{"id":1681,"date":"2024-04-24T14:57:08","date_gmt":"2024-04-24T14:57:08","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=1681"},"modified":"2025-11-02T03:09:48","modified_gmt":"2025-11-02T03:09:48","slug":"introduction-to-derivatives-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/introduction-to-derivatives-background-youll-need-2\/","title":{"raw":"Introduction to Derivatives: Background You'll Need 2","rendered":"Introduction to Derivatives: Background You&#8217;ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n\t<li>Multiply polynomials\u00a0<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Multiplying Polynomials<\/h2>\r\n<p>While adding and subtracting polynomials is straightforward, multiplication is a bit more intricate. It hinges on the distributive property, which necessitates multiplying every term of the first polynomial by each term of the second. For the special case of binomials, we have a helpful shortcut\u2014the FOIL method, which stands for First, Outer, Inner, Last. This and other special multiplication patterns, like squaring a binomial, are tools that can make the process more efficient and are valuable to recognize.<\/p>\r\n<h3>Multiplying Polynomials Using the Distributive Property<\/h3>\r\n<p>The distributive property is the key to multiplying polynomials. It requires us to distribute, or multiply, each term of one polynomial by every term of the other polynomial.<\/p>\r\n<section class=\"textbox questionHelp\">\r\n<p><strong>How To: Use the Distributive Property to Multiply Polynomials<\/strong><\/p>\r\n<ol>\r\n\t<li><strong>Distribute<\/strong>: Multiply every term in the first polynomial by every term in the second polynomial.<\/li>\r\n\t<li><strong>Combine<\/strong>: Look for like terms \u2014 terms with the same variables raised to the same power \u2014 and sum them.<\/li>\r\n\t<li><strong>Simplify<\/strong>: Arrange the resulting polynomial in standard form, terms written in descending order of their degree.<\/li>\r\n<\/ol>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Find the product.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\left(2x+1\\right)\\left(3{x}^{2}-x+4\\right)[\/latex]<\/p>\r\n<p>[reveal-answer q=\"752165\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"752165\"]<\/p>\r\n<p style=\"text-align: center;\">[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]<\/p>\r\n<p>We can use a table to keep track of our work, as shown in the table 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.<\/p>\r\n<table style=\"height: 45px;\" 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<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 115px; height: 15px;\">\u00a0<\/td>\r\n<td style=\"width: 159px; height: 15px;\"><strong>[latex]3{x}^{2}[\/latex]<\/strong><\/td>\r\n<td style=\"width: 154px; height: 15px;\"><strong>[latex]-x[\/latex]<\/strong><\/td>\r\n<td style=\"width: 116px; height: 15px;\"><strong>[latex]+4[\/latex]<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 115px; height: 15px;\"><strong>[latex]2x[\/latex]<\/strong><\/td>\r\n<td style=\"width: 159px; height: 15px;\">[latex]6{x}^{3}\\\\[\/latex]<\/td>\r\n<td style=\"width: 154px; height: 15px;\">[latex]-2{x}^{2}[\/latex]<\/td>\r\n<td style=\"width: 116px; height: 15px;\">[latex]8x[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 115px; height: 15px;\"><strong>[latex]+1[\/latex]<\/strong><\/td>\r\n<td style=\"width: 159px; height: 15px;\">[latex]3{x}^{2}[\/latex]<\/td>\r\n<td style=\"width: 154px; height: 15px;\">[latex]-x[\/latex]<\/td>\r\n<td style=\"width: 116px; height: 15px;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox proTip\">\r\n<p>When multiplying a binomial and another polynomial with two or more terms, be sure to multiply each term in the first to each term in the second.<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_number=1]288285[\/ohm_question]<\/p>\r\n<\/section>\r\n<p>Watch this video to see more examples of how to use the distributive property to multiply polynomials.