{"id":10838,"date":"2017-06-05T21:25:30","date_gmt":"2017-06-05T21:25:30","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=10838"},"modified":"2017-09-24T00:04:17","modified_gmt":"2017-09-24T00:04:17","slug":"multiplying-two-binomails-using-the-foil-method","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/monroecc-prealgebra\/chapter\/multiplying-two-binomails-using-the-foil-method\/","title":{"raw":"Vertical and FOIL Methods for Multiplying Two Binomials","rendered":"Vertical and FOIL Methods for Multiplying Two Binomials"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use the FOIL method to multiply two binomials<\/li>\r\n \t<li>Use the vertical method to multiply two binomials<\/li>\r\n<\/ul>\r\n<\/div>\r\nRemember that when you multiply a binomial by a binomial you get four terms. Sometimes you can combine like terms to get a trinomial, but sometimes there are no like terms to combine. Let's look at the last example again and pay particular attention to how we got the four terms.\r\n<p style=\"text-align: center;\">[latex]\\left(x+2\\right)\\left(x-y\\right)[\/latex]\r\n[latex]{x}^{2}-\\mathit{\\text{xy}}+2x - 2y[\/latex]<\/p>\r\nWhere did the first term, [latex]{x}^{2}[\/latex], come from?\r\n\r\nIt is the product of [latex]x\\text{ and }x[\/latex], the <strong>first<\/strong> terms in [latex]\\left(x+2\\right)\\text{and}\\left(x-y\\right)[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224439\/CNX_BMath_Figure_10_03_016_img.png\" alt=\"Parentheses x plus 2 times parentheses x minus y is shown. There is a red arrow from the first x to the second. Beside this, \" \/>\r\nThe next term, [latex]-\\mathit{\\text{xy}}[\/latex], is the product of [latex]x\\text{ and }-y[\/latex], the two <strong>outer<\/strong> terms.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224440\/CNX_BMath_Figure_10_03_017_img.png\" alt=\"Parentheses x plus 2 times parentheses x minus y is shown. There is a black arrow from the first x to the second x. There is a red arrow from the first x to the y. Beside this, \" \/>\r\nThe third term, [latex]+2x[\/latex], is the product of [latex]2\\text{ and }x[\/latex], the two <strong>inner<\/strong> terms.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224441\/CNX_BMath_Figure_10_03_018_img.png\" alt=\"Parentheses x plus 2 times parentheses x minus y is shown. There is a black arrow from the first x to the second x. There is a black arrow from the first x to the y. There is a red arrow from the 2 to the x. Below that, \" \/>\r\nAnd the last term, [latex]-2y[\/latex], came from multiplying the two <strong>last<\/strong> terms.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224441\/CNX_BMath_Figure_10_03_019_img.png\" alt=\"Parentheses x plus 2 times parentheses x minus y is shown. There is a black arrow from the first x to the second x. There is a black arrow from the first x to the y. There is a black arrow from the 2 to the x. There is a red arrow from the 2 to the y. Above that, \" \/>\r\nWe abbreviate \"First, Outer, Inner, Last\" as FOIL. The letters stand for \u2018First, Outer, Inner, Last\u2019. The word FOIL is easy to remember and ensures we find all four products. We might say we use the FOIL method to multiply two binomials.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224442\/CNX_BMath_Figure_10_03_025_img.png\" alt=\"Parentheses a plus b times parentheses c plus d is shown. Above a is first, above b is last, above c is first, above d is last. There is a brace connecting a and d that says outer. There is a brace connecting b and c that says inner.\" \/>\r\nLet's look at [latex]\\left(x+3\\right)\\left(x+7\\right)[\/latex] again. Now we will work through an example where we use the FOIL pattern to multiply two binomials.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224444\/CNX_BMath_Figure_10_03_063_img.png\" alt=\".\" \/>\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply using the FOIL method: [latex]\\left(x+6\\right)\\left(x+9\\right)[\/latex]\r\n\r\nSolution\r\n<table id=\"eip-id1168469487011\" class=\"unnumbered unstyled\" summary=\"The first line says, \">\r\n<tbody>\r\n<tr>\r\n<td><strong>Step 1<\/strong>: Multiply the <strong>First<\/strong> terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224445\/CNX_BMath_Figure_10_03_054_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Step 2<\/strong>: Multiply the <strong>Outer<\/strong> terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224447\/CNX_BMath_Figure_10_03_054_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Step 3<\/strong>: Multiply the <strong>Inner<\/strong> terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224448\/CNX_BMath_Figure_10_03_054_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Step 4<\/strong>: Multiply the <strong>Last<\/strong> terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224450\/CNX_BMath_Figure_10_03_054_img-04.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Step 5<\/strong>: Combine like terms, when possible.<\/td>\r\n<td>[latex]x^2+15x+54[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146211[\/ohm_question]\r\n\r\n<\/div>\r\nWe summarize the steps of the FOIL method below. The FOIL method only applies to multiplying binomials, not other polynomials!