{"id":342,"date":"2021-06-04T00:05:46","date_gmt":"2021-06-04T00:05:46","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/read-multiply-any-two-polynomials\/"},"modified":"2022-01-03T23:14:44","modified_gmt":"2022-01-03T23:14:44","slug":"read-multiply-any-two-polynomials","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/read-multiply-any-two-polynomials\/","title":{"raw":"8.4.4: Multiplying Polynomials","rendered":"8.4.4: Multiplying Polynomials"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcome<\/h3>\r\n<ul>\r\n \t<li>Multiply polynomials<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Multiplying Polynomials<\/h2>\r\nIn the last two sections we learned how to use the Distributive Property to multiply monomials and binomials. The Distributive Property can be expanded to find the product of any two polynomials; each term in the first polynomial must be multiplied into each term in the second polynomial.\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\nFind the product:\u00a0 [latex](3x+4)(2x^2-3x-8)[\/latex]\r\n<h4>Solution<\/h4>\r\nDistributive Property:\r\n\r\n[latex]\\begin{equation}\\begin{aligned}&amp;\\;\\;\\;\\;(\\color{red}{3x}\\color{blue}{+4})(2x^2-3x-8)\\\\&amp;=\\color{red}{3x}(2x^2)+\\color{red}{3x}(-3x)+\\color{red}{3x}(-8)\\color{blue}{+4}(2x^2)\\color{blue}{+4}(-3x)\\color{blue}{+4}(-8)\\\\&amp;=6x^3-9x^2-24x+8x^2-12x-32\\\\&amp;=6x^3-x^2-36x-32\\end{aligned}\\end{equation}[\/latex]\r\n\r\nAnswer\r\n\r\n[latex](3x+4)(2x^2-3x-8)=6x^3-x^2-36x-32[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\nFind the product:\u00a0 [latex](5x^2-x+3)(2x^2+4x-7)[\/latex]\r\n<h4>Solution<\/h4>\r\nDistributive Property:\r\n\r\n[latex]\\begin{equation}\\begin{aligned}&amp;\\;\\;\\;\\;(\\color{red}{5x^2}\\color{blue}{-x}+\\color{green}{3})(2x^2+4x-7)\\\\&amp;=\\color{red}{5x^2}(2x^2)+\\color{red}{5x^2}(4x)+\\color{red}{5x^2}(-7)\\color{blue}{-x}(2x^2)\\color{blue}{-x}(4x)\\color{blue}{-x}(-7)+\\color{green}{3}(2x^2)+\\color{green}{3}(4x)+\\color{green}{3}(-7)\\\\&amp;=10x^4+20x^3-35x^2-2x^3-4x^2+7x+6x^2+12x-21\\\\&amp;=10x^4+20x^3-2x^3-35x^2-4x^2+6x^2+7x+12x-21\\\\&amp;=10x^4+18x^3-33x^2+19x-21\\end{aligned}\\end{equation}[\/latex]\r\n\r\nAnswer\r\n\r\n[latex](5x^2-x+3)(2x^2+4x-7)=10x^4+18x^3-33x^2+19x-21[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nFind the product:\u00a0 [latex](x+5)(x^2-3x+4)[\/latex]\r\n\r\n[reveal-answer q=\"hjm611\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm611\"][latex]x^3+2x^2-11x+20[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nFind the product:\u00a0 [latex](2x^2-3x+1)(x^2-4x+3)[\/latex]\r\n\r\n[reveal-answer q=\"hjm931\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm931\"][latex]2x^4-11x^3+19x^2-13x+3[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146218[\/ohm_question]\r\n\r\n&nbsp;\r\n\r\n[ohm_question]146217[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nThe vertical method can also be extended to any size polynomial. The next example shows the multiplication of a trinomial and a binomial.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply using the Vertical Method: [latex]\\left(x+3\\right)\\left(2{x}^{2}-5x+8\\right)[\/latex]\r\n<h4>Solution<\/h4>\r\nIt is easier to put the polynomial with fewer terms on the bottom because we get fewer partial products this way.\r\n<table id=\"eip-id1168469786876\" class=\"unnumbered unstyled\" style=\"height: 142px; width: 494px;\" summary=\"A vertical multiplication problem is shown. 2 x squared minus 5x plus 8 times x plus 3 is written, with a line beneath it. The next line says, \" width=\"546\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 369.688px;\"><\/td>\r\n<td style=\"width: 207.234px;\"><img class=\"alignright\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224533\/CNX_BMath_Figure_10_03_062_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 369.688px;\">Multiply [latex]\\left(2{x}^{2}-5x+8\\right)[\/latex] by [latex]3[\/latex].<\/td>\r\n<td style=\"width: 207.234px; text-align: right;\">[latex]6x^2-15x+24[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 369.688px;\">Multiply [latex]\\left(2{x}^{2}-5x+8\\right)[\/latex] by [latex]x[\/latex] .