<\/p>\r\n<section class=\"textbox watchIt\">\r\n<p>https:\/\/youtu.be\/bwTmApTV_8o<\/p>\r\n<\/section>\r\n<h3>Using FOIL to Multiply Binomials<\/h3>\r\n\r\n\r\n[caption id=\"\" align=\"alignright\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21205149\/CNX_CAT_Figure_01_04_003.jpg\" alt=\"Two quantities in parentheses are being multiplied, the first being: a times x plus b and the second being: c times x plus d. This expression equals ac times x squared plus ad times x plus bc times x plus bd. The terms ax and cx are labeled: First Terms. The terms ax and d are labeled: Outer Terms. The terms b and cx are labeled: Inner Terms. The terms b and d are labeled: Last Terms.\" width=\"487\" height=\"191\" \/> Illustration of the FOIL method for multiplying two binomials[\/caption]\r\n\r\n\r\n<p>FOIL is a mnemonic that stands for First, Outer, Inner, Last, and it\u2019s a technique used to multiply two binomials efficiently. Each word in the acronym FOIL represents a pair of terms to be multiplied together:<\/p>\r\n<ul>\r\n\t<li><strong>First<\/strong> terms from each binomial<\/li>\r\n\t<li><strong>Outer<\/strong> terms from each binomial<\/li>\r\n\t<li><strong>Inner<\/strong> terms from each binomial<\/li>\r\n\t<li><strong>Last<\/strong> terms from each binomial<\/li>\r\n<\/ul>\r\n<p>The products from these multiplications are then added together to get the final expanded form of the polynomial.<\/p>\r\n<section class=\"textbox questionHelp\">\r\n<p><strong>How To: Multiplying Binomials Using FOIL<\/strong><\/p>\r\n<ol>\r\n\t<li>Multiply the <strong>First<\/strong> terms of each binomial.<\/li>\r\n\t<li>Multiply the <strong>Outer<\/strong> terms.<\/li>\r\n\t<li>Multiply the <strong>Inner<\/strong> terms.<\/li>\r\n\t<li>Multiply the <strong>Last<\/strong> terms.<\/li>\r\n\t<li>Sum all the products.<\/li>\r\n\t<li>Combine any like terms to simplify the expression.<\/li>\r\n<\/ol>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Use the FOIL method to find the product of the following:<\/p>\r\n<center>[latex]\\left(2x-18\\right)\\left(3x + 3\\right)[\/latex]<\/center>\r\n<p>[reveal-answer q=\"698991\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"698991\"]<br \/>\r\nFind the product of the first terms.<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21205151\/CNX_CAT_Figure_01_04_004.jpg\" alt=\"\" width=\"487\" height=\"52\" \/> First term multiplication[\/caption]\r\n\r\n\r\n<p>Find the product of the outer terms.<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21205154\/CNX_CAT_Figure_01_04_005.jpg\" alt=\"\" width=\"487\" height=\"52\" \/> Outer term multiplication[\/caption]\r\n\r\n\r\n<p>Find the product of the inner terms.<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21205156\/CNX_CAT_Figure_01_04_006.jpg\" alt=\"\" width=\"487\" height=\"52\" \/> Inner term multiplication[\/caption]\r\n\r\n\r\n<p>Find the product of the last terms.<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21205158\/CNX_CAT_Figure_01_04_007.jpg\" alt=\"\" width=\"487\" height=\"52\" \/> Last term multiplication[\/caption]\r\n\r\n\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{cc}6{x}^{2}+6x - 54x - 54\\hfill &amp; \\text{Add the products}.\\hfill \\\\ 6{x}^{2}+\\left(6x - 54x\\right)-54\\hfill &amp; \\text{Combine like terms}.\\hfill \\\\ 6{x}^{2}-48x - 54\\hfill &amp; \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_number=1]288286[\/ohm_question]<\/p>\r\n<\/section>\r\n<h3>Perfect Square Trinomials<\/h3>\r\n<p>When squaring a binomial, the result is known as a <strong>perfect square trinomial<\/strong>, which has a recognizable form. Instead of multiplying the binomial by itself, we can use a formula that simplifies the process.<\/p>\r\n<section class=\"textbox example\">\r\n<p>Here are some examples of perfect square trinomials and the general pattern they follow:<\/p>\r\n<center>[latex]\\begin{array}{ccc}\\hfill \\text{ }{\\left(x+5\\right)}^{2}&amp; =&amp; \\text{ }{x}^{2}+10x+25\\hfill \\\\ \\hfill {\\left(x - 3\\right)}^{2}&amp; =&amp; \\text{ }{x}^{2}-6x+9\\hfill \\\\ \\hfill {\\left(4x - 1\\right)}^{2}&amp; =&amp; 4{x}^{2}-8x+1\\hfill \\end{array}[\/latex]<\/center><\/section>\r\n<p>The pattern here is clear: the first and last terms of the trinomial are the squares of the first and last terms of the binomial, respectively. The middle term is twice the product of the two terms in the binomial. The sign of the middle term matches the sign of the binomial.