\r\n<div class=\"textbox shaded\">\r\n<h3>Use the FOIL method for multiplying two binomials<\/h3>\r\n<ol id=\"eip-id1168469711482\" class=\"stepwise\">\r\n \t<li>Multiply the <strong>First<\/strong> terms.<\/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>Combine like terms, when possible.<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224442\/CNX_BMath_Figure_10_03_025_img.png\" alt=\"Parentheses a plus b times parentheses c plus d is shown. Above a is first, above b is last, above c is first, above d is last. There is a brace connecting a and d that says outer. There is a brace connecting b and c that says inner.\" width=\"187\" height=\"137\" \/><\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]\\left(y - 8\\right)\\left(y+6\\right)[\/latex]\r\n[reveal-answer q=\"120940\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"120940\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168466233448\" class=\"unnumbered unstyled\" summary=\"The first line says, \">\r\n<tbody>\r\n<tr>\r\n<td><strong>Step 1<\/strong>: Multiply the <strong>First<\/strong> terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224453\/CNX_BMath_Figure_10_03_055_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Step 2<\/strong>: Multiply the <strong>Outer<\/strong> terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224455\/CNX_BMath_Figure_10_03_055_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Step 3<\/strong>: Multiply the <strong>Inner<\/strong> terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224456\/CNX_BMath_Figure_10_03_055_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Step 4<\/strong>: Multiply the <strong>Last<\/strong> terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224501\/CNX_BMath_Figure_10_03_055_img-04.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Step 5<\/strong>: Combine like terms<\/td>\r\n<td>[latex]y^2-2y-48[\/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=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146212[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]\\left(2a+3\\right)\\left(3a - 1\\right)[\/latex]\r\n[reveal-answer q=\"807420\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"807420\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468512120\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses 2a plus 3 times parentheses 3a minus 1. The next line shows the same thing, but with arrows pointing from the 2a to the 3a, from the 2a to the 1, from the 3 to the 3a, and from the 3 to the 1. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex](2a+3)(3a-1)[\/latex]<\/td>\r\n<\/tr>\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\/24224505\/CNX_BMath_Figure_10_03_056_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the <strong>First<\/strong> terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224506\/CNX_BMath_Figure_10_03_056_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the <strong>Outer<\/strong> terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224507\/CNX_BMath_Figure_10_03_056_img-04.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the <strong>Inner<\/strong> terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224510\/CNX_BMath_Figure_10_03_056_img-05.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the <strong>Last<\/strong> terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224511\/CNX_BMath_Figure_10_03_056_img-06.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]6a^2+7a-3[\/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=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146213[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]\\left(5x-y\\right)\\left(2x - 7\\right)[\/latex]\r\n[reveal-answer q=\"840187\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"840187\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468525268\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses 5x minus y times parentheses 2x minus 7. The next line shows the same thing, but with arrows pointing from the 5x to the 2x, from the 5x to the 7, from the y to the 2x, and from the y to the 7. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex](5x-y)(2x-7)[\/latex]<\/td>\r\n<\/tr>\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\/24224515\/CNX_BMath_Figure_10_03_057_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the <strong>First<\/strong> terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224516\/CNX_BMath_Figure_10_03_057_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the <strong>Outer<\/strong> terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224517\/CNX_BMath_Figure_10_03_057_img-04.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the <strong>Inner<\/strong> terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224518\/CNX_BMath_Figure_10_03_057_img-05.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the <strong>Last<\/strong> terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224519\/CNX_BMath_Figure_10_03_057_img-06.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms. There are none.<\/td>\r\n<td>[latex]10x^2-35x-2xy+7y[\/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=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146215[\/ohm_question]\r\n\r\n<\/div>\r\nFor another example of using the FOIL method to multiply two binomials watch the next video.