{"id":4720,"date":"2017-06-07T18:56:00","date_gmt":"2017-06-07T18:56:00","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/chapter\/read-multiply-binomials\/"},"modified":"2017-08-15T20:37:36","modified_gmt":"2017-08-15T20:37:36","slug":"read-multiply-binomials","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/chapter\/read-multiply-binomials\/","title":{"raw":"Multiply Binomials (Including Using FOIL)","rendered":"Multiply Binomials (Including Using FOIL)"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Apply the FOIL method to multiply two binomials<\/li>\r\n \t<li>Use a table to multiply two binomials<\/li>\r\n \t<li>Simplify the product of two binomials given a wide variety of variables, constants, signs, and arrangement of terms in the binomial<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2><\/h2>\r\n[caption id=\"attachment_4589\" align=\"aligncenter\" width=\"335\"]<img class=\"wp-image-4589\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185557\/Screen-Shot-2016-06-03-at-2.08.13-PM-243x300.png\" alt=\"Crane made from aluminum foil\" width=\"335\" height=\"414\" \/> Foil Crane[\/caption]\r\n\r\n<span style=\"color: #000000\">In the last section we finished with an example of multiplying two binomials,[latex]\\left(x+4\\right)\\left(x+2\\right)[\/latex]. In this section we will provide examples of how to use two different methods to multiply to binomials. Keep in mind as you read through the page that simplify and multiply are used interchangeably.<\/span>\r\n<h2>FOIL<\/h2>\r\nSome people use the FOIL method to keep track of which pairs of terms have been multiplied when you are multiplying two binomials. This is not the same thing you use to wrap up leftovers, but an acronym for <strong>First, Outer, Inner, Last.<\/strong> Let's go back to the example from the previous page, where we were asked to multiply the two binomials: [latex]\\left(x+4\\right)\\left(x+2\\right)[\/latex]. \u00a0The following steps show you how to apply this method to multiplying two binomials.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\text{First}\\text{ term in each binomial}: \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left(x+4\\right)\\left(x+2\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\left(x\\right)=x^{2}\\\\\\text{Outer terms}:\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left(x+4\\right)\\left(x+2\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\left(2\\right)=2x\\\\\\text{Inner terms}:\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left(x+4\\right)\\left(x+2\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4\\left(x\\right)=4x\\\\\\text{Last terms in each binomial}:\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left(x+4\\right)\\left(x+2\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4\\left(2\\right)=8\\end{array}[\/latex]<\/p>\r\nWhen you add the four results, you get the same answer,\u00a0[latex]x^{2}+2x+4x+8=x^{2}+6x+8[\/latex].\r\nThe last step in multiplying polynomials is to combine like terms. Remember that a polynomial is simplified only when there are no like terms remaining.\r\n<div class=\"textbox shaded\"><img class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07184952\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"41\" height=\"36\" \/>Caution! Note that the FOIL method only works for multiplying two binomials together. It sill not work for multiplying a binomial and a trinomial, or two trinomials.<\/div>\r\n\r\n[caption id=\"attachment_4595\" align=\"alignleft\" width=\"138\"]<img class=\"wp-image-4595\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185559\/Screen-Shot-2016-06-03-at-4.09.32-PM-299x300.png\" alt=\"two tomatoes sitting next to each other with two different phonetic pronunciations for the word tomato underneath\" width=\"138\" height=\"139\" \/> Order Doesn't Matter When You Multiply[\/caption]\r\n\r\nOne of the neat things about multiplication is that\u00a0terms can be multiplied in either order. The expression [latex]\\left(x+2\\right)\\left(x+4\\right)[\/latex] has the same product as [latex]\\left(x+4\\right)\\left(x+2\\right)[\/latex], [latex]x^{2}+6x+8[\/latex]. (Work it out and see.) The order in which you multiply binomials does not matter. What matters is that you multiply each term in one binomial by each term in the other binomial.\r\n<h2><\/h2>\r\n<span style=\"color: #000000\">Polynomials can take many forms. \u00a0So far we have seen examples of binomials with variable terms on the left and constant terms on the right, such as this binomial [latex]\\left(2r-3\\right)[\/latex]. \u00a0Variables may also be on the right of the constant term, as in this binomial [latex]\\left(5+r\\right)[\/latex]. \u00a0In the next example, we will show that multiplying binomials in this form requires one extra step at the end.<\/span>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\n<span style=\"color: #000000\">Find the product.