises, use long division to divide. Specify the quotient and the remainder.<\/p>\n<p>3. [latex]\\left({x}^{2}+5x - 1\\right)\\div \\left(x - 1\\right)[\/latex]<\/p>\n<p>4.\u00a0[latex]\\left(2{x}^{2}-9x - 5\\right)\\div \\left(x - 5\\right)[\/latex]<\/p>\n<p>5. [latex]\\left(3{x}^{2}+23x+14\\right)\\div \\left(x+7\\right)[\/latex]<\/p>\n<p>6.\u00a0[latex]\\left(4{x}^{2}-10x+6\\right)\\div \\left(4x+2\\right)[\/latex]<\/p>\n<p>7. [latex]\\left(6{x}^{2}-25x - 25\\right)\\div \\left(6x+5\\right)[\/latex]<\/p>\n<p>8.\u00a0[latex]\\left(-{x}^{2}-1\\right)\\div \\left(x+1\\right)[\/latex]<\/p>\n<p>9. [latex]\\left(2{x}^{2}-3x+2\\right)\\div \\left(x+2\\right)[\/latex]<\/p>\n<p>10.\u00a0[latex]\\left({x}^{3}-126\\right)\\div \\left(x - 5\\right)[\/latex]<\/p>\n<p>11. [latex]\\left(3{x}^{2}-5x+4\\right)\\div \\left(3x+1\\right)[\/latex]<\/p>\n<p>12.\u00a0[latex]\\left({x}^{3}-3{x}^{2}+5x - 6\\right)\\div \\left(x - 2\\right)[\/latex]<\/p>\n<p>13. [latex]\\left(2{x}^{3}+3{x}^{2}-4x+15\\right)\\div \\left(x+3\\right)[\/latex]<\/p>\n<p>For the following exercises, use synthetic division to find the quotient.<\/p>\n<p>14. [latex]\\left(3{x}^{3}-2{x}^{2}+x - 4\\right)\\div \\left(x+3\\right)[\/latex]<\/p>\n<p>15. [latex]\\left(2{x}^{3}-6{x}^{2}-7x+6\\right)\\div \\left(x - 4\\right)[\/latex]<\/p>\n<p>16.\u00a0[latex]\\left(6{x}^{3}-10{x}^{2}-7x - 15\\right)\\div \\left(x+1\\right)[\/latex]<\/p>\n<p>17. [latex]\\left(4{x}^{3}-12{x}^{2}-5x - 1\\right)\\div \\left(2x+1\\right)[\/latex]<\/p>\n<p>18.\u00a0[latex]\\left(9{x}^{3}-9{x}^{2}+18x+5\\right)\\div \\left(3x - 1\\right)[\/latex]<\/p>\n<p>19. [latex]\\left(3{x}^{3}-2{x}^{2}+x - 4\\right)\\div \\left(x+3\\right)[\/latex]<\/p>\n<p>20.\u00a0[latex]\\left(-6{x}^{3}+{x}^{2}-4\\right)\\div \\left(2x - 3\\right)[\/latex]<\/p>\n<p>21. [latex]\\left(2{x}^{3}+7{x}^{2}-13x - 3\\right)\\div \\left(2x - 3\\right)[\/latex]<\/p>\n<p>22.\u00a0[latex]\\left(3{x}^{3}-5{x}^{2}+2x+3\\right)\\div \\left(x+2\\right)[\/latex]<\/p>\n<p>23. [latex]\\left(4{x}^{3}-5{x}^{2}+13\\right)\\div \\left(x+4\\right)[\/latex]<\/p>\n<p>24.\u00a0[latex]\\left({x}^{3}-3x+2\\right)\\div \\left(x+2\\right)[\/latex]<\/p>\n<p>25. [latex]\\left({x}^{3}-21{x}^{2}+147x - 343\\right)\\div \\left(x - 7\\right)[\/latex]<\/p>\n<p>26.\u00a0[latex]\\left({x}^{3}-15{x}^{2}+75x - 125\\right)\\div \\left(x - 5\\right)[\/latex]<\/p>\n<p>27. [latex]\\left(9{x}^{3}-x+2\\right)\\div \\left(3x - 1\\right)[\/latex]<\/p>\n<p>28.\u00a0[latex]\\left(6{x}^{3}-{x}^{2}+5x+2\\right)\\div \\left(3x+1\\right)[\/latex]<\/p>\n<p>29. [latex]\\left({x}^{4}+{x}^{3}-3{x}^{2}-2x+1\\right)\\div \\left(x+1\\right)[\/latex]<\/p>\n<p>30.\u00a0[latex]\\left({x}^{4}-3{x}^{2}+1\\right)\\div \\left(x - 1\\right)[\/latex]<\/p>\n<p>31. [latex]\\left({x}^{4}+2{x}^{3}-3{x}^{2}+2x+6\\right)\\div \\left(x+3\\right)[\/latex]<\/p>\n<p>32.\u00a0[latex]\\left({x}^{4}-10{x}^{3}+37{x}^{2}-60x+36\\right)\\div \\left(x - 2\\right)[\/latex]<\/p>\n<p>33. [latex]\\left({x}^{4}-8{x}^{3}+24{x}^{2}-32x+16\\right)\\div \\left(x - 2\\right)[\/latex]<\/p>\n<p>34.