\r\n\r\nhttps:\/\/youtu.be\/0HzsAjucUaw\r\n<h2>Multiplying Two Binomials Using the Vertical Method<\/h2>\r\nThe FOIL method is usually the quickest method for multiplying two binomials, but it works <em>only<\/em> for binomials. You can use the Distributive Property to find the product of any two polynomials. Another method that works for all polynomials is the Vertical Method. It is very much like the method you use to multiply whole numbers. Look carefully at this example of multiplying two-digit numbers.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224522\/CNX_BMath_Figure_10_03_058_img.png\" alt=\"A vertical multiplication problem is shown. 23 times 46 is written with a line underneath. Beneath the line is 138. Beside 138 is written \" \/>\r\nYou start by multiplying [latex]23[\/latex] by [latex]6[\/latex] to get [latex]138[\/latex].\r\n\r\nThen you multiply [latex]23[\/latex] by [latex]4[\/latex], lining up the partial product in the correct columns.\r\n\r\nLast, you add the partial products.\r\n\r\nNow we'll apply this same method to multiply two binomials.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply using the vertical method: [latex]\\left(5x - 1\\right)\\left(2x - 7\\right)[\/latex]\r\n\r\n[reveal-answer q=\"985210\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"985210\"]\r\n\r\nSolution\r\nIt does not matter which binomial goes on the top. Line up the columns when you multiply as we did when we multiplied [latex]23\\left(46\\right)[\/latex].\r\n<table id=\"eip-id1168469481295\" class=\"unnumbered unstyled\" summary=\"A vertical multiplication problem is shown. 2x minus 7 times 5x minus 1 is written with a line underneath. 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\/24224523\/CNX_BMath_Figure_10_03_059_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply [latex]2x - 7[\/latex] by [latex]-1[\/latex] .<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224524\/CNX_BMath_Figure_10_03_059_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply [latex]2x - 7[\/latex] by [latex]5x[\/latex] .<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224526\/CNX_BMath_Figure_10_03_059_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add like terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224527\/CNX_BMath_Figure_10_03_059_img-04.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice the partial products are the same as the terms in the FOIL method.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224528\/CNX_BMath_Figure_10_03_060_img.png\" alt=\"On the left, 5x minus 1 times 2x minus 7 is shown. Below that is 10 x squared minus 35x minus 2x plus 7. The first two terms are in blue, the second two in red. Beneath that is 10 x squared minus 37x plus 7. On the right, a vertical multiplication problem is shown. 2xx minus 7 times 5x minus 1 is written with a line underneath. Beneath the line is a red negative 2x plus 7. Beneath that is 10 x squared minus 35 x in blue. Beneath that, there is another line. Beneath that line is 10 x squared minus 37x plus 7.\" \/>[\/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]146216[\/ohm_question]\r\n\r\n<\/div>\r\nWe have now used three methods for multiplying binomials. Be sure to practice each method, and try to decide which one you prefer. The three methods are listed here to help you remember them.\r\n<div class=\"textbox shaded\">\r\n<h3>Multiplying Two Binomials<\/h3>\r\nTo multiply binomials, use the:\r\n<ul id=\"eip-id1170323966125\">\r\n \t<li>Distributive Property<\/li>\r\n \t<li>FOIL Method<\/li>\r\n \t<li>Vertical Method<\/li>\r\n<\/ul>\r\nRemember, FOIL only works when multiplying two binomials.\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use the FOIL method to multiply two binomials<\/li>\n<li>Use the vertical method to multiply two binomials<\/li>\n<\/ul>\n<\/div>\n<p>Remember that when you multiply a binomial by a binomial you get four terms. Sometimes you can combine like terms to get a trinomial, but sometimes there are no like terms to combine. Let&#8217;s look at the last example again and pay particular attention to how we got the four terms.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(x+2\\right)\\left(x-y\\right)[\/latex]<br \/>\n[latex]{x}^{2}-\\mathit{\\text{xy}}+2x - 2y[\/latex]<\/p>\n<p>Where did the first term, [latex]{x}^{2}[\/latex], come from?<\/p>\n<p>It is the product of [latex]x\\text{ and }x[\/latex], the <strong>first<\/strong> terms in [latex]\\left(x+2\\right)\\text{and}\\left(x-y\\right)[\/latex].<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224439\/CNX_BMath_Figure_10_03_016_img.png\" alt=\"Parentheses x plus 2 times parentheses x minus y is shown. There is a red arrow from the first x to the second. Beside this,\" \/><br \/>\nThe next term, [latex]-\\mathit{\\text{xy}}[\/latex], is the product of [latex]x\\text{ and }-y[\/latex], the two <strong>outer<\/strong> terms.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224440\/CNX_BMath_Figure_10_03_017_img.png\" alt=\"Parentheses x plus 2 times parentheses x minus y is shown. There is a black arrow from the first x to the second x. There is a red arrow from the first x to the y. Beside this,\" \/><br \/>\nThe third term, [latex]+2x[\/latex], is the product of [latex]2\\text{ and }x[\/latex], the two <strong>inner<\/strong> terms.