[latex]\\left(3\u2013s\\right)\\left(1-s\\right)[\/latex]<\/span>\r\n<span style=\"color: #000000\"> [reveal-answer q=\"531601\"]Show Solution[\/reveal-answer]<\/span>\r\n<span style=\"color: #000000\"> [hidden-answer a=\"531601\"]<\/span>\r\n\r\n<span style=\"color: #000000\">Notice how the binomials have the variable on the right instead of the left. \u00a0There is nothing different in the way you find the product. \u00a0At the end we will reorganize terms so they are in descending order as a matter of convention.<\/span>\r\n\r\n<span style=\"color: #000000\">[latex]\\left(3\u2013s\\right)\\left(1\u2013s\\right)[\/latex]<\/span>\r\n\r\n<span style=\"color: #000000\">Use a table this time.<\/span>\r\n<table style=\"width: 40%\">\r\n<tbody>\r\n<tr>\r\n<td><span style=\"color: #000000\">\u00a0<\/span><\/td>\r\n<th><span style=\"color: #000000\">[latex]3[\/latex]<\/span><\/th>\r\n<th><span style=\"color: #000000\">[latex]-s[\/latex]<\/span><\/th>\r\n<\/tr>\r\n<tr>\r\n<th><span style=\"color: #000000\">[latex]1[\/latex]<\/span><\/th>\r\n<td><span style=\"color: #000000\">[latex]3[\/latex]<\/span><\/td>\r\n<td><span style=\"color: #000000\">[latex]-s[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<th><span style=\"color: #000000\">[latex]-s[\/latex]<\/span><\/th>\r\n<td><span style=\"color: #000000\">[latex]-3s[\/latex]<\/span><\/td>\r\n<td><span style=\"color: #000000\">[latex]s^2[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<span style=\"color: #000000\">Notice how the <em>s<\/em> term is now positive. Collect the terms and simplify.<\/span>\r\n<p style=\"text-align: center\"><span style=\"color: #000000\">[latex]\\begin{array}{c}\\left(3\u2013s\\right)\\left(1\u2013s\\right)\\\\\\text{ }\\\\=3-3s-s+s^2\\\\\\text{ }\\\\=3-4s+s^2\\end{array}[\/latex]<\/span><\/p>\r\n<span style=\"color: #000000\">As a matter of convention, we will organize the terms so the one with greatest degree comes first. Pay close attention to the signs on the terms when you reorganize them. The 3 is positive, so we will use a plus in front of it, and the 4 is negative so we use a minus in front of it.<\/span>\r\n<p style=\"text-align: center\"><span style=\"color: #000000\">[latex]\\begin{array}{c}\\left(3\u2013s\\right)\\left(1\u2013s\\right)\\\\\\text{ }\\\\=s^{2}-4s+3\\end{array}[\/latex]<\/span><\/p>\r\n\r\n<h4><span style=\"color: #000000\">Answer<\/span><\/h4>\r\n<span style=\"color: #000000\">[latex]\\left(3\u2013s\\right)\\left(1\u2013s\\right)=s^2-4s+3[\/latex]<\/span>\r\n\r\n<span style=\"color: #000000\">[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<span style=\"color: #000000\">In the next example, you will see that sometimes there are constants in front of the variable. They will get multiplied together just as we have done before.<\/span>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\n<span style=\"color: #000000\">Simplify [latex]\\left(4x\u201310\\right)\\left(2x+3\\right)[\/latex] using the FOIL acronym.<\/span>\r\n\r\n<span style=\"color: #000000\">[reveal-answer q=\"930433\"]Show Solution[\/reveal-answer]<\/span>\r\n<span style=\"color: #000000\"> [hidden-answer a=\"930433\"]<\/span>\r\n<p style=\"text-align: center\"><span style=\"color: #000000\">[latex]\\begin{array}{l}\\text{First}:\\,\\,\\,\\,\\,4x\\left(2x\\right)=8x^{2}\\\\\\text{Outer}:\\,\\,\\,4x\\left(3\\right)=12x\\\\\\text{Inner}:\\,\\,\\,\u221210\\left(2x\\right)=-20x\\\\\\text{Last}:\\,\\,\\,\\,\\,-10\\left(3\\right)=-30\\end{array}[\/latex]<\/span><\/p>\r\n<span style=\"color: #000000\">Be careful about including the negative sign on the [latex]\u201110[\/latex], since 10 is subtracted.<\/span>\r\n\r\n<span style=\"color: #000000\">Combine like terms.<\/span>\r\n<p style=\"text-align: center\"><span style=\"color: #000000\">[latex]\\begin{array}{c}8x^{2}+12x\u201320x\u201330\\\\\\text{ }\\\\=8x^{2}-8x\u201330\\end{array}[\/latex]<\/span><\/p>\r\n\r\n<h4><span style=\"color: #000000\">Answer<\/span><\/h4>\r\n<span style=\"color: #000000\">[latex]\\left(4x\u201310\\right)\\left(2x+3\\right)=8x^{2}\u20138x\u201330[\/latex]<\/span>\r\n\r\n<span style=\"color: #000000\">[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<span style=\"color: #000000\">The video that follows gives another example of multiplying two binomials using the FOIL acronym. Remember this method only works when you are multiplying two binomials.<\/span>\r\n\r\nhttps:\/\/youtu.be\/_MrdEFnXNGA\r\n<h2>The Table Method<\/h2>\r\nYou may see a binomial multiplied by itself written as\u00a0[latex]{\\left(x+3\\right)}^{2}[\/latex] instead of\u00a0[latex]\\left(x+3\\right)\\left(x+3\\right)[\/latex]. \u00a0To find this product, let's use another method. We will place the terms of each binomial along the top row and first column of a table, like this:\r\n<table class=\"lines\" style=\"width: 20%\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>[latex]+3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]+3[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow multiply the term in each column by the term in each row to get the terms of the resulting polynomial. Note how we keep the signs on the terms, even when they are positive, this will help us write the new polynomial.