{"id":16285,"date":"2019-10-02T21:57:11","date_gmt":"2019-10-02T21:57:11","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-multiply-any-two-polynomials\/"},"modified":"2020-10-22T09:29:03","modified_gmt":"2020-10-22T09:29:03","slug":"read-multiply-any-two-polynomials","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/read-multiply-any-two-polynomials\/","title":{"raw":"12.2.e - Multiplying Polynomials","rendered":"12.2.e &#8211; 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\nAnother type of polynomial multiplication problem is the product of a binomial and trinomial. Although the FOIL method cannot be used since there are more than two terms in a trinomial, you still use the Distributive Property and the Vertical Method to organize the individual products. Using the distributive property, each term in the binomial must be multiplied by each of the terms in the trinomial.\u00a0The most important part of the process is finding a way to organize terms.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply using the Distributive Property: [latex]\\left(x+3\\right)\\left(2{x}^{2}-5x+8\\right)[\/latex]\r\n\r\nSolution\r\n<table id=\"eip-id1168467363028\" class=\"unnumbered unstyled\" 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, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224530\/CNX_BMath_Figure_10_03_061_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute.<\/td>\r\n<td>[latex]x\\color{red}{(2x^2-5x+8)}+3\\color{red}{(2x^2-5x+8)}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]2{x}^{3}-5{x}^{2}+8x+6{x}^{2}-15x+24[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]2{x}^{3}+{x}^{2}-7x+24[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind the product. \u00a0[latex]\\left(3x+6\\right)\\left(5x^{2}+3x+10\\right)[\/latex]\r\n[reveal-answer q=\"637359\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"637359\"]\r\n\r\nDistribute the trinomial to each term in the binomial.\r\n\r\n[latex]3x\\left(5x^{2}+3x+10\\right)+6\\left(5x^{2}+3x+10\\right)[\/latex]\r\n\r\nUse the distributive property to distribute the monomials to each term in the trinomials.\r\n\r\n[latex]3x\\left(5x^{2}\\right)+3x\\left(3x\\right)+3x\\left(10\\right)+6\\left(5x^{2}\\right)+6\\left(3x\\right)+6\\left(10\\right)[\/latex]\r\n\r\nMultiply.\r\n\r\n[latex]15x^{3}+9x^{2}+30x+30x^{2}+18x+60[\/latex]\r\n\r\nGroup like terms.\r\n\r\n[latex]15x^{3}+\\left(9x^{2}+30x^{2}\\right)+\\left(30x+18x\\right)+60[\/latex]\r\n\r\nCombine like terms.\r\n\r\n[latex]\\left(3x+6\\right)\\left(5x^{2}+3x+10\\right)=15x^{3}+39x^{2}+48x+60[\/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]146218[\/ohm_question]\r\n\r\n<\/div>\r\nAs you can see, multiplying a binomial by a trinomial leads to a lot of individual terms! Using the same problems as above, we will use the vertical method to organize all the terms produced by multiplying two polynomials with more than two terms.\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[reveal-answer q=\"80068\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"80068\"]\r\n\r\nSolution\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\" 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, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td><img 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>Multiply [latex]\\left(2{x}^{2}-5x+8\\right)[\/latex] by [latex]3[\/latex].<\/td>\r\n<td>[latex]6x^2-15x+24[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply [latex]\\left(2{x}^{2}-5x+8\\right)[\/latex] by [latex]x[\/latex] .<\/td>\r\n<td>[latex]2x^3-5x^2+8x[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add like terms.<\/td>\r\n<td>[latex]2x^3+x^2-7x+24[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nMultiply.\u00a0[latex]\\left(3x+6\\right)\\left(5x^{2}+3x+10\\right)[\/latex]\r\n[reveal-answer q=\"262750\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"262750\"]\r\n\r\nSet up the problem in a vertical form, and begin by multiplying [latex]3x+6[\/latex] by [latex]+10[\/latex]. Place the products underneath, as shown.\r\n\r\n[latex]\\begin{array}{r}3x+\\,\\,\\,6\\,\\\\\\underline{\\times\\,\\,\\,\\,\\,\\,5x^{2}+\\,\\,3x+10}\\\\+30x+60\\,\\end{array}[\/latex]\r\n\r\nNow multiply [latex]3x+6[\/latex] by [latex]+3x[\/latex]. Notice that [latex]\\left(6\\right)\\left(3x\\right)=18x[\/latex]; since this term is like [latex]30x[\/latex], place it directly beneath it.\r\n\r\n[latex]\\begin{array}{r}3x\\,\\,\\,\\,\\,\\,+\\,\\,\\,6\\,\\,\\\\\\underline{\\times\\,\\,\\,\\,\\,\\,5x^{2}\\,\\,\\,\\,\\,\\,+3x\\,\\,\\,\\,\\,\\,+10}\\\\+30x\\,\\,\\,\\,\\,+60\\,\\,\\\\+9x^{2}\\,\\,\\,+18x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]\r\n\r\nFinally, multiply [latex]3x+6[\/latex] by [latex]5x^{2}[\/latex]. Notice that [latex]30x^{2}[\/latex]\u00a0is placed underneath [latex]9x^{2}[\/latex].