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224441\/CNX_BMath_Figure_10_03_018_img.png\" alt=\"Parentheses x plus 2 times parentheses x minus y is shown. There is a black arrow from the first x to the second x. There is a black arrow from the first x to the y. There is a red arrow from the 2 to the x. Below that,\" \/><br \/>\nAnd the last term, [latex]-2y[\/latex], came from multiplying the two <strong>last<\/strong> terms.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224441\/CNX_BMath_Figure_10_03_019_img.png\" alt=\"Parentheses x plus 2 times parentheses x minus y is shown. There is a black arrow from the first x to the second x. There is a black arrow from the first x to the y. There is a black arrow from the 2 to the x. There is a red arrow from the 2 to the y. Above that,\" \/><br \/>\nWe abbreviate &#8220;First, Outer, Inner, Last&#8221; as FOIL. The letters stand for \u2018First, Outer, Inner, Last\u2019. The word FOIL is easy to remember and ensures we find all four products. We might say we use the FOIL method to multiply two binomials.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224442\/CNX_BMath_Figure_10_03_025_img.png\" alt=\"Parentheses a plus b times parentheses c plus d is shown. Above a is first, above b is last, above c is first, above d is last. There is a brace connecting a and d that says outer. There is a brace connecting b and c that says inner.\" \/><br \/>\nLet&#8217;s look at [latex]\\left(x+3\\right)\\left(x+7\\right)[\/latex] again. Now we will work through an example where we use the FOIL pattern to multiply two binomials.<br \/>\n<img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224444\/CNX_BMath_Figure_10_03_063_img.png\" alt=\".\" \/><\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply using the FOIL method: [latex]\\left(x+6\\right)\\left(x+9\\right)[\/latex]<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168469487011\" class=\"unnumbered unstyled\" summary=\"The first line says,\">\n<tbody>\n<tr>\n<td><strong>Step 1<\/strong>: Multiply the <strong>First<\/strong> terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224445\/CNX_BMath_Figure_10_03_054_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><strong>Step 2<\/strong>: Multiply the <strong>Outer<\/strong> terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224447\/CNX_BMath_Figure_10_03_054_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><strong>Step 3<\/strong>: Multiply the <strong>Inner<\/strong> terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224448\/CNX_BMath_Figure_10_03_054_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><strong>Step 4<\/strong>: Multiply the <strong>Last<\/strong> terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224450\/CNX_BMath_Figure_10_03_054_img-04.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><strong>Step 5<\/strong>: Combine like terms, when possible.<\/td>\n<td>[latex]x^2+15x+54[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146211\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146211&theme=oea&iframe_resize_id=ohm146211&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>We summarize the steps of the FOIL method below. The FOIL method only applies to multiplying binomials, not other polynomials!<\/p>\n<div class=\"textbox shaded\">\n<h3>Use the FOIL method for multiplying two binomials<\/h3>\n<ol id=\"eip-id1168469711482\" class=\"stepwise\">\n<li>Multiply the <strong>First<\/strong> terms.<\/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>Combine like terms, when possible.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224442\/CNX_BMath_Figure_10_03_025_img.png\" alt=\"Parentheses a plus b times parentheses c plus d is shown. Above a is first, above b is last, above c is first, above d is last. There is a brace connecting a and d that says outer. There is a brace connecting b and c that says inner.\" width=\"187\" height=\"137\" \/><\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]\\left(y - 8\\right)\\left(y+6\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q120940\">Show Solution<\/span><\/p>\n<div id=\"q120940\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168466233448\" class=\"unnumbered unstyled\" summary=\"The first line says,\">\n<tbody>\n<tr>\n<td><strong>Step 1<\/strong>: Multiply the <strong>First<\/strong> terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224453\/CNX_BMath_Figure_10_03_055_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><strong>Step 2<\/strong>: Multiply the <strong>Outer<\/strong> terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224455\/CNX_BMath_Figure_10_03_055_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><strong>Step 3<\/strong>: Multiply the <strong>Inner<\/strong> terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224456\/CNX_BMath_Figure_10_03_055_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><strong>Step 4<\/strong>: Multiply the <strong>Last<\/strong> terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224501\/CNX_BMath_Figure_10_03_055_img-04.