{"id":4713,"date":"2017-06-07T18:55:50","date_gmt":"2017-06-07T18:55:50","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/chapter\/read-subtract-polynomials\/"},"modified":"2017-08-15T20:31:42","modified_gmt":"2017-08-15T20:31:42","slug":"read-subtract-polynomials","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/chapter\/read-subtract-polynomials\/","title":{"raw":"Subtract Polynomials","rendered":"Subtract Polynomials"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Find the opposite of a polynomial<\/li>\r\n \t<li>Subtract polynomials using both horizontal and vertical organization<\/li>\r\n<\/ul>\r\n<\/div>\r\n\r\n[caption id=\"attachment_4552\" align=\"aligncenter\" width=\"385\"]<img class=\"wp-image-4552\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185546\/Screen-Shot-2016-06-02-at-1.14.57-PM-300x198.png\" alt=\"Man in a leather jacket and slicked back hair looking very rebellious sitting next to a woman in a pink dress looking very proper.\" width=\"385\" height=\"254\" \/> Opposites[\/caption]\r\n<h2>Find the opposite of a polynomial<\/h2>\r\n<img class=\"wp-image-4554 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185549\/Screen-Shot-2016-06-02-at-1.22.59-PM.png\" alt=\"SCale with a(b+c) on one side and ab+ac on the other adn an equal sign in between the two sides of the scale\" width=\"165\" height=\"123\" \/>When you are solving equations, it may come up that you need to subtract polynomials. This means subtracting each term of a polynomial, which requires\u00a0changing the sign of each term in a polynomial. Recall that changing the sign\u00a0of 3 gives [latex]\u22123[\/latex], and changing the sign\u00a0of [latex]\u22123[\/latex] gives 3. Just as changing the sign\u00a0of a number is found by multiplying the number by [latex]\u22121[\/latex], we can change the sign\u00a0of a polynomial by multiplying it by [latex]\u22121[\/latex]. Think of this in the same way as you would the distributive property. \u00a0You are distributing [latex]\u22121[\/latex] to each term in the polynomial. \u00a0Changing the sign of a polynomial is also called finding the opposite.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind the opposite of [latex]9x^{2}+10x+5[\/latex].\r\n\r\n[reveal-answer q=\"161313\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"161313\"]Find the opposite by multiplying by [latex]\u22121[\/latex].\r\n<p style=\"text-align: center\">[latex]\\left(-1\\right)\\left(9x^{2}+10x+5\\right)[\/latex]<\/p>\r\nDistribute [latex]\u22121[\/latex] to each term in the polynomial.\r\n<p style=\"text-align: center\">[latex]\\left(-1\\right)9x^{2}+\\left(-1\\right)10x+\\left(-1\\right)5[\/latex]<\/p>\r\nYour new terms all have the opposite sign:\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\left(-1\\right)9x^{2}=-9x^{2}\\\\\\text{ }\\\\\\left(-1\\right)10x=-10x\\\\\\text{ }\\\\\\left(-1\\right)5=-5\\end{array}[\/latex]<\/p>\r\nNow you can rewrite the polynomial with the new sign on each term:\r\n<p style=\"text-align: center\">[latex]-9x^{2}-10x-5[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nThe opposite of [latex]9x^{2}+10x+5[\/latex] is [latex]-9x^{2}-10x-5[\/latex]\r\n\r\nYou can also write:\r\n\r\n[latex]\\left(-1\\right)\\left(9x^{2}+10x+5\\right)=-9x^{2}-10x-5[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\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=\"46\" height=\"41\" \/>Be careful when there are negative terms\u00a0or subtractions in the polynomial already. \u00a0Just remember that you are changing the sign, so if it is negative, it will become positive.<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind the opposite of [latex]3p^{2}\u20135p+7[\/latex].\r\n\r\n[reveal-answer q=\"278382\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"278382\"]Find the opposite by multiplying by [latex]-1[\/latex].\r\n<p style=\"text-align: center\">[latex]\\left(-1\\right)\\left(3p^{2}-5p+7\\right)[\/latex]<\/p>\r\nDistribute [latex]-1[\/latex] to each term in the polynomial by multiplying each coefficient by [latex]-1[\/latex].