\r\n\r\n[latex]\\begin{array}{r}3x\\,\\,\\,\\,\\,\\,+\\,\\,\\,6\\,\\,\\\\\\underline{\\times\\,\\,\\,\\,\\,\\,5x^{2}\\,\\,\\,\\,\\,\\,+3x\\,\\,\\,\\,\\,\\,+10}\\\\+30x\\,\\,\\,\\,\\,+60\\,\\,\\\\+9x^{2}\\,\\,\\,+18x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\underline{+15x^{3}+30x^{2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,}\\end{array}[\/latex]\r\n\r\nNow add like terms.\r\n\r\n[latex]\\begin{array}{r}3x\\,\\,\\,\\,\\,\\,+\\,\\,\\,6\\,\\,\\\\\\underline{\\times\\,\\,\\,\\,\\,\\,5x^{2}\\,\\,\\,\\,\\,\\,+3x\\,\\,\\,\\,\\,\\,+10}\\\\+30x\\,\\,\\,\\,\\,+60\\,\\,\\\\+9x^{2}\\,\\,\\,+18x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\underline{+15x^{3}\\,\\,\\,\\,\\,\\,+30x^{2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,}\\\\+15x^{3}\\,\\,\\,\\,\\,\\,+39x^{2}\\,\\,\\,\\,+48x\\,\\,\\,\\,\\,+60\\end{array}[\/latex]\r\n\r\nThe answer is [latex]15x^{3}+39x^{2}+48x+60[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146217[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nNotice that although the two problems were solved using different strategies, the product is the same. Both the horizontal and vertical methods apply the distributive property to multiply a binomial by a trinomial.\r\n\r\nIn our next example, we will multiply a binomial and a trinomial that contains subtraction. Pay attention to the signs on the terms. Forgetting a negative sign is the easiest mistake to make in this case.\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\r\n[reveal-answer q=\"485882\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"485882\"]\r\n\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[\/hidden-answer]\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, as shown below. 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[reveal-answer q=\"654814\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"654814\"]\r\n\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[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we show more examples of multiplying polynomials.\r\n\r\nhttps:\/\/youtu.be\/bBKbldmlbqI\r\n<h2>Summary<\/h2>\r\nMultiplication of binomials and polynomials requires use of the distributive property as well as the commutative and associative properties of multiplication. 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 simplified form if all of its like terms have been combined.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcome<\/h3>\n<ul>\n<li>Multiply polynomials<\/li>\n<\/ul>\n<\/div>\n<p>Another type of polynomial multiplication problem is the product of a binomial and trinomial. Although the FOIL method cannot be used since there are more than two terms in a trinomial, you still use the Distributive Property and the Vertical Method to organize the individual products. Using the distributive property, each term in the binomial must be multiplied by each of the terms in the trinomial.\u00a0The most important part of the process is finding a way to organize terms.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply using the Distributive Property: [latex]\\left(x+3\\right)\\left(2{x}^{2}-5x+8\\right)[\/latex]<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168467363028\" class=\"unnumbered unstyled\" 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><\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224530\/CNX_BMath_Figure_10_03_061_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td>[latex]x\\color{red}{(2x^2-5x+8)}+3\\color{red}{(2x^2-5x+8)}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]2{x}^{3}-5{x}^{2}+8x+6{x}^{2}-15x+24[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]2{x}^{3}+{x}^{2}-7x+24[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find the product. \u00a0[latex]\\left(3x+6\\right)\\left(5x^{2}+3x+10\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q637359\">Show Solution<\/span><\/p>\n<div id=\"q637359\" class=\"hidden-answer\" style=\"display: none\">\n<p>Distribute the trinomial to each term in the binomial.<\/p>\n<p>[latex]3x\\left(5x^{2}+3x+10\\right)+6\\left(5x^{2}+3x+10\\right)[\/latex]<\/p>\n<p>Use the distributive property to distribute the monomials to each term in the trinomials.<\/p>\n<p>[latex]3x\\left(5x^{2}\\right)+3x\\left(3x\\right)+3x\\left(10\\right)+6\\left(5x^{2}\\right)+6\\left(3x\\right)+6\\left(10\\right)[\/latex]<\/p>\n<p>Multiply.<\/p>\n<p>[latex]15x^{3}+9x^{2}+30x+30x^{2}+18x+60[\/latex]<\/p>\n<p>Group like terms.<\/p>\n<p>[latex]15x^{3}+\\left(9x^{2}+30x^{2}\\right)+\\left(30x+18x\\right)+60[\/latex]<\/p>\n<p>Combine like terms.<\/p>\n<p>[latex]\\left(3x+6\\right)\\left(5x^{2}+3x+10\\right)=15x^{3}+39x^{2}+48x+60[\/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=\"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<\/div>\n<p>As you can see, multiplying a binomial by a trinomial leads to a lot of individual terms! Using the same problems as above, we will use the vertical method to organize all the terms produced by multiplying two polynomials with more than two terms.