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><strong>Step 5<\/strong>: Combine like terms<\/td>\n<td>[latex]y^2-2y-48[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\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=\"ohm146212\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146212&theme=oea&iframe_resize_id=ohm146212&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]\\left(2a+3\\right)\\left(3a - 1\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q807420\">Show Solution<\/span><\/p>\n<div id=\"q807420\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468512120\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses 2a plus 3 times parentheses 3a minus 1. The next line shows the same thing, but with arrows pointing from the 2a to the 3a, from the 2a to the 1, from the 3 to the 3a, and from the 3 to the 1. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex](2a+3)(3a-1)[\/latex]<\/td>\n<\/tr>\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\/24224505\/CNX_BMath_Figure_10_03_056_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply the <strong>First<\/strong> terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224506\/CNX_BMath_Figure_10_03_056_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply the <strong>Outer<\/strong> terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224507\/CNX_BMath_Figure_10_03_056_img-04.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply the <strong>Inner<\/strong> terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224510\/CNX_BMath_Figure_10_03_056_img-05.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply the <strong>Last<\/strong> terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224511\/CNX_BMath_Figure_10_03_056_img-06.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]6a^2+7a-3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\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=\"ohm146213\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146213&theme=oea&iframe_resize_id=ohm146213&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]\\left(5x-y\\right)\\left(2x - 7\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q840187\">Show Solution<\/span><\/p>\n<div id=\"q840187\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468525268\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses 5x minus y times parentheses 2x minus 7. The next line shows the same thing, but with arrows pointing from the 5x to the 2x, from the 5x to the 7, from the y to the 2x, and from the y to the 7. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex](5x-y)(2x-7)[\/latex]<\/td>\n<\/tr>\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\/24224515\/CNX_BMath_Figure_10_03_057_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply the <strong>First<\/strong> terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224516\/CNX_BMath_Figure_10_03_057_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply the <strong>Outer<\/strong> terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224517\/CNX_BMath_Figure_10_03_057_img-04.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply the <strong>Inner<\/strong> terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224518\/CNX_BMath_Figure_10_03_057_img-05.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply the <strong>Last<\/strong> terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224519\/CNX_BMath_Figure_10_03_057_img-06.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Combine like terms. There are none.<\/td>\n<td>[latex]10x^2-35x-2xy+7y[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\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=\"ohm146215\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146215&theme=oea&iframe_resize_id=ohm146215&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>For another example of using the FOIL method to multiply two binomials watch the next video.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Find The Product of Two Binomials  (09x-52)\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/0HzsAjucUaw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Multiplying Two Binomials Using the Vertical Method<\/h2>\n<p>The FOIL method is usually the quickest method for multiplying two binomials, but it works <em>only<\/em> for binomials. You can use the Distributive Property to find the product of any two polynomials. Another method that works for all polynomials is the Vertical Method. It is very much like the method you use to multiply whole numbers. Look carefully at this example of multiplying two-digit numbers.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224522\/CNX_BMath_Figure_10_03_058_img.png\" alt=\"A vertical multiplication problem is shown. 23 times 46 is written with a line underneath. Beneath the line is 138. Beside 138 is written\" \/><br \/>\nYou start by multiplying [latex]23[\/latex] by [latex]6[\/latex] to get [latex]138[\/latex].<\/p>\n<p>Then you multiply [latex]23[\/latex] by [latex]4[\/latex], lining up the partial product in the correct columns.<\/p>\n<p>Last, you add the partial products.<\/p>\n<p>Now we&#8217;ll apply this same method to multiply two binomials.