\r\n<p style=\"text-align: center\">[latex]\\left(-1\\right)3p^{2}+\\left(-1\\right)\\left(-5p\\right)+\\left(-1\\right)7[\/latex]<\/p>\r\nYour new terms all have the opposite sign:\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\left(-1\\right)3p^{2}=-3p^{2}\\\\\\text{ }\\\\\\left(-1\\right)\\left(-5p\\right)=5p\\\\\\text{ }\\\\\\left(-1\\right)7=-7\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left\">Now you can rewrite the polynomial with the new sign on each term:<\/p>\r\n<p style=\"text-align: center\">[latex]-3p^{2}+5p-7[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nThe opposite of [latex]3p^{2}-5p+7[\/latex] is [latex]-3p^{2}+5p-7[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNotice that in finding the opposite of a polynomial, you change the sign of <i>each term<\/i> in the polynomial, then rewrite the polynomial with the new signs on each term.\r\n<h2>Subtract polynomials<\/h2>\r\nWhen you subtract one polynomial from another, you will first find the opposite of the polynomial being subtracted, then combine like terms. The easiest mistake to make when subtracting one polynomial from another is to forget to change the sign of EVERY term in the polynomial being subtracted.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSubtract. [latex]\\left(15x^{2}+12x+20\\right)\u2013\\left(9x^{2}+10x+5\\right)[\/latex]\r\n\r\n[reveal-answer q=\"267023\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"267023\"]Change\u00a0the sign of <i>each<\/i> term in the polynomial [latex]9x^{2}+10x+5[\/latex]! All the terms are positive, so they will all become negative.\r\n<p style=\"text-align: center\">[latex]\\left(15x^{2}+12x+20\\right)-9x^{2}-10x-5[\/latex]<\/p>\r\nRegroup to match like terms, remember to check\u00a0the sign of each term.\r\n<p style=\"text-align: center\">[latex]15x^{2}-9x^{2}+12x-10x+20-5[\/latex]<\/p>\r\nCombine like terms.\r\n<p style=\"text-align: center\">[latex]6x^{2}+2x+15[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(15x^{2}+12x+20\\right)-\\left(9x^{2}+10x+5\\right)=6x^{2}+2x+15[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\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=\"52\" height=\"47\" \/>When polynomials include a lot of terms, it can be easy to lose track of the signs. Be careful to transfer them correctly, especially when subtracting a negative term.<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSubtract. [latex]\\left(14x^{3}+3x^{2}\u20135x+14\\right)\u2013\\left(7x^{3}+5x^{2}\u20138x+10\\right)[\/latex]\r\n\r\n[reveal-answer q=\"783926\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"783926\"]Change the sign of each term in the polynomial [latex]7x^{3}+5x^{2}\u20138x+10[\/latex]\r\n<p style=\"text-align: center\">[latex]\\left(14x^{3}+3x^{2}-5x+14\\right)-7x^{3}-5x^{2}+8x-10[\/latex]<\/p>\r\nRegroup to put like terms together and combine like terms.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\underbrace{14x^{3}-7x^{3}}+\\underbrace{3x^{2}-5x^{2}}-\\underbrace{5x+8x}+\\underbrace{14-10}\\\\=7x^{3}\\,\\,\\,\\,\\,\\,\\,\\,\\,=-2x^{2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=3x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=4\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left\">Write the resulting polynomial with each term's sign in front.<\/p>\r\n<p style=\"text-align: center\">[latex]7x^{3}-2x^{2}+3x+4[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(14x^{3}+3x^{2}-5x+14\\right)-\\left(7x^{3}+5x^{2}-8x+10\\right)=7x^{3}-2x^{2}+3x+4[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWhen you have many terms,\u00a0like in the example above, try the vertical approach from the previous page to keep your terms organized. \u00a0However you choose to combine polynomials is up to you\u2014the key point is to identify like terms, keep track of their signs, and be able to organize them accurately.