\r\n<table style=\"width: 20%\">\r\n<thead>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>[latex]+3[\/latex]<\/td>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td><span style=\"color: #0000ff\">[latex]x\\cdot{x}=x^2[\/latex]<\/span><\/td>\r\n<td><span style=\"color: #ff0000\">[latex]3\\cdot{x}=+3x[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]+3[\/latex]<\/td>\r\n<td><span style=\"color: #ff0000\">[latex]x\\cdot{3}=+3x[\/latex]<\/span><\/td>\r\n<td><span style=\"color: #ff00ff\">\u00a0[latex]3\\cdot{3}=+9[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow we can write the terms of the polynomial from the entries in the table:\r\n<p style=\"text-align: center\">[latex]\\left(x+3\\right)^{2}[\/latex]<\/p>\r\n<p style=\"text-align: center\">=\u00a0<span style=\"color: #0000ff\">[latex]x^2[\/latex]<\/span> + <span style=\"color: #ff0000\">[latex]3x[\/latex]<\/span> + <span style=\"color: #ff0000\">[latex]3x[\/latex]<\/span> + <span style=\"color: #ff00ff\">[latex]9[\/latex]<\/span><\/p>\r\n<p style=\"text-align: center\">= <span style=\"color: #0000ff\">[latex]x^{2}[\/latex]<\/span> + <span style=\"color: #ff0000\">[latex]6x[\/latex]<\/span> + <span style=\"color: #ff00ff\">[latex]9[\/latex]<\/span>.<\/p>\r\n<p style=\"text-align: center\">Pretty cool, huh?<\/p>\r\nSo far, we have shown two\u00a0methods for multiplying two binomials together. \u00a0Why are we focusing\u00a0so much on binomials? \u00a0They are one of the most well studied and widely used polynomials, so there is a lot of information out there about them. In the previous example, we saw the result of squaring a binomial that was a sum of two terms. In the next example we will find the product of squaring a binomial that is the difference of two terms.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSquare the binomial difference\u00a0[latex]\\left(x\u20137\\right)[\/latex]\r\n[reveal-answer q=\"293164\"]Show solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"293164\"]\r\n\r\nWrite the product of the binomial.\r\n<p style=\"text-align: center\">[latex]{\\left(x-7\\right)}^2=\\left(x\u20137\\right)\\left(x\u20137\\right)[\/latex]<\/p>\r\n<p style=\"text-align: left\">Let's use the table method, just because. Note how we carry the negative sign with the\u00a0[latex]7[\/latex].<\/p>\r\n\r\n<table style=\"width: 20%\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>\u00a0[latex]x[\/latex]<\/td>\r\n<td>\u00a0[latex]-7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0[latex]x[\/latex]<\/td>\r\n<td>\u00a0\u00a0[latex]x^2[\/latex]<\/td>\r\n<td>\u00a0\u00a0[latex]-7x[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0\u00a0[latex]-7[\/latex]<\/td>\r\n<td>\u00a0\u00a0[latex]-7x[\/latex]<\/td>\r\n<td>\u00a0\u00a0[latex]49[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nCollect the terms, and simplify. Note how we keep the sign on each term.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}x^2-7x-7x+49\\\\\\text{ }\\\\=x^2-14x+49\\end{array}[\/latex]<\/p>\r\nAnswer\r\n[latex]x^2-14x+49[\/latex]\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n\r\n<img class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07184952\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"51\" height=\"45\" \/>Caution! It is VERY important to remember the caution from the exponents section about squaring a binomial:\r\n<p style=\"text-align: center\">You can't move the exponent into a grouped sum because of the order of operations!!!!!<\/p>\r\n<p style=\"text-align: center\"><strong>INCORRECT:<\/strong> [latex]\\left(2+x\\right)^{2}\\neq2^{2}+x^{2}[\/latex]<\/p>\r\n<p style=\"text-align: center\"><strong>\u00a0CORRECT:<\/strong> [latex]\\left(2+x\\right)^{2}=\\left(2+x\\right)\\left(2+x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\nIn the video that follows, you will see another examples of using a table to multiply two binomials.\r\n\r\nhttps:\/\/youtu.be\/tWsLJ_pn5mQ\r\n<h2>Further Examples<\/h2>\r\nThe next couple of examples show you some different forms binomials can take. \u00a0In the first, we will square a binomial that has a coefficient in front of the variable, like the product in the first example on this page. In the second we will find\u00a0the product of two binomials that have the variable on the right instead of the left.\u00a0We will use both\u00a0the FOIL method and the table method to simplify.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind the product.\u00a0[latex]\\left(2x+6\\right)^{2}[\/latex]\r\n[reveal-answer q=\"255359\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"255359\"]We will use the FOIL method.