<\/td>\r\n<td style=\"width: 207.234px; text-align: right;\">[latex]2x^3-5x^2+\\;8x\\;\\;\\;\\;\\;\\;\\;\\;[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 369.688px;\">Add like terms.<\/td>\r\n<td style=\"width: 207.234px; text-align: right;\">[latex]2x^3\\;+\\;\\;x^2-\\;7x+24[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146217[\/ohm_question]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nWatching signs and keeping track of all the terms require organization and attention to detail.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind the product.\r\n\r\n[latex]\\left(2x+1\\right)\\left(3{x}^{2}-x+4\\right)[\/latex]\r\n<h4>Solution<\/h4>\r\n[latex]\\begin{array}{cc}2x\\left(3{x}^{2}-x+4\\right)+1\\left(3{x}^{2}-x+4\\right) \\hfill &amp; \\text{Use the distributive property}.\\hfill \\\\ \\left(6{x}^{3}-2{x}^{2}+8x\\right)+\\left(3{x}^{2}-x+4\\right)\\hfill &amp; \\text{Multiply}.\\hfill \\\\ 6{x}^{3}+\\left(-2{x}^{2}+3{x}^{2}\\right)+\\left(8x-x\\right)+4\\hfill &amp; \\text{Combine like terms}.\\hfill \\\\ 6{x}^{3}+{x}^{2}+7x+4 \\hfill &amp; \\text{Simplify}.\\hfill \\end{array}[\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div>\r\n\r\nAnother way to keep track of all the terms involved in the above product is to use a table. Write one polynomial across the top and the other down the side. For each box in the table, multiply the term for that row by the term for that column. Then add all of the terms together, combine like terms, and simplify. Notice how we kept the sign on each term; for example, we are subtracting [latex]x[\/latex] from [latex]3x^2[\/latex], so we place [latex]-x[\/latex] in the table.\r\n<table style=\"width: 30%;\" summary=\"A table with 3 rows and 4 columns. The first entry of the first row is empty, the others are labeled: three times x squared, negative x, and positive four. The first entry of the second row is labeled: two times x. The second entry reads: six times x cubed. The third entry reads: negative two times x squared. The fourth entry reads: eight times x. The first entry of the third row reads: positive one. The second entry reads: three times x squared. The third entry reads: negative x. The fourth entry reads: four.\">\r\n<thead>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]3{x}^{2}[\/latex]<\/td>\r\n<td>[latex]-x[\/latex]<\/td>\r\n<td>[latex]+4[\/latex]<\/td>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]2x[\/latex]<\/td>\r\n<td>[latex]6{x}^{3}[\/latex]<\/td>\r\n<td>[latex]-2{x}^{2}[\/latex]<\/td>\r\n<td>[latex]8x[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]+1[\/latex]<\/td>\r\n<td>[latex]3{x}^{2}[\/latex]<\/td>\r\n<td>[latex]-x[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nMultiply. \u00a0[latex]\\left(2p-1\\right)\\left(3p^{2}-3p+1\\right)[\/latex]\r\n<h4>Solution<\/h4>\r\nDistribute\u00a0[latex]2p[\/latex] and\u00a0[latex]-1[\/latex] to each term in the trinomial.\r\n<p style=\"text-align: left;\">[latex]2p\\left(3p^{2}-3p+1\\right)-1\\left(3p^{2}-3p+1\\right)[\/latex]<\/p>\r\n<p style=\"text-align: left;\">[latex]2p\\left(3p^{2}\\right)+2p\\left(-3p\\right)+2p\\left(1\\right)-1\\left(3p^{2}\\right)-1\\left(-3p\\right)-1\\left(1\\right)[\/latex]<\/p>\r\nMultiply. Notice that the subtracted\u00a0[latex]1[\/latex] and the subtracted\u00a0[latex]3p[\/latex] have a positive product that is added.\r\n<p style=\"text-align: left;\">[latex]6p^{3}-6p^{2}+2p-3p^{2}+3p-1[\/latex]<\/p>\r\nCombine like terms.\r\n<p style=\"text-align: left;\">[latex]6p^{3}-9p^{2}+5p-1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nMultiply. \u00a0[latex]\\left(2y-5\\right)\\left(y^{2}-2y+3\\right)[\/latex]\r\n\r\n[reveal-answer q=\"hjm988\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm988\"]\r\n\r\n<span style=\"font-size: 1rem; text-align: initial;\">[latex]2y^3-9y^2+16y-15[\/latex]<\/span>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nThe following video shows more examples of multiplying polynomials.