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSubtract. [latex]\\left(14x^{3}+3x^{2}\u20135x+14\\right)\u2013\\left(7x^{3}+5x^{2}\u20138x+10\\right)[\/latex]\r\n\r\n[reveal-answer q=\"29114\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"29114\"]Reorganizing using the vertical approach.\r\n<p style=\"text-align: center\">[latex]14x^{3}+3x^{2}-5x+14-\\left(7x^{3}+5x^{2}-8x+10\\right)[\/latex]<\/p>\r\nChange the signs, and combine like terms.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}14x^{3}+3x^{2}-5x+14\\,\\,\\,\\,\\\\\\underline{-7x^{3}-5x^{2}+8x-10}\\\\=7x^{3}-2x^{2}+3x+4\\,\\,\\,\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(14x^{3}+3x^{2}-5x+14\\right)-\\left(7x^{3}+5x^{2}-8x+10\\right)=7x^{3}-2x^{2}+3x+4[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, you will see more examples of subtracting polynomials.\r\n\r\nhttps:\/\/youtu.be\/xq-zVm25VC0\r\n<h2>Summary<\/h2>\r\nWe have seen that subtracting a polynomial means changing the sign of each term in the polynomial and then reorganizing all the terms to make it easier to combine those that are alike. \u00a0How you organize this process is up to you, but we have shown two ways here. \u00a0One method is to place the terms next to each other horizontally, putting like terms next to each other to make combining them easier. \u00a0The other method was to place the polynomial being subtracted underneath the other after changing the signs of each term. In this method it is important to align like terms and use a blank space when there is no like term.\r\n<h2><\/h2>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Find the opposite of a polynomial<\/li>\n<li>Subtract polynomials using both horizontal and vertical organization<\/li>\n<\/ul>\n<\/div>\n<div id=\"attachment_4552\" style=\"width: 395px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4552\" class=\"wp-image-4552\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185546\/Screen-Shot-2016-06-02-at-1.14.57-PM-300x198.png\" alt=\"Man in a leather jacket and slicked back hair looking very rebellious sitting next to a woman in a pink dress looking very proper.\" width=\"385\" height=\"254\" \/><\/p>\n<p id=\"caption-attachment-4552\" class=\"wp-caption-text\">Opposites<\/p>\n<\/div>\n<h2>Find the opposite of a polynomial<\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-4554 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185549\/Screen-Shot-2016-06-02-at-1.22.59-PM.png\" alt=\"SCale with a(b+c) on one side and ab+ac on the other adn an equal sign in between the two sides of the scale\" width=\"165\" height=\"123\" \/>When you are solving equations, it may come up that you need to subtract polynomials. This means subtracting each term of a polynomial, which requires\u00a0changing the sign of each term in a polynomial. Recall that changing the sign\u00a0of 3 gives [latex]\u22123[\/latex], and changing the sign\u00a0of [latex]\u22123[\/latex] gives 3. Just as changing the sign\u00a0of a number is found by multiplying the number by [latex]\u22121[\/latex], we can change the sign\u00a0of a polynomial by multiplying it by [latex]\u22121[\/latex]. Think of this in the same way as you would the distributive property. \u00a0You are distributing [latex]\u22121[\/latex] to each term in the polynomial. \u00a0Changing the sign of a polynomial is also called finding the opposite.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find the opposite of [latex]9x^{2}+10x+5[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q161313\">Show Solution<\/span><\/p>\n<div id=\"q161313\" class=\"hidden-answer\" style=\"display: none\">Find the opposite by multiplying by [latex]\u22121[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\left(-1\\right)\\left(9x^{2}+10x+5\\right)[\/latex]<\/p>\n<p>Distribute [latex]\u22121[\/latex] to each term in the polynomial.