<\/p>\r\n<section class=\"textbox questionHelp\">\r\n<p><strong>How To: Squaring a Binomial Using the Perfect Square Trinomial Formula<\/strong><\/p>\r\n<ol>\r\n\t<li>Square the first term of the binomial.<\/li>\r\n\t<li>Square the second term of the binomial.<\/li>\r\n\t<li>Double the product of the two terms for the middle term.<\/li>\r\n\t<li>Combine these to form your perfect square trinomial.<\/li>\r\n<\/ol>\r\n<center>[latex]{\\left(x+a\\right)}^{2}=\\left(x+a\\right)\\left(x+a\\right)={x}^{2}+2ax+{a}^{2}[\/latex]<\/center><\/section>\r\n<section class=\"textbox example\">\r\n<p>Expand [latex]{\\left(3x - 8\\right)}^{2}[\/latex]. [reveal-answer q=\"733978\"]Show Solution[\/reveal-answer] [hidden-answer a=\"733978\"] Begin by squaring the first term and the last term. For the middle term of the trinomial, double the product of the two terms.<\/p>\r\n<div style=\"text-align: center;\">[latex]{\\left(3x\\right)}^{2}-2\\left(3x\\right)\\left(8\\right)+{\\left(-8\\right)}^{2}[\/latex]<\/div>\r\n<p style=\"text-align: center;\">[latex]9{x}^{2}-48x+64[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]288287[\/ohm_question]<\/p>\r\n<\/section>\r\n<h3>Difference of Squares<\/h3>\r\n<p>When we multiply a binomial by another with the same terms but opposite signs, we arrive at a product known as the <strong>difference of squares<\/strong>. This occurs because the middle terms of the binomials, which are opposites, cancel each other out. The result is the square of the first term minus the square of the second term.<\/p>\r\n<section class=\"textbox example\">\r\n<p>Let\u2019s see what happens when we multiply [latex]\\left(x+1\\right)\\left(x - 1\\right)[\/latex] using the FOIL method.<\/p>\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill \\left(x+1\\right)\\left(x - 1\\right)&amp; =&amp; {x}^{2}-x+x - 1\\hfill \\\\ &amp; =&amp; {x}^{2}-1\\hfill \\end{array}[\/latex]<\/div>\r\n<p>The [latex]+x[\/latex] and [latex]\u2212x[\/latex] terms cancel out, leaving us with the difference of squares.<\/p>\r\n<p>Here are more examples:<\/p>\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill \\left(x+5\\right)\\left(x - 5\\right)&amp; =&amp; {x}^{2}-25\\hfill \\\\ \\hfill \\left(x+11\\right)\\left(x - 11\\right)&amp; =&amp; {x}^{2}-121\\hfill \\\\ \\hfill \\left(2x+3\\right)\\left(2x - 3\\right)&amp; =&amp; 4{x}^{2}-9\\hfill \\end{array}[\/latex]<\/div>\r\n<p>In each case, the middle terms cancel, and we are left with the square of the first term minus the square of the second term.<\/p>\r\n<\/section>\r\n<section class=\"textbox proTip\">\r\n<p>When a binomial is multiplied by a binomial with the same terms separated by the opposite sign, the result is the square of the first term minus the square of the last term.<\/p>\r\n<div style=\"text-align: center;\">[latex]\\left(a+b\\right)\\left(a-b\\right)={a}^{2}-{b}^{2}[\/latex]<\/div>\r\n<\/section>\r\n<section class=\"textbox questionHelp\">\r\n<p><strong>How To: Calculating the Difference of Squares<\/strong><\/p>\r\n<ol>\r\n\t<li>Square the first term of each binomial.<\/li>\r\n\t<li>Square the second term of each binomial.<\/li>\r\n\t<li>Subtract the square of the second term from the square of the first term to find the difference of squares.<\/li>\r\n<\/ol>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Multiply [latex]\\left(9x+4\\right)\\left(9x - 4\\right)[\/latex]. [reveal-answer q=\"366563\"]Show Solution[\/reveal-answer] [hidden-answer a=\"366563\"] Square the first term to get [latex]{\\left(9x\\right)}^{2}=81{x}^{2}[\/latex]. <br \/>\r\nSquare the last term to get [latex]{4}^{2}=16[\/latex]. <br \/>\r\nSubtract the square of the last term from the square of the first term to find the product of [latex]81{x}^{2}-16[\/latex]. [\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]288288[\/ohm_question]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Multiply polynomials\u00a0<\/li>\n<\/ul>\n<\/section>\n<h2>Multiplying Polynomials<\/h2>\n<p>While adding and subtracting polynomials is straightforward, multiplication is a bit more intricate. It hinges on the distributive property, which necessitates multiplying every term of the first polynomial by each term of the second. For the special case of binomials, we have a helpful shortcut\u2014the FOIL method, which stands for First, Outer, Inner, Last. This and other special multiplication patterns, like squaring a binomial, are tools that can make the process more efficient and are valuable to recognize.