\u00a0[latex]\\left({x}^{4}+5{x}^{3}-3{x}^{2}-13x+10\\right)\\div \\left(x+5\\right)[\/latex]<\/p>\n<p>35. [latex]\\left({x}^{4}-12{x}^{3}+54{x}^{2}-108x+81\\right)\\div \\left(x - 3\\right)[\/latex]<\/p>\n<p>36.\u00a0[latex]\\left(4{x}^{4}-2{x}^{3}-4x+2\\right)\\div \\left(2x - 1\\right)[\/latex]<\/p>\n<p>37. [latex]\\left(4{x}^{4}+2{x}^{3}-4{x}^{2}+2x+2\\right)\\div \\left(2x+1\\right)[\/latex]<\/p>\n<p>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<p>38. Factor is [latex]{x}^{2}-x+3[\/latex]<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010745\/CNX_PreCalc_Figure_03_05_2012.jpg\" alt=\"Graph of a polynomial that has a x-intercept at -1.\" \/><\/p>\n<p>39. Factor is [latex]\\left({x}^{2}+2x+4\\right)[\/latex]<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010745\/CNX_PreCalc_Figure_03_05_2022.jpg\" alt=\"Graph of a polynomial that has a x-intercept at 1.\" \/><\/p>\n<p>40. Factor is [latex]{x}^{2}+2x+5[\/latex]<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010745\/CNX_PreCalc_Figure_03_05_2032.jpg\" alt=\"Graph of a polynomial that has a x-intercept at 2.\" \/><\/p>\n<p>41. Factor is [latex]{x}^{2}+x+1[\/latex]<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010745\/CNX_PreCalc_Figure_03_05_2042.jpg\" alt=\"Graph of a polynomial that has a x-intercept at 5.\" \/><\/p>\n<p>42.\u00a0Factor is [latex]{x}^{2}+2x+2[\/latex]<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010746\/CNX_PreCalc_Figure_03_05_2052.jpg\" alt=\"Graph of a polynomial that has a x-intercept at -3.\" \/><\/p>\n<p>For the following exercises, use synthetic division to find the quotient and remainder.<\/p>\n<p>43. [latex]\\frac{4{x}^{3}-33}{x - 2}[\/latex]<\/p>\n<p>44.\u00a0[latex]\\frac{2{x}^{3}+25}{x+3}[\/latex]<\/p>\n<p>45. [latex]\\frac{3{x}^{3}+2x - 5}{x - 1}[\/latex]<\/p>\n<p>46.\u00a0[latex]\\frac{-4{x}^{3}-{x}^{2}-12}{x+4}[\/latex]<\/p>\n<p>47. [latex]\\frac{{x}^{4}-22}{x+2}[\/latex]<\/p>\n<p>For the following exercises, use a calculator with CAS to answer the questions.<\/p>\n<p>48. 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 <em>k<\/em> = 4?<\/p>\n<p>49. 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\u00a0<em>k<\/em> = 7?<\/p>\n<p>50.\u00a0Consider [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\u00a0<em>k<\/em> = 4?<\/p>\n<p>51. Consider [latex]\\frac{{x}^{k}}{x+1}[\/latex] with [latex]k=1, 2, 3[\/latex]. What do you expect the result to be if\u00a0<em>k<\/em> = 4?<\/p>\n<p>52.\u00a0Consider [latex]\\frac{{x}^{k}}{x - 1}[\/latex] with [latex]k=1, 2, 3[\/latex]. What do you expect the result to be if\u00a0<em>k<\/em> = 4?<\/p>\n<p>For the following exercises, use synthetic division to determine the quotient involving a complex number.<\/p>\n<p>53. [latex]\\frac{x+1}{x-i}[\/latex]<\/p>\n<p>54.\u00a0[latex]\\frac{{x}^{2}+1}{x-i}[\/latex]<\/p>\n<p>55. [latex]\\frac{x+1}{x+i}[\/latex]<\/p>\n<p>56.