<\/p>\n<p style=\"text-align: center\">[latex]\\left(-1\\right)9x^{2}+\\left(-1\\right)10x+\\left(-1\\right)5[\/latex]<\/p>\n<p>Your new terms all have the opposite sign:<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\left(-1\\right)9x^{2}=-9x^{2}\\\\\\text{ }\\\\\\left(-1\\right)10x=-10x\\\\\\text{ }\\\\\\left(-1\\right)5=-5\\end{array}[\/latex]<\/p>\n<p>Now you can rewrite the polynomial with the new sign on each term:<\/p>\n<p style=\"text-align: center\">[latex]-9x^{2}-10x-5[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>The opposite of [latex]9x^{2}+10x+5[\/latex] is [latex]-9x^{2}-10x-5[\/latex]<\/p>\n<p>You can also write:<\/p>\n<p>[latex]\\left(-1\\right)\\left(9x^{2}+10x+5\\right)=-9x^{2}-10x-5[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\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=\"46\" height=\"41\" \/>Be careful when there are negative terms\u00a0or subtractions in the polynomial already. \u00a0Just remember that you are changing the sign, so if it is negative, it will become positive.<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find the opposite of [latex]3p^{2}\u20135p+7[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q278382\">Show Solution<\/span><\/p>\n<div id=\"q278382\" class=\"hidden-answer\" style=\"display: none\">Find the opposite by multiplying by [latex]-1[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\left(-1\\right)\\left(3p^{2}-5p+7\\right)[\/latex]<\/p>\n<p>Distribute [latex]-1[\/latex] to each term in the polynomial by multiplying each coefficient by [latex]-1[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\left(-1\\right)3p^{2}+\\left(-1\\right)\\left(-5p\\right)+\\left(-1\\right)7[\/latex]<\/p>\n<p>Your new terms all have the opposite sign:<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\left(-1\\right)3p^{2}=-3p^{2}\\\\\\text{ }\\\\\\left(-1\\right)\\left(-5p\\right)=5p\\\\\\text{ }\\\\\\left(-1\\right)7=-7\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left\">Now you can rewrite the polynomial with the new sign on each term:<\/p>\n<p style=\"text-align: center\">[latex]-3p^{2}+5p-7[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>The opposite of [latex]3p^{2}-5p+7[\/latex] is [latex]-3p^{2}+5p-7[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Notice that in finding the opposite of a polynomial, you change the sign of <i>each term<\/i> in the polynomial, then rewrite the polynomial with the new signs on each term.<\/p>\n<h2>Subtract polynomials<\/h2>\n<p>When you subtract one polynomial from another, you will first find the opposite of the polynomial being subtracted, then combine like terms. The easiest mistake to make when subtracting one polynomial from another is to forget to change the sign of EVERY term in the polynomial being subtracted.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Subtract. [latex]\\left(15x^{2}+12x+20\\right)\u2013\\left(9x^{2}+10x+5\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q267023\">Show Solution<\/span><\/p>\n<div id=\"q267023\" class=\"hidden-answer\" style=\"display: none\">Change\u00a0the sign of <i>each<\/i> term in the polynomial [latex]9x^{2}+10x+5[\/latex]! All the terms are positive, so they will all become negative.<\/p>\n<p style=\"text-align: center\">[latex]\\left(15x^{2}+12x+20\\right)-9x^{2}-10x-5[\/latex]<\/p>\n<p>Regroup to match like terms, remember to check\u00a0the sign of each term.<\/p>\n<p style=\"text-align: center\">[latex]15x^{2}-9x^{2}+12x-10x+20-5[\/latex]<\/p>\n<p>Combine like terms.<\/p>\n<p style=\"text-align: center\">[latex]6x^{2}+2x+15[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(15x^{2}+12x+20\\right)-\\left(9x^{2}+10x+5\\right)=6x^{2}+2x+15[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\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=\"52\" height=\"47\" \/>When polynomials include a lot of terms, it can be easy to lose track of the signs. Be careful to transfer them correctly, especially when subtracting a negative term.