\r\n[latex]\\left(2x+6\\right)^{2}=\\left(2x+6\\right)\\left(2x+6\\right)[\/latex]\r\n[latex]\\begin{array}{l}\\text{First}:\\,\\,\\,\\,\\,2x\\left(2x\\right)=4x^{2}\\\\\\text{Outer}:\\,\\,\\,2x\\left(6\\right)=12x\\\\\\text{Inner}:\\,\\,\\,6\\left(2x\\right)=12x\\\\\\text{Last}:\\,\\,\\,\\,\\,6\\left(6\\right)=36\\end{array}[\/latex]\r\n\r\nNow you can collect the terms and simplify:\r\n[latex]\\begin{array}{c}4x^2+12x+12x+36\\\\\\text{ }\\\\=4x^2+24x+36\\end{array}[\/latex]\r\n\r\nAnswer\r\n\r\n[latex](2x+6)^{2}=4x^{2}+24x+36[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the last example, we want to show you another common form a binomial can take, each of the terms in the two binomials is the same, but the signs are different. You will see that in this case, the middle term will disappear.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nMultiply the binomials. [latex]\\left(x+8\\right)\\left(x\u20138\\right)[\/latex]\r\n[reveal-answer q=\"812247\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"812247\"]\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\text{First}:\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\left(x\\right)=x^{2}\\\\\\text{Outer}:\\,\\,\\,\\,\\,\\,x\\left(-8\\right)=-8x\\\\\\text{Inner}:\\,\\,\\,\\,\\,\\,\\,8\\left(x\\right)=+8x\\\\\\text{Last}:\\,\\,\\,\\,\\,\\,\\,\\,\\,8\\left(-8\\right)=-64\\end{array}[\/latex]<\/p>\r\nAdd the terms. Note how the two x terms are opposites, so they sum to zero.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}x^{2}\\underbrace{-8x+8x}-64\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=0\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\text{ }\\\\=x^2-64\\\\\\text{ }\\\\\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(x+8\\right)\\left(x-8\\right)=x^{2}-64[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Think About It<\/h3>\r\n<p style=\"text-align: left\">There are predictable outcomes when you square a binomial sum or difference. In general terms, for a binomial difference,<\/p>\r\n<p style=\"text-align: center\">[latex]\\left(a-b\\right)^{2}=\\left(a-b\\right)\\left(a-b\\right)[\/latex],<\/p>\r\n<p style=\"text-align: left\">the resulting product, after being simplified, will look like this:<\/p>\r\n<p style=\"text-align: center\">[latex]a^2-2ab+b^2[\/latex].<\/p>\r\n<p style=\"text-align: left\">The product of a binomial sum will have the following predictable outcome:<\/p>\r\n<p style=\"text-align: center\">[latex]\\left(a+b\\right)^{2}=\\left(a+b\\right)\\left(a+b\\right)=a^2+2ab+b^2[\/latex].<\/p>\r\n<p style=\"text-align: left\">Note that a and b in these generalizations could be integers, fractions, or variables with any kind of constant. \u00a0You will learn more about predictable patterns from products of binomials in later math classes.<\/p>\r\n\r\n<\/div>\r\nIn this section we showed two ways to find the product of two binomials, the FOIL method, and by using a table. Some of the forms a product of two binomials can take are listed here:\r\n<ul>\r\n \t<li>[latex]\\left(x+5\\right)\\left(2x-3\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\left(x+7\\right)^{2}[\/latex]<\/li>\r\n \t<li>[latex]\\left(x-1\\right)^{2}[\/latex]<\/li>\r\n \t<li>[latex]\\left(2-y\\right)\\left(5+y\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\left(x+9\\right)\\left(x-9\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\left(2x-4\\right)\\left(x+3\\right)[\/latex]<\/li>\r\n<\/ul>\r\nAnd this is just a small list, the possible combinations are endless. For each of the products in the list, using one of the two methods presented here will work to simplify.\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 simplified form if all of its like terms have been combined.","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Apply the FOIL method to multiply two binomials<\/li>\n<li>Use a table to multiply two binomials<\/li>\n<li>Simplify the product of two binomials given a wide variety of variables, constants, signs, and arrangement of terms in the binomial<\/li>\n<\/ul>\n<\/div>\n<h2><\/h2>\n<div id=\"attachment_4589\" style=\"width: 345px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4589\" class=\"wp-image-4589\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185557\/Screen-Shot-2016-06-03-at-2.08.13-PM-243x300.png\" alt=\"Crane made from aluminum foil\" width=\"335\" height=\"414\" \/><\/p>\n<p id=\"caption-attachment-4589\" class=\"wp-caption-text\">Foil Crane<\/p>\n<\/div>\n<p><span style=\"color: #000000\">In the last section we finished with an example of multiplying two binomials,[latex]\\left(x+4\\right)\\left(x+2\\right)[\/latex]. In this section we will provide examples of how to use two different methods to multiply to binomials. Keep in mind as you read through the page that simplify and multiply are used interchangeably.<\/span><\/p>\n<h2>FOIL<\/h2>\n<p>Some people use the FOIL method to keep track of which pairs of terms have been multiplied when you are multiplying two binomials. This is not the same thing you use to wrap up leftovers, but an acronym for <strong>First, Outer, Inner, Last.