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply using the vertical method: [latex]\\left(5x - 1\\right)\\left(2x - 7\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q985210\">Show Solution<\/span><\/p>\n<div id=\"q985210\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nIt does not matter which binomial goes on the top. Line up the columns when you multiply as we did when we multiplied [latex]23\\left(46\\right)[\/latex].<\/p>\n<table id=\"eip-id1168469481295\" class=\"unnumbered unstyled\" summary=\"A vertical multiplication problem is shown. 2x minus 7 times 5x minus 1 is written with a line underneath. 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\/24224523\/CNX_BMath_Figure_10_03_059_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply [latex]2x - 7[\/latex] by [latex]-1[\/latex] .<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224524\/CNX_BMath_Figure_10_03_059_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply [latex]2x - 7[\/latex] by [latex]5x[\/latex] .<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224526\/CNX_BMath_Figure_10_03_059_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Add like terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224527\/CNX_BMath_Figure_10_03_059_img-04.png\" alt=\".\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice the partial products are the same as the terms in the FOIL method.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224528\/CNX_BMath_Figure_10_03_060_img.png\" alt=\"On the left, 5x minus 1 times 2x minus 7 is shown. Below that is 10 x squared minus 35x minus 2x plus 7. The first two terms are in blue, the second two in red. Beneath that is 10 x squared minus 37x plus 7. On the right, a vertical multiplication problem is shown. 2xx minus 7 times 5x minus 1 is written with a line underneath. Beneath the line is a red negative 2x plus 7. Beneath that is 10 x squared minus 35 x in blue. Beneath that, there is another line. Beneath that line is 10 x squared minus 37x plus 7.\" \/><\/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=\"ohm146216\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146216&theme=oea&iframe_resize_id=ohm146216&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>We have now used three methods for multiplying binomials. Be sure to practice each method, and try to decide which one you prefer. The three methods are listed here to help you remember them.<\/p>\n<div class=\"textbox shaded\">\n<h3>Multiplying Two Binomials<\/h3>\n<p>To multiply binomials, use the:<\/p>\n<ul id=\"eip-id1170323966125\">\n<li>Distributive Property<\/li>\n<li>FOIL Method<\/li>\n<li>Vertical Method<\/li>\n<\/ul>\n<p>Remember, FOIL only works when multiplying two binomials.<\/p>\n<\/div>\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-10838\">\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>Question ID 146215, 146213, 146212, 146211. <strong>Authored 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>Find The Product of Two Binomials (09x-52). <strong>Authored by<\/strong>: James Sousa (mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/0HzsAjucUaw\">https:\/\/youtu.be\/0HzsAjucUaw<\/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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <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\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/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":21046,"menu_order":43,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"original\",\"description\":\"Question ID 146215, 146213, 146212, 146211\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Find The Product of Two Binomials (09x-52)\",\"author\":\"James Sousa (mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/0HzsAjucUaw\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"9b49e184-8291-4a60-bca0-02a1936fafa8","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-10838","chapter","type-chapter","status-publish","hentry"],"part":5884,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/monroecc-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/10838","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/monroecc-prealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/monroecc-prealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/monroecc-prealgebra\/wp-json\/wp\/v2\/users\/21046"}],"version-history":[{"count":24,"href":"https:\/\/courses.lumenlearning.com\/monroecc-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/10838\/revisions"}],"predecessor-version":[{"id":15171,"href":"https:\/\/courses.lumenlearning.com\/monroecc-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/10838\/revisions\/15171"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/monroecc-prealgebra\/wp-json\/pressbooks\/v2\/parts\/5884"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/monroecc-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/10838\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/monroecc-prealgebra\/wp-json\/wp\/v2\/media?parent=10838"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/monroecc-prealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=10838"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/monroecc-prealgebra\/wp-json\/wp\/v2\/contributor?post=10838"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/monroecc-prealgebra\/wp-json\/wp\/v2\/license?post=10838"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}