<\/p>\n<h3>Multiplying Polynomials Using the Distributive Property<\/h3>\n<p>The distributive property is the key to multiplying polynomials. It requires us to distribute, or multiply, each term of one polynomial by every term of the other polynomial.<\/p>\n<section class=\"textbox questionHelp\">\n<p><strong>How To: Use the Distributive Property to Multiply Polynomials<\/strong><\/p>\n<ol>\n<li><strong>Distribute<\/strong>: Multiply every term in the first polynomial by every term in the second polynomial.<\/li>\n<li><strong>Combine<\/strong>: Look for like terms \u2014 terms with the same variables raised to the same power \u2014 and sum them.<\/li>\n<li><strong>Simplify<\/strong>: Arrange the resulting polynomial in standard form, terms written in descending order of their degree.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">\n<p>Find the product.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(2x+1\\right)\\left(3{x}^{2}-x+4\\right)[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q752165\">Show Solution<\/button><\/p>\n<div id=\"q752165\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[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>We can use a table to keep track of our work, as shown in the table 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.<\/p>\n<table style=\"height: 45px;\" 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<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"width: 115px; height: 15px;\">\u00a0<\/td>\n<td style=\"width: 159px; height: 15px;\"><strong>[latex]3{x}^{2}[\/latex]<\/strong><\/td>\n<td style=\"width: 154px; height: 15px;\"><strong>[latex]-x[\/latex]<\/strong><\/td>\n<td style=\"width: 116px; height: 15px;\"><strong>[latex]+4[\/latex]<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 115px; height: 15px;\"><strong>[latex]2x[\/latex]<\/strong><\/td>\n<td style=\"width: 159px; height: 15px;\">[latex]6{x}^{3}\\\\[\/latex]<\/td>\n<td style=\"width: 154px; height: 15px;\">[latex]-2{x}^{2}[\/latex]<\/td>\n<td style=\"width: 116px; height: 15px;\">[latex]8x[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 115px; height: 15px;\"><strong>[latex]+1[\/latex]<\/strong><\/td>\n<td style=\"width: 159px; height: 15px;\">[latex]3{x}^{2}[\/latex]<\/td>\n<td style=\"width: 154px; height: 15px;\">[latex]-x[\/latex]<\/td>\n<td style=\"width: 116px; height: 15px;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\">\n<p>When multiplying a binomial and another polynomial with two or more terms, be sure to multiply each term in the first to each term in the second.<\/p>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm288285\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288285&theme=lumen&iframe_resize_id=ohm288285&source=tnh&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<p>Watch this video to see more examples of how to use the distributive property to multiply polynomials.<\/p>\n<section class=\"textbox watchIt\">\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Multiplying Using the Distributive Property\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/bwTmApTV_8o?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/section>\n<h3>Using FOIL to Multiply Binomials<\/h3>\n<figure style=\"width: 487px\" class=\"wp-caption alignright\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21205149\/CNX_CAT_Figure_01_04_003.jpg\" alt=\"Two quantities in parentheses are being multiplied, the first being: a times x plus b and the second being: c times x plus d. This expression equals ac times x squared plus ad times x plus bc times x plus bd. The terms ax and cx are labeled: First Terms. The terms ax and d are labeled: Outer Terms. The terms b and cx are labeled: Inner Terms. The terms b and d are labeled: Last Terms.\" width=\"487\" height=\"191\" \/><figcaption class=\"wp-caption-text\">Illustration of the FOIL method for multiplying two binomials<\/figcaption><\/figure>\n<p>FOIL is a mnemonic that stands for First, Outer, Inner, Last, and it\u2019s a technique used to multiply two binomials efficiently. Each word in the acronym FOIL represents a pair of terms to be multiplied together:<\/p>\n<ul>\n<li><strong>First<\/strong> terms from each binomial<\/li>\n<li><strong>Outer<\/strong> terms from each binomial<\/li>\n<li><strong>Inner<\/strong> terms from each binomial<\/li>\n<li><strong>Last<\/strong> terms from each binomial<\/li>\n<\/ul>\n<p>The products from these multiplications are then added together to get the final expanded form of the polynomial.