<\/strong> Let&#8217;s go back to the example from the previous page, where we were asked to multiply the two binomials: [latex]\\left(x+4\\right)\\left(x+2\\right)[\/latex]. \u00a0The following steps show you how to apply this method to multiplying two binomials.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\text{First}\\text{ term in each binomial}: \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left(x+4\\right)\\left(x+2\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\left(x\\right)=x^{2}\\\\\\text{Outer terms}:\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left(x+4\\right)\\left(x+2\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\left(2\\right)=2x\\\\\\text{Inner terms}:\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left(x+4\\right)\\left(x+2\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4\\left(x\\right)=4x\\\\\\text{Last terms in each binomial}:\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left(x+4\\right)\\left(x+2\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4\\left(2\\right)=8\\end{array}[\/latex]<\/p>\n<p>When you add the four results, you get the same answer,\u00a0[latex]x^{2}+2x+4x+8=x^{2}+6x+8[\/latex].<br \/>\nThe last step in multiplying polynomials is to combine like terms. Remember that a polynomial is simplified only when there are no like terms remaining.<\/p>\n<div class=\"textbox shaded\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07184952\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"41\" height=\"36\" \/>Caution! Note that the FOIL method only works for multiplying two binomials together. It sill not work for multiplying a binomial and a trinomial, or two trinomials.<\/div>\n<div id=\"attachment_4595\" style=\"width: 148px\" class=\"wp-caption alignleft\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4595\" class=\"wp-image-4595\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185559\/Screen-Shot-2016-06-03-at-4.09.32-PM-299x300.png\" alt=\"two tomatoes sitting next to each other with two different phonetic pronunciations for the word tomato underneath\" width=\"138\" height=\"139\" \/><\/p>\n<p id=\"caption-attachment-4595\" class=\"wp-caption-text\">Order Doesn&#8217;t Matter When You Multiply<\/p>\n<\/div>\n<p>One of the neat things about multiplication is that\u00a0terms can be multiplied in either order. The expression [latex]\\left(x+2\\right)\\left(x+4\\right)[\/latex] has the same product as [latex]\\left(x+4\\right)\\left(x+2\\right)[\/latex], [latex]x^{2}+6x+8[\/latex]. (Work it out and see.) The order in which you multiply binomials does not matter. What matters is that you multiply each term in one binomial by each term in the other binomial.<\/p>\n<h2><\/h2>\n<p><span style=\"color: #000000\">Polynomials can take many forms. \u00a0So far we have seen examples of binomials with variable terms on the left and constant terms on the right, such as this binomial [latex]\\left(2r-3\\right)[\/latex]. \u00a0Variables may also be on the right of the constant term, as in this binomial [latex]\\left(5+r\\right)[\/latex]. \u00a0In the next example, we will show that multiplying binomials in this form requires one extra step at the end.<\/span><\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p><span style=\"color: #000000\">Find the product.[latex]\\left(3\u2013s\\right)\\left(1-s\\right)[\/latex]<\/span><br \/>\n<span style=\"color: #000000\"> <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q531601\">Show Solution<\/span><\/span><br \/>\n<span style=\"color: #000000\"> <\/p>\n<div id=\"q531601\" class=\"hidden-answer\" style=\"display: none\"><\/span><\/p>\n<p><span style=\"color: #000000\">Notice how the binomials have the variable on the right instead of the left. \u00a0There is nothing different in the way you find the product. \u00a0At the end we will reorganize terms so they are in descending order as a matter of convention.<\/span><\/p>\n<p><span style=\"color: #000000\">[latex]\\left(3\u2013s\\right)\\left(1\u2013s\\right)[\/latex]<\/span><\/p>\n<p><span style=\"color: #000000\">Use a table this time.<\/span><\/p>\n<table style=\"width: 40%\">\n<tbody>\n<tr>\n<td><span style=\"color: #000000\">\u00a0<\/span><\/td>\n<th><span style=\"color: #000000\">[latex]3[\/latex]<\/span><\/th>\n<th><span style=\"color: #000000\">[latex]-s[\/latex]<\/span><\/th>\n<\/tr>\n<tr>\n<th><span style=\"color: #000000\">[latex]1[\/latex]<\/span><\/th>\n<td><span style=\"color: #000000\">[latex]3[\/latex]<\/span><\/td>\n<td><span style=\"color: #000000\">[latex]-s[\/latex]<\/span><\/td>\n<\/tr>\n<tr>\n<th><span style=\"color: #000000\">[latex]-s[\/latex]<\/span><\/th>\n<td><span style=\"color: #000000\">[latex]-3s[\/latex]<\/span><\/td>\n<td><span style=\"color: #000000\">[latex]s^2[\/latex]<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span style=\"color: #000000\">Notice how the <em>s<\/em> term is now positive. Collect the terms and simplify.