<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Subtract. [latex]\\left(14x^{3}+3x^{2}\u20135x+14\\right)\u2013\\left(7x^{3}+5x^{2}\u20138x+10\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q783926\">Show Solution<\/span><\/p>\n<div id=\"q783926\" class=\"hidden-answer\" style=\"display: none\">Change the sign of each term in the polynomial [latex]7x^{3}+5x^{2}\u20138x+10[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\left(14x^{3}+3x^{2}-5x+14\\right)-7x^{3}-5x^{2}+8x-10[\/latex]<\/p>\n<p>Regroup to put like terms together and combine like terms.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\underbrace{14x^{3}-7x^{3}}+\\underbrace{3x^{2}-5x^{2}}-\\underbrace{5x+8x}+\\underbrace{14-10}\\\\=7x^{3}\\,\\,\\,\\,\\,\\,\\,\\,\\,=-2x^{2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=3x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=4\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left\">Write the resulting polynomial with each term&#8217;s sign in front.<\/p>\n<p style=\"text-align: center\">[latex]7x^{3}-2x^{2}+3x+4[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(14x^{3}+3x^{2}-5x+14\\right)-\\left(7x^{3}+5x^{2}-8x+10\\right)=7x^{3}-2x^{2}+3x+4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>When you have many terms,\u00a0like in the example above, try the vertical approach from the previous page to keep your terms organized. \u00a0However you choose to combine polynomials is up to you\u2014the key point is to identify like terms, keep track of their signs, and be able to organize them accurately.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Subtract. [latex]\\left(14x^{3}+3x^{2}\u20135x+14\\right)\u2013\\left(7x^{3}+5x^{2}\u20138x+10\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q29114\">Show Solution<\/span><\/p>\n<div id=\"q29114\" class=\"hidden-answer\" style=\"display: none\">Reorganizing using the vertical approach.<\/p>\n<p style=\"text-align: center\">[latex]14x^{3}+3x^{2}-5x+14-\\left(7x^{3}+5x^{2}-8x+10\\right)[\/latex]<\/p>\n<p>Change the signs, and combine like terms.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}14x^{3}+3x^{2}-5x+14\\,\\,\\,\\,\\\\\\underline{-7x^{3}-5x^{2}+8x-10}\\\\=7x^{3}-2x^{2}+3x+4\\,\\,\\,\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(14x^{3}+3x^{2}-5x+14\\right)-\\left(7x^{3}+5x^{2}-8x+10\\right)=7x^{3}-2x^{2}+3x+4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, you will see more examples of subtracting polynomials.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Subtracting Polynomials\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/xq-zVm25VC0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Summary<\/h2>\n<p>We have seen that subtracting a polynomial means changing the sign of each term in the polynomial and then reorganizing all the terms to make it easier to combine those that are alike. \u00a0How you organize this process is up to you, but we have shown two ways here. \u00a0One method is to place the terms next to each other horizontally, putting like terms next to each other to make combining them easier. \u00a0The other method was to place the polynomial being subtracted underneath the other after changing the signs of each term. In this method it is important to align like terms and use a blank space when there is no like term.<\/p>\n<h2><\/h2>\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-4713\">\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: Opposites. <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>Image: 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: Distributive Property Scales, Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Subtracting Polynomials. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/xq-zVm25VC0\">https:\/\/youtu.be\/xq-zVm25VC0<\/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, 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":12,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Screenshot: Opposites\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Image: Caution\",\"author\":\"\",\"organization\":\"Lumen 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