<\/p>\n<section class=\"textbox questionHelp\">\n<p><strong>How To: Multiplying Binomials Using FOIL<\/strong><\/p>\n<ol>\n<li>Multiply the <strong>First<\/strong> terms of each binomial.<\/li>\n<li>Multiply the <strong>Outer<\/strong> terms.<\/li>\n<li>Multiply the <strong>Inner<\/strong> terms.<\/li>\n<li>Multiply the <strong>Last<\/strong> terms.<\/li>\n<li>Sum all the products.<\/li>\n<li>Combine any like terms to simplify the expression.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">\n<p>Use the FOIL method to find the product of the following:<\/p>\n<div style=\"text-align: center;\">[latex]\\left(2x-18\\right)\\left(3x + 3\\right)[\/latex]<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q698991\">Show Solution<\/button><\/p>\n<div id=\"q698991\" class=\"hidden-answer\" style=\"display: none\">\nFind the product of the first terms.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21205151\/CNX_CAT_Figure_01_04_004.jpg\" alt=\"\" width=\"487\" height=\"52\" \/><figcaption class=\"wp-caption-text\">First term multiplication<\/figcaption><\/figure>\n<p>Find the product of the outer terms.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21205154\/CNX_CAT_Figure_01_04_005.jpg\" alt=\"\" width=\"487\" height=\"52\" \/><figcaption class=\"wp-caption-text\">Outer term multiplication<\/figcaption><\/figure>\n<p>Find the product of the inner terms.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21205156\/CNX_CAT_Figure_01_04_006.jpg\" alt=\"\" width=\"487\" height=\"52\" \/><figcaption class=\"wp-caption-text\">Inner term multiplication<\/figcaption><\/figure>\n<p>Find the product of the last terms.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21205158\/CNX_CAT_Figure_01_04_007.jpg\" alt=\"\" width=\"487\" height=\"52\" \/><figcaption class=\"wp-caption-text\">Last term multiplication<\/figcaption><\/figure>\n<p style=\"text-align: center;\">[latex]\\begin{array}{cc}6{x}^{2}+6x - 54x - 54\\hfill & \\text{Add the products}.\\hfill \\\\ 6{x}^{2}+\\left(6x - 54x\\right)-54\\hfill & \\text{Combine like terms}.\\hfill \\\\ 6{x}^{2}-48x - 54\\hfill & \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm288286\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288286&theme=lumen&iframe_resize_id=ohm288286&source=tnh&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<h3>Perfect Square Trinomials<\/h3>\n<p>When squaring a binomial, the result is known as a <strong>perfect square trinomial<\/strong>, which has a recognizable form. Instead of multiplying the binomial by itself, we can use a formula that simplifies the process.<\/p>\n<section class=\"textbox example\">\n<p>Here are some examples of perfect square trinomials and the general pattern they follow:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill \\text{ }{\\left(x+5\\right)}^{2}& =& \\text{ }{x}^{2}+10x+25\\hfill \\\\ \\hfill {\\left(x - 3\\right)}^{2}& =& \\text{ }{x}^{2}-6x+9\\hfill \\\\ \\hfill {\\left(4x - 1\\right)}^{2}& =& 4{x}^{2}-8x+1\\hfill \\end{array}[\/latex]<\/div>\n<\/section>\n<p>The pattern here is clear: the first and last terms of the trinomial are the squares of the first and last terms of the binomial, respectively. The middle term is twice the product of the two terms in the binomial. The sign of the middle term matches the sign of the binomial.<\/p>\n<section class=\"textbox questionHelp\">\n<p><strong>How To: Squaring a Binomial Using the Perfect Square Trinomial Formula<\/strong><\/p>\n<ol>\n<li>Square the first term of the binomial.<\/li>\n<li>Square the second term of the binomial.<\/li>\n<li>Double the product of the two terms for the middle term.<\/li>\n<li>Combine these to form your perfect square trinomial.<\/li>\n<\/ol>\n<div style=\"text-align: center;\">[latex]{\\left(x+a\\right)}^{2}=\\left(x+a\\right)\\left(x+a\\right)={x}^{2}+2ax+{a}^{2}[\/latex]<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Expand [latex]{\\left(3x - 8\\right)}^{2}[\/latex]. <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q733978\">Show Solution<\/button> <\/p>\n<div id=\"q733978\" class=\"hidden-answer\" style=\"display: none\"> Begin by squaring the first term and the last term. For the middle term of the trinomial, double the product of the two terms.