<\/span><\/p>\n<p style=\"text-align: center\"><span style=\"color: #000000\">[latex]\\begin{array}{c}\\left(3\u2013s\\right)\\left(1\u2013s\\right)\\\\\\text{ }\\\\=3-3s-s+s^2\\\\\\text{ }\\\\=3-4s+s^2\\end{array}[\/latex]<\/span><\/p>\n<p><span style=\"color: #000000\">As a matter of convention, we will organize the terms so the one with greatest degree comes first. Pay close attention to the signs on the terms when you reorganize them. The 3 is positive, so we will use a plus in front of it, and the 4 is negative so we use a minus in front of it.<\/span><\/p>\n<p style=\"text-align: center\"><span style=\"color: #000000\">[latex]\\begin{array}{c}\\left(3\u2013s\\right)\\left(1\u2013s\\right)\\\\\\text{ }\\\\=s^{2}-4s+3\\end{array}[\/latex]<\/span><\/p>\n<h4><span style=\"color: #000000\">Answer<\/span><\/h4>\n<p><span style=\"color: #000000\">[latex]\\left(3\u2013s\\right)\\left(1\u2013s\\right)=s^2-4s+3[\/latex]<\/span><\/p>\n<p><span style=\"color: #000000\"><\/div>\n<\/div>\n<p><\/span><\/p>\n<\/div>\n<p><span style=\"color: #000000\">In the next example, you will see that sometimes there are constants in front of the variable. They will get multiplied together just as we have done before.<\/span><\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p><span style=\"color: #000000\">Simplify [latex]\\left(4x\u201310\\right)\\left(2x+3\\right)[\/latex] using the FOIL acronym.<\/span><\/p>\n<p><span style=\"color: #000000\"><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q930433\">Show Solution<\/span><\/span><br \/>\n<span style=\"color: #000000\"> <\/p>\n<div id=\"q930433\" class=\"hidden-answer\" style=\"display: none\"><\/span><\/p>\n<p style=\"text-align: center\"><span style=\"color: #000000\">[latex]\\begin{array}{l}\\text{First}:\\,\\,\\,\\,\\,4x\\left(2x\\right)=8x^{2}\\\\\\text{Outer}:\\,\\,\\,4x\\left(3\\right)=12x\\\\\\text{Inner}:\\,\\,\\,\u221210\\left(2x\\right)=-20x\\\\\\text{Last}:\\,\\,\\,\\,\\,-10\\left(3\\right)=-30\\end{array}[\/latex]<\/span><\/p>\n<p><span style=\"color: #000000\">Be careful about including the negative sign on the [latex]\u201110[\/latex], since 10 is subtracted.<\/span><\/p>\n<p><span style=\"color: #000000\">Combine like terms.<\/span><\/p>\n<p style=\"text-align: center\"><span style=\"color: #000000\">[latex]\\begin{array}{c}8x^{2}+12x\u201320x\u201330\\\\\\text{ }\\\\=8x^{2}-8x\u201330\\end{array}[\/latex]<\/span><\/p>\n<h4><span style=\"color: #000000\">Answer<\/span><\/h4>\n<p><span style=\"color: #000000\">[latex]\\left(4x\u201310\\right)\\left(2x+3\\right)=8x^{2}\u20138x\u201330[\/latex]<\/span><\/p>\n<p><span style=\"color: #000000\"><\/div>\n<\/div>\n<p><\/span><\/p>\n<\/div>\n<p><span style=\"color: #000000\">The video that follows gives another example of multiplying two binomials using the FOIL acronym. Remember this method only works when you are multiplying two binomials.<\/span><\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Multiply Binomials Using the FOIL Acronym\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/_MrdEFnXNGA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>The Table Method<\/h2>\n<p>You may see a binomial multiplied by itself written as\u00a0[latex]{\\left(x+3\\right)}^{2}[\/latex] instead of\u00a0[latex]\\left(x+3\\right)\\left(x+3\\right)[\/latex]. \u00a0To find this product, let&#8217;s use another method. We will place the terms of each binomial along the top row and first column of a table, like this:<\/p>\n<table class=\"lines\" style=\"width: 20%\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]x[\/latex]<\/td>\n<td>[latex]+3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]+3[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Now multiply the term in each column by the term in each row to get the terms of the resulting polynomial. Note how we keep the signs on the terms, even when they are positive, this will help us write the new polynomial.<\/p>\n<table style=\"width: 20%\">\n<thead>\n<tr>\n<td><\/td>\n<td>[latex]x[\/latex]<\/td>\n<td>[latex]+3[\/latex]<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td><span style=\"color: #0000ff\">[latex]x\\cdot{x}=x^2[\/latex]<\/span><\/td>\n<td><span style=\"color: #ff0000\">[latex]3\\cdot{x}=+3x[\/latex]<\/span><\/td>\n<\/tr>\n<tr>\n<td>[latex]+3[\/latex]<\/td>\n<td><span style=\"color: #ff0000\">[latex]x\\cdot{3}=+3x[\/latex]<\/span><\/td>\n<td><span style=\"color: #ff00ff\">\u00a0[latex]3\\cdot{3}=+9[\/latex]<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Now we can write the terms of the polynomial from the entries in the table:<\/p>\n<p style=\"text-align: center\">[latex]\\left(x+3\\right)^{2}[\/latex]<\/p>\n<p style=\"text-align: center\">=\u00a0<span style=\"color: #0000ff\">[latex]x^2[\/latex]<\/span> + <span style=\"color: #ff0000\">[latex]3x[\/latex]<\/span> + <span style=\"color: #ff0000\">[latex]3x[\/latex]<\/span> + <span style=\"color: #ff00ff\">[latex]9[\/latex]<\/span><\/p>\n<p style=\"text-align: center\">= <span style=\"color: #0000ff\">[latex]x^{2}[\/latex]<\/span> + <span style=\"color: #ff0000\">[latex]6x[\/latex]<\/span> + <span style=\"color: #ff00ff\">[latex]9[\/latex]<\/span>.