\r\n\r\nhttps:\/\/youtu.be\/bBKbldmlbqI\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nMultiply:\u00a0 [latex]2x(x+3)(x-3)[\/latex]\r\n\r\n[reveal-answer q=\"hjm493\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm493\"][latex]2x^2+18x[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nMultiply:\u00a0 [latex]x^3(x+4)(x^2-3x+1))[\/latex]\r\n\r\n[reveal-answer q=\"hjm106\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm106\"]\r\n\r\n[latex]x^6+x^5-11x^4+4x^3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2>Summary<\/h2>\r\nMultiplication of binomials and polynomials requires an understanding of the distributive property, rules for exponents, and a keen eye for collecting like terms. Whether the polynomials are monomials, binomials, or trinomials, carefully multiply each term in one polynomial by each term in the other polynomial. Be careful to watch the addition and subtraction signs and negative coefficients. A product is written in standard form when all of its like terms have been combined and the resulting terms are written in descending order.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcome<\/h3>\n<ul>\n<li>Multiply polynomials<\/li>\n<\/ul>\n<\/div>\n<h2>Multiplying Polynomials<\/h2>\n<p>In the last two sections we learned how to use the Distributive Property to multiply monomials and binomials. The Distributive Property can be expanded to find the product of any two polynomials; each term in the first polynomial must be multiplied into each term in the second polynomial.<\/p>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<p>Find the product:\u00a0 [latex](3x+4)(2x^2-3x-8)[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p>Distributive Property:<\/p>\n<p>[latex]\\begin{equation}\\begin{aligned}&\\;\\;\\;\\;(\\color{red}{3x}\\color{blue}{+4})(2x^2-3x-8)\\\\&=\\color{red}{3x}(2x^2)+\\color{red}{3x}(-3x)+\\color{red}{3x}(-8)\\color{blue}{+4}(2x^2)\\color{blue}{+4}(-3x)\\color{blue}{+4}(-8)\\\\&=6x^3-9x^2-24x+8x^2-12x-32\\\\&=6x^3-x^2-36x-32\\end{aligned}\\end{equation}[\/latex]<\/p>\n<p>Answer<\/p>\n<p>[latex](3x+4)(2x^2-3x-8)=6x^3-x^2-36x-32[\/latex]<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<p>Find the product:\u00a0 [latex](5x^2-x+3)(2x^2+4x-7)[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p>Distributive Property:<\/p>\n<p>[latex]\\begin{equation}\\begin{aligned}&\\;\\;\\;\\;(\\color{red}{5x^2}\\color{blue}{-x}+\\color{green}{3})(2x^2+4x-7)\\\\&=\\color{red}{5x^2}(2x^2)+\\color{red}{5x^2}(4x)+\\color{red}{5x^2}(-7)\\color{blue}{-x}(2x^2)\\color{blue}{-x}(4x)\\color{blue}{-x}(-7)+\\color{green}{3}(2x^2)+\\color{green}{3}(4x)+\\color{green}{3}(-7)\\\\&=10x^4+20x^3-35x^2-2x^3-4x^2+7x+6x^2+12x-21\\\\&=10x^4+20x^3-2x^3-35x^2-4x^2+6x^2+7x+12x-21\\\\&=10x^4+18x^3-33x^2+19x-21\\end{aligned}\\end{equation}[\/latex]<\/p>\n<p>Answer<\/p>\n<p>[latex](5x^2-x+3)(2x^2+4x-7)=10x^4+18x^3-33x^2+19x-21[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Find the product:\u00a0 [latex](x+5)(x^2-3x+4)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm611\">Show Answer<\/span><\/p>\n<div id=\"qhjm611\" class=\"hidden-answer\" style=\"display: none\">[latex]x^3+2x^2-11x+20[\/latex]<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Find the product:\u00a0 [latex](2x^2-3x+1)(x^2-4x+3)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm931\">Show Answer<\/span><\/p>\n<div id=\"qhjm931\" class=\"hidden-answer\" style=\"display: none\">[latex]2x^4-11x^3+19x^2-13x+3[\/latex]<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146218\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146218&theme=oea&iframe_resize_id=ohm146218&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146217\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146217&theme=oea&iframe_resize_id=ohm146217&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The vertical method can also be extended to any size polynomial. The next example shows the multiplication of a trinomial and a binomial.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply using the Vertical Method: [latex]\\left(x+3\\right)\\left(2{x}^{2}-5x+8\\right)[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p>It is easier to put the polynomial with fewer terms on the bottom because we get fewer partial products this way.<\/p>\n<table id=\"eip-id1168469786876\" class=\"unnumbered unstyled\" style=\"height: 142px; width: 494px; width: 546px;\" summary=\"A vertical multiplication problem is shown. 