<\/p>\n<div style=\"text-align: center;\">[latex]{\\left(3x\\right)}^{2}-2\\left(3x\\right)\\left(8\\right)+{\\left(-8\\right)}^{2}[\/latex]<\/div>\n<p style=\"text-align: center;\">[latex]9{x}^{2}-48x+64[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm288287\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288287&theme=lumen&iframe_resize_id=ohm288287&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<h3>Difference of Squares<\/h3>\n<p>When we multiply a binomial by another with the same terms but opposite signs, we arrive at a product known as the <strong>difference of squares<\/strong>. This occurs because the middle terms of the binomials, which are opposites, cancel each other out. The result is the square of the first term minus the square of the second term.<\/p>\n<section class=\"textbox example\">\n<p>Let\u2019s see what happens when we multiply [latex]\\left(x+1\\right)\\left(x - 1\\right)[\/latex] using the FOIL method.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill \\left(x+1\\right)\\left(x - 1\\right)& =& {x}^{2}-x+x - 1\\hfill \\\\ & =& {x}^{2}-1\\hfill \\end{array}[\/latex]<\/div>\n<p>The [latex]+x[\/latex] and [latex]\u2212x[\/latex] terms cancel out, leaving us with the difference of squares.<\/p>\n<p>Here are more examples:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill \\left(x+5\\right)\\left(x - 5\\right)& =& {x}^{2}-25\\hfill \\\\ \\hfill \\left(x+11\\right)\\left(x - 11\\right)& =& {x}^{2}-121\\hfill \\\\ \\hfill \\left(2x+3\\right)\\left(2x - 3\\right)& =& 4{x}^{2}-9\\hfill \\end{array}[\/latex]<\/div>\n<p>In each case, the middle terms cancel, and we are left with the square of the first term minus the square of the second term.<\/p>\n<\/section>\n<section class=\"textbox proTip\">\n<p>When a binomial is multiplied by a binomial with the same terms separated by the opposite sign, the result is the square of the first term minus the square of the last term.<\/p>\n<div style=\"text-align: center;\">[latex]\\left(a+b\\right)\\left(a-b\\right)={a}^{2}-{b}^{2}[\/latex]<\/div>\n<\/section>\n<section class=\"textbox questionHelp\">\n<p><strong>How To: Calculating the Difference of Squares<\/strong><\/p>\n<ol>\n<li>Square the first term of each binomial.<\/li>\n<li>Square the second term of each binomial.<\/li>\n<li>Subtract the square of the second term from the square of the first term to find the difference of squares.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">\n<p>Multiply [latex]\\left(9x+4\\right)\\left(9x - 4\\right)[\/latex]. <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q366563\">Show Solution<\/button> <\/p>\n<div id=\"q366563\" class=\"hidden-answer\" style=\"display: none\"> Square the first term to get [latex]{\\left(9x\\right)}^{2}=81{x}^{2}[\/latex]. <br \/>\nSquare the last term to get [latex]{4}^{2}=16[\/latex]. <br \/>\nSubtract the square of the last term from the square of the first term to find the product of [latex]81{x}^{2}-16[\/latex]. <\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm288288\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288288&theme=lumen&iframe_resize_id=ohm288288&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n","protected":false},"author":15,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":503,"module-header":"background_you_need","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1681"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":34,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1681\/revisions"}],"predecessor-version":[{"id":4853,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1681\/revisions\/4853"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/503"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1681\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=1681"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=1681"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=1681"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=1681"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}