\u00a0[latex]\\frac{{x}^{2}+1}{x+i}[\/latex]<\/p>\n<p>57. [latex]\\frac{{x}^{3}+1}{x-i}[\/latex]<\/p>\n<p>For the following exercises, use the given length and area of a rectangle to express the width algebraically.<\/p>\n<p>58. Length is [latex]x+5[\/latex], area is [latex]2{x}^{2}+9x - 5[\/latex].<\/p>\n<p>59. Length is [latex]2x\\text{ }+\\text{ }5[\/latex], area is [latex]4{x}^{3}+10{x}^{2}+6x+15[\/latex]<\/p>\n<p>60.\u00a0Length is [latex]3x - 4[\/latex], area is [latex]6{x}^{4}-8{x}^{3}+9{x}^{2}-9x - 4[\/latex]<\/p>\n<p>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<p>61. 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<p>62.\u00a0Volume is [latex]18{x}^{3}-21{x}^{2}-40x+48[\/latex], length is [latex]3x - 4[\/latex],\u00a0width is [latex]3x - 4[\/latex].<\/p>\n<p>63. Volume is [latex]10{x}^{3}+27{x}^{2}+2x - 24[\/latex], length is [latex]5x - 4[\/latex],\u00a0width is [latex]2x+3[\/latex].<\/p>\n<p>64.\u00a0Volume is [latex]10{x}^{3}+30{x}^{2}-8x - 24[\/latex], length is 2, width is [latex]x+3[\/latex].<\/p>\n<p>For the following exercises, use the given volume and radius of a cylinder to express the height of the cylinder algebraically.<\/p>\n<p>65. Volume is [latex]\\pi \\left(25{x}^{3}-65{x}^{2}-29x - 3\\right)[\/latex], radius is [latex]5x+1[\/latex].<\/p>\n<p>66.\u00a0Volume is [latex]\\pi \\left(4{x}^{3}+12{x}^{2}-15x - 50\\right)[\/latex], radius is [latex]2x+5[\/latex].<\/p>\n<p>67. 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\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-13864\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section><hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-13864-1\">National Park Service. \"Lincoln Memorial Building Statistics.\" <a href=\"http:\/\/www.nps.gov\/linc\/historyculture\/lincoln-memorial-building-statistics.htm\" target=\"_blank\" rel=\"noopener\">http:\/\/www.nps.gov\/linc\/historyculture\/lincoln-memorial-building-statistics.htm<\/a>. Accessed 4\/3\/2014 <a href=\"#return-footnote-13864-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":97803,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-13864","chapter","type-chapter","status-publish","hentry"],"part":10733,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/chapters\/13864","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/wp\/v2\/users\/97803"}],"version-history":[{"count":16,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/chapters\/13864\/revisions"}],"predecessor-version":[{"id":17573,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/chapters\/13864\/revisions\/17573"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/parts\/10733"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/chapters\/13864\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/wp\/v2\/media?parent=13864"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=13864"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/wp\/v2\/contributor?post=13864"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/wp\/v2\/license?post=13864"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}