2 x squared minus 5x plus 8 times x plus 3 is written, with a line beneath it. The next line says,\">\n<tbody>\n<tr>\n<td style=\"width: 369.688px;\"><\/td>\n<td style=\"width: 207.234px;\"><img decoding=\"async\" class=\"alignright\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224533\/CNX_BMath_Figure_10_03_062_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 369.688px;\">Multiply [latex]\\left(2{x}^{2}-5x+8\\right)[\/latex] by [latex]3[\/latex].<\/td>\n<td style=\"width: 207.234px; text-align: right;\">[latex]6x^2-15x+24[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 369.688px;\">Multiply [latex]\\left(2{x}^{2}-5x+8\\right)[\/latex] by [latex]x[\/latex] .<\/td>\n<td style=\"width: 207.234px; text-align: right;\">[latex]2x^3-5x^2+\\;8x\\;\\;\\;\\;\\;\\;\\;\\;[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 369.688px;\">Add like terms.<\/td>\n<td style=\"width: 207.234px; text-align: right;\">[latex]2x^3\\;+\\;\\;x^2-\\;7x+24[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146217\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146217&theme=oea&iframe_resize_id=ohm146217&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Watching signs and keeping track of all the terms require organization and attention to detail.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find the product.<\/p>\n<p>[latex]\\left(2x+1\\right)\\left(3{x}^{2}-x+4\\right)[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p>[latex]\\begin{array}{cc}2x\\left(3{x}^{2}-x+4\\right)+1\\left(3{x}^{2}-x+4\\right) \\hfill & \\text{Use the distributive property}.\\hfill \\\\ \\left(6{x}^{3}-2{x}^{2}+8x\\right)+\\left(3{x}^{2}-x+4\\right)\\hfill & \\text{Multiply}.\\hfill \\\\ 6{x}^{3}+\\left(-2{x}^{2}+3{x}^{2}\\right)+\\left(8x-x\\right)+4\\hfill & \\text{Combine like terms}.\\hfill \\\\ 6{x}^{3}+{x}^{2}+7x+4 \\hfill & \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<div>\n<p>Another way to keep track of all the terms involved in the above product is to use a table. Write one polynomial across the top and the other down the side. For each box in the table, multiply the term for that row by the term for that column. Then add all of the terms together, combine like terms, and simplify. Notice how we kept the sign on each term; for example, we are subtracting [latex]x[\/latex] from [latex]3x^2[\/latex], so we place [latex]-x[\/latex] in the table.<\/p>\n<table style=\"width: 30%;\" summary=\"A table with 3 rows and 4 columns. The first entry of the first row is empty, the others are labeled: three times x squared, negative x, and positive four. The first entry of the second row is labeled: two times x. The second entry reads: six times x cubed. The third entry reads: negative two times x squared. The fourth entry reads: eight times x. The first entry of the third row reads: positive one. The second entry reads: three times x squared. The third entry reads: negative x. The fourth entry reads: four.\">\n<thead>\n<tr>\n<td><\/td>\n<td>[latex]3{x}^{2}[\/latex]<\/td>\n<td>[latex]-x[\/latex]<\/td>\n<td>[latex]+4[\/latex]<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]2x[\/latex]<\/td>\n<td>[latex]6{x}^{3}[\/latex]<\/td>\n<td>[latex]-2{x}^{2}[\/latex]<\/td>\n<td>[latex]8x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]+1[\/latex]<\/td>\n<td>[latex]3{x}^{2}[\/latex]<\/td>\n<td>[latex]-x[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Multiply. \u00a0[latex]\\left(2p-1\\right)\\left(3p^{2}-3p+1\\right)[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p>Distribute\u00a0[latex]2p[\/latex] and\u00a0[latex]-1[\/latex] to each term in the trinomial.<\/p>\n<p style=\"text-align: left;\">[latex]2p\\left(3p^{2}-3p+1\\right)-1\\left(3p^{2}-3p+1\\right)[\/latex]<\/p>\n<p style=\"text-align: left;\">[latex]2p\\left(3p^{2}\\right)+2p\\left(-3p\\right)+2p\\left(1\\right)-1\\left(3p^{2}\\right)-1\\left(-3p\\right)-1\\left(1\\right)[\/latex]<\/p>\n<p>Multiply. Notice that the subtracted\u00a0[latex]1[\/latex] and the subtracted\u00a0[latex]3p[\/latex] have a positive product that is added.<\/p>\n<p style=\"text-align: left;\">[latex]6p^{3}-6p^{2}+2p-3p^{2}+3p-1[\/latex]<\/p>\n<p>Combine like terms.