<\/p>\n<p style=\"text-align: center\">Pretty cool, huh?<\/p>\n<p>So far, we have shown two\u00a0methods for multiplying two binomials together. \u00a0Why are we focusing\u00a0so much on binomials? \u00a0They are one of the most well studied and widely used polynomials, so there is a lot of information out there about them. In the previous example, we saw the result of squaring a binomial that was a sum of two terms. In the next example we will find the product of squaring a binomial that is the difference of two terms.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Square the binomial difference\u00a0[latex]\\left(x\u20137\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q293164\">Show solution<\/span><\/p>\n<div id=\"q293164\" class=\"hidden-answer\" style=\"display: none\">\n<p>Write the product of the binomial.<\/p>\n<p style=\"text-align: center\">[latex]{\\left(x-7\\right)}^2=\\left(x\u20137\\right)\\left(x\u20137\\right)[\/latex]<\/p>\n<p style=\"text-align: left\">Let&#8217;s use the table method, just because. Note how we carry the negative sign with the\u00a0[latex]7[\/latex].<\/p>\n<table style=\"width: 20%\">\n<tbody>\n<tr>\n<td><\/td>\n<td>\u00a0[latex]x[\/latex]<\/td>\n<td>\u00a0[latex]-7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>\u00a0[latex]x[\/latex]<\/td>\n<td>\u00a0\u00a0[latex]x^2[\/latex]<\/td>\n<td>\u00a0\u00a0[latex]-7x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>\u00a0\u00a0[latex]-7[\/latex]<\/td>\n<td>\u00a0\u00a0[latex]-7x[\/latex]<\/td>\n<td>\u00a0\u00a0[latex]49[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Collect the terms, and simplify. Note how we keep the sign on each term.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}x^2-7x-7x+49\\\\\\text{ }\\\\=x^2-14x+49\\end{array}[\/latex]<\/p>\n<p>Answer<br \/>\n[latex]x^2-14x+49[\/latex]\n<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07184952\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"51\" height=\"45\" \/>Caution! It is VERY important to remember the caution from the exponents section about squaring a binomial:<\/p>\n<p style=\"text-align: center\">You can&#8217;t move the exponent into a grouped sum because of the order of operations!!!!!<\/p>\n<p style=\"text-align: center\"><strong>INCORRECT:<\/strong> [latex]\\left(2+x\\right)^{2}\\neq2^{2}+x^{2}[\/latex]<\/p>\n<p style=\"text-align: center\"><strong>\u00a0CORRECT:<\/strong> [latex]\\left(2+x\\right)^{2}=\\left(2+x\\right)\\left(2+x\\right)[\/latex]<\/p>\n<\/div>\n<p>In the video that follows, you will see another examples of using a table to multiply two binomials.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Multiply Binomials Using a Table\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/tWsLJ_pn5mQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Further Examples<\/h2>\n<p>The next couple of examples show you some different forms binomials can take. \u00a0In the first, we will square a binomial that has a coefficient in front of the variable, like the product in the first example on this page. In the second we will find\u00a0the product of two binomials that have the variable on the right instead of the left.\u00a0We will use both\u00a0the FOIL method and the table method to simplify.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find the product.\u00a0[latex]\\left(2x+6\\right)^{2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q255359\">Show Solution<\/span><\/p>\n<div id=\"q255359\" class=\"hidden-answer\" style=\"display: none\">We will use the FOIL method.<br \/>\n[latex]\\left(2x+6\\right)^{2}=\\left(2x+6\\right)\\left(2x+6\\right)[\/latex]<br \/>\n[latex]\\begin{array}{l}\\text{First}:\\,\\,\\,\\,\\,2x\\left(2x\\right)=4x^{2}\\\\\\text{Outer}:\\,\\,\\,2x\\left(6\\right)=12x\\\\\\text{Inner}:\\,\\,\\,6\\left(2x\\right)=12x\\\\\\text{Last}:\\,\\,\\,\\,\\,6\\left(6\\right)=36\\end{array}[\/latex]<\/p>\n<p>Now you can collect the terms and simplify:<br \/>\n[latex]\\begin{array}{c}4x^2+12x+12x+36\\\\\\text{ }\\\\=4x^2+24x+36\\end{array}[\/latex]<\/p>\n<p>Answer<\/p>\n<p>[latex](2x+6)^{2}=4x^{2}+24x+36[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the last example, we want to show you another common form a binomial can take, each of the terms in the two binomials is the same, but the signs are different. You will see that in this case, the middle term will disappear.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Multiply the binomials. [latex]\\left(x+8\\right)\\left(x\u20138\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q812247\">Show Solution<\/span><\/p>\n<div id=\"q812247\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\text{First}:\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\left(x\\right)=x^{2}\\\\\\text{Outer}:\\,\\,\\,\\,\\,\\,x\\left(-8\\right)=-8x\\\\\\text{Inner}:\\,\\,\\,\\,\\,\\,\\,8\\left(x\\right)=+8x\\\\\\text{Last}:\\,\\,\\,\\,\\,\\,\\,\\,\\,8\\left(-8\\right)=-64\\end{array}[\/latex]<\/p>\n<p>Add the terms. Note how the two x terms are opposites, so they sum to zero.