<\/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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q80068\">Show Solution<\/span><\/p>\n<div id=\"q80068\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nIt 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\" 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><\/td>\n<td><img decoding=\"async\" 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>Multiply [latex]\\left(2{x}^{2}-5x+8\\right)[\/latex] by [latex]3[\/latex].<\/td>\n<td>[latex]6x^2-15x+24[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply [latex]\\left(2{x}^{2}-5x+8\\right)[\/latex] by [latex]x[\/latex] .<\/td>\n<td>[latex]2x^3-5x^2+8x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add like terms.<\/td>\n<td>[latex]2x^3+x^2-7x+24[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Multiply.\u00a0[latex]\\left(3x+6\\right)\\left(5x^{2}+3x+10\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q262750\">Show Solution<\/span><\/p>\n<div id=\"q262750\" class=\"hidden-answer\" style=\"display: none\">\n<p>Set up the problem in a vertical form, and begin by multiplying [latex]3x+6[\/latex] by [latex]+10[\/latex]. Place the products underneath, as shown.<\/p>\n<p>[latex]\\begin{array}{r}3x+\\,\\,\\,6\\,\\\\\\underline{\\times\\,\\,\\,\\,\\,\\,5x^{2}+\\,\\,3x+10}\\\\+30x+60\\,\\end{array}[\/latex]<\/p>\n<p>Now multiply [latex]3x+6[\/latex] by [latex]+3x[\/latex]. Notice that [latex]\\left(6\\right)\\left(3x\\right)=18x[\/latex]; since this term is like [latex]30x[\/latex], place it directly beneath it.<\/p>\n<p>[latex]\\begin{array}{r}3x\\,\\,\\,\\,\\,\\,+\\,\\,\\,6\\,\\,\\\\\\underline{\\times\\,\\,\\,\\,\\,\\,5x^{2}\\,\\,\\,\\,\\,\\,+3x\\,\\,\\,\\,\\,\\,+10}\\\\+30x\\,\\,\\,\\,\\,+60\\,\\,\\\\+9x^{2}\\,\\,\\,+18x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Finally, multiply [latex]3x+6[\/latex] by [latex]5x^{2}[\/latex]. Notice that [latex]30x^{2}[\/latex]\u00a0is placed underneath [latex]9x^{2}[\/latex].<\/p>\n<p>[latex]\\begin{array}{r}3x\\,\\,\\,\\,\\,\\,+\\,\\,\\,6\\,\\,\\\\\\underline{\\times\\,\\,\\,\\,\\,\\,5x^{2}\\,\\,\\,\\,\\,\\,+3x\\,\\,\\,\\,\\,\\,+10}\\\\+30x\\,\\,\\,\\,\\,+60\\,\\,\\\\+9x^{2}\\,\\,\\,+18x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\underline{+15x^{3}+30x^{2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,}\\end{array}[\/latex]<\/p>\n<p>Now add like terms.<\/p>\n<p>[latex]\\begin{array}{r}3x\\,\\,\\,\\,\\,\\,+\\,\\,\\,6\\,\\,\\\\\\underline{\\times\\,\\,\\,\\,\\,\\,5x^{2}\\,\\,\\,\\,\\,\\,+3x\\,\\,\\,\\,\\,\\,+10}\\\\+30x\\,\\,\\,\\,\\,+60\\,\\,\\\\+9x^{2}\\,\\,\\,+18x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\underline{+15x^{3}\\,\\,\\,\\,\\,\\,+30x^{2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,}\\\\+15x^{3}\\,\\,\\,\\,\\,\\,+39x^{2}\\,\\,\\,\\,+48x\\,\\,\\,\\,\\,+60\\end{array}[\/latex]<\/p>\n<p>The answer is [latex]15x^{3}+39x^{2}+48x+60[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\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<\/div>\n<p>&nbsp;<\/p>\n<p>Notice that although the two problems were solved using different strategies, the product is the same. Both the horizontal and vertical methods apply the distributive property to multiply a binomial by a trinomial.<\/p>\n<p>In our next example, we will multiply a binomial and a trinomial that contains subtraction. Pay attention to the signs on the terms. Forgetting a negative sign is the easiest mistake to make in this case.<\/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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q485882\">Show Solution<\/span><\/p>\n<div id=\"q485882\" class=\"hidden-answer\" style=\"display: none\">\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<\/div>\n<\/div>\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, as shown below. 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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q654814\">Show Solution<\/span><\/p>\n<div id=\"q654814\" class=\"hidden-answer\" style=\"display: none\">\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>\n<\/div>\n<p>In the following video, we show 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<h2>Summary<\/h2>\n<p>Multiplication of binomials and polynomials requires use of the distributive property as well as the commutative and associative properties of multiplication. 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 simplified form if all of its like terms have been combined.<\/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-16285\">\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>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <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>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>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":169554,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"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\":\"Unit 11: Exponents and 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