<\/p>\n<p style=\"text-align: left;\">[latex]6p^{3}-9p^{2}+5p-1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Multiply. \u00a0[latex]\\left(2y-5\\right)\\left(y^{2}-2y+3\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm988\">Show Answer<\/span><\/p>\n<div id=\"qhjm988\" class=\"hidden-answer\" style=\"display: none\">\n<p><span style=\"font-size: 1rem; text-align: initial;\">[latex]2y^3-9y^2+16y-15[\/latex]<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The following video shows more examples of multiplying polynomials.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"(New Version Available) Polynomial Multiplication Involving Binomials and Trinomials\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/bBKbldmlbqI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Multiply:\u00a0 [latex]2x(x+3)(x-3)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm493\">Show Answer<\/span><\/p>\n<div id=\"qhjm493\" class=\"hidden-answer\" style=\"display: none\">[latex]2x^2+18x[\/latex]<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Multiply:\u00a0 [latex]x^3(x+4)(x^2-3x+1))[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm106\">Show Answer<\/span><\/p>\n<div id=\"qhjm106\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x^6+x^5-11x^4+4x^3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Summary<\/h2>\n<p>Multiplication of binomials and polynomials requires an understanding of the distributive property, rules for exponents, and a keen eye for collecting like terms. Whether the polynomials are monomials, binomials, or trinomials, carefully multiply each term in one polynomial by each term in the other polynomial. Be careful to watch the addition and subtraction signs and negative coefficients. A product is written in standard form when all of its like terms have been combined and the resulting terms are written in descending order.<\/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-342\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Try It hjm493; hjm106; hjm988; hjm931; hjm611; Examples 1 and 2.. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Adapted and revised: College Algebra. <strong>Authored by<\/strong>: Abramson, Jay Et al.. <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>. <strong>License Terms<\/strong>: Download for free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/li><li>Adapted and revised: Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Polynomial Multiplication Involving Binomials and Trinomials. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/bBKbldmlbqI\">https:\/\/youtu.be\/bBKbldmlbqI<\/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>","protected":false},"author":23485,"menu_order":8,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Adapted and revised: College Algebra\",\"author\":\"Abramson, Jay Et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\"},{\"type\":\"cc\",\"description\":\"Adapted and revised: Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology\",\"url\":\"http:\/\/nrocnetwork.org\/dm-opentext\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Try It hjm493; hjm106; hjm988; hjm931; hjm611; Examples 1 and 2.\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Polynomial Multiplication Involving Binomials and Trinomials\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/bBKbldmlbqI\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"900491ffb15747d7abbbe31f06329d3c, 252573316f0f4f81847e5257936ac19f","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-342","chapter","type-chapter","status-publish","hentry"],"part":663,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/342","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/users\/23485"}],"version-history":[{"count":9,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/342\/revisions"}],"predecessor-version":[{"id":2041,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/342\/revisions\/2041"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/parts\/663"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/342\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/media?parent=342"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=342"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/contributor?post=342"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/license?post=342"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}