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}x^{2}\\underbrace{-8x+8x}-64\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=0\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\text{ }\\\\=x^2-64\\\\\\text{ }\\\\\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(x+8\\right)\\left(x-8\\right)=x^{2}-64[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Think About It<\/h3>\n<p style=\"text-align: left\">There are predictable outcomes when you square a binomial sum or difference. In general terms, for a binomial difference,<\/p>\n<p style=\"text-align: center\">[latex]\\left(a-b\\right)^{2}=\\left(a-b\\right)\\left(a-b\\right)[\/latex],<\/p>\n<p style=\"text-align: left\">the resulting product, after being simplified, will look like this:<\/p>\n<p style=\"text-align: center\">[latex]a^2-2ab+b^2[\/latex].<\/p>\n<p style=\"text-align: left\">The product of a binomial sum will have the following predictable outcome:<\/p>\n<p style=\"text-align: center\">[latex]\\left(a+b\\right)^{2}=\\left(a+b\\right)\\left(a+b\\right)=a^2+2ab+b^2[\/latex].<\/p>\n<p style=\"text-align: left\">Note that a and b in these generalizations could be integers, fractions, or variables with any kind of constant. \u00a0You will learn more about predictable patterns from products of binomials in later math classes.<\/p>\n<\/div>\n<p>In this section we showed two ways to find the product of two binomials, the FOIL method, and by using a table. Some of the forms a product of two binomials can take are listed here:<\/p>\n<ul>\n<li>[latex]\\left(x+5\\right)\\left(2x-3\\right)[\/latex]<\/li>\n<li>[latex]\\left(x+7\\right)^{2}[\/latex]<\/li>\n<li>[latex]\\left(x-1\\right)^{2}[\/latex]<\/li>\n<li>[latex]\\left(2-y\\right)\\left(5+y\\right)[\/latex]<\/li>\n<li>[latex]\\left(x+9\\right)\\left(x-9\\right)[\/latex]<\/li>\n<li>[latex]\\left(2x-4\\right)\\left(x+3\\right)[\/latex]<\/li>\n<\/ul>\n<p>And this is just a small list, the possible combinations are endless. For each of the products in the list, using one of the two methods presented here will work to simplify.<\/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 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-4720\">\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>Screenshot: Tomato, Tomato. <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><li>Multiply Binomials Using the FOIL Acronym. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/_MrdEFnXNGA\">https:\/\/youtu.be\/_MrdEFnXNGA<\/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>Multiply Binomials Using a Table. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/tWsLJ_pn5mQ\">https:\/\/youtu.be\/tWsLJ_pn5mQ<\/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>Screenshot: Caution. <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><li>Screenshot: Foil Crane. <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>Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <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><\/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":17533,"menu_order":14,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Screenshot: Tomato, Tomato\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Multiply Binomials Using the FOIL Acronym\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/_MrdEFnXNGA\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Multiply Binomials Using a Table\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/tWsLJ_pn5mQ\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Screenshot: Caution\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and Education\",\"url\":\"http:\/\/nrocnetwork.org\/dm-opentext\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Screenshot: Foil Crane\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"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-4720","chapter","type-chapter","status-publish","hentry"],"part":4674,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/pressbooks\/v2\/chapters\/4720","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/pressbooks\/v2\/chapters\/4720\/revisions"}],"predecessor-version":[{"id":5035,"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/pressbooks\/v2\/chapters\/4720\/revisions\/5035"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/pressbooks\/v2\/parts\/4674"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/pressbooks\/v2\/chapters\/4720\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/wp\/v2\/media?parent=4720"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/pressbooks\/v2\/chapter-type?post=4720"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/wp\/v2\/contributor?post=4720"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/wp-json\/wp\/v2\/license?post=4720"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}