{"id":10814,"date":"2017-06-05T21:00:17","date_gmt":"2017-06-05T21:00:17","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=10814"},"modified":"2020-10-22T09:28:20","modified_gmt":"2020-10-22T09:28:20","slug":"adding-and-subtracting-polynomials","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/adding-and-subtracting-polynomials\/","title":{"raw":"12.2.a - Adding and Subtracting Polynomials","rendered":"12.2.a &#8211; Adding and Subtracting Polynomials"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Add and subtract monomials<\/li>\r\n \t<li>Add and subtract polynomials<\/li>\r\n<\/ul>\r\n<\/div>\r\n\r\n[caption id=\"attachment_4439\" align=\"aligncenter\" width=\"488\"]<img class=\"wp-image-4439\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/07\/27223107\/Apple_and_Orange_-_they_do_not_compare-300x206.jpg\" alt=\"Apple sitting next to an Orange\" width=\"488\" height=\"335\" \/> Apple and Orange[\/caption]\r\n\r\n&nbsp;\r\n<h3>Combining Like Terms<\/h3>\r\nA polynomial may need to be simplified. One way to simplify a polynomial is to combine the <b>like terms<\/b> if there are any. Two or more terms in a polynomial are like terms if they have the same variable (or variables) with the same exponent. For example, [latex]3x^{2}[\/latex] and [latex]-5x^{2}[\/latex] are like terms: They both have [latex]x[\/latex] as the variable, and the exponent is [latex]2[\/latex] for each. However, [latex]3x^{2}[\/latex]\u00a0and [latex]3x[\/latex]\u00a0are not like terms, because their exponents are different.\r\n\r\nHere are some examples of terms that are alike and some that are unlike.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Term<\/td>\r\n<td>Like Terms<\/td>\r\n<td>UNLike Terms<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]a[\/latex]<\/td>\r\n<td>[latex]3a, \\,\\,\\,-2a,\\,\\,\\, \\frac{1}{2}a[\/latex]<\/td>\r\n<td>[latex]a^2,\\,\\,\\,\\frac{1}{a},\\,\\,\\, \\sqrt{a}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]a^2[\/latex]<\/td>\r\n<td>[latex]-5a^2,\\,\\,\\,\\frac{1}{4}a^2,\\,\\,\\, 0.56a^2[\/latex]<\/td>\r\n<td>[latex]\\frac{1}{a^2},\\,\\,\\,\\sqrt{a^2},\\,\\,\\, a^3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]ab[\/latex]<\/td>\r\n<td>[latex]7ab,\\,\\,\\,0.23ab,\\,\\,\\,\\frac{2}{3}ab,\\,\\,\\,-ab[\/latex]<\/td>\r\n<td>[latex]a^2b,\\,\\,\\,\\frac{1}{ab},\\,\\,\\,\\sqrt{ab} [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]ab^2[\/latex]<\/td>\r\n<td>\u00a0[latex]4ab^2,\\,\\,\\, \\frac{ab^2}{7},\\,\\,\\,0.4ab^2,\\,\\,\\, -a^2b[\/latex]<\/td>\r\n<td>\u00a0[latex]a^2b,\\,\\,\\, ab,\\,\\,\\,\\sqrt{ab^2},\\,\\,\\,\\frac{1}{ab^2}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nWhich of these terms are like terms?\r\n<p style=\"text-align: center\">[latex]7x^{3}, 7x, 7y, -8x^{3}, 9y, -3x^{2}, 8y^{2}[\/latex]<\/p>\r\n[reveal-answer q=\"413363\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"413363\"]Like terms must have the same variables, so first identify which terms use the same variables.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}x:7x^{3}, 7x, -8x^{3}, -3x^{2}\\\\y:7y, 9y, 8y^{2}\\end{array}[\/latex]<\/p>\r\nLike terms must also have the same exponents. Identify which terms with the same variables also use the same exponents.\r\n\r\nThe <em>x<\/em> terms [latex]7x^{3}[\/latex]\u00a0and [latex]-8x^{3}[\/latex]\u00a0have the same exponent.\r\n\r\nThe <em>y<\/em> terms [latex]7y[\/latex] and [latex]9y[\/latex] have the same exponent.\r\n<h4>Answer<\/h4>\r\n[latex]7x^{3}[\/latex] and [latex]-8x^{3}[\/latex] are like terms.\r\n\r\n[latex]7y[\/latex] and [latex]9y[\/latex]\u00a0are like terms.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nYou can use the distributive property to simplify the sum of like terms. Recall that the distributive property of addition states that the product of a number and a sum (or difference) is equal to the sum (or difference) of the products.\r\n<p style=\"text-align: center\">[latex]2\\left(3+6\\right)=2\\left(3\\right)+2\\left(6\\right)[\/latex]<\/p>\r\nBoth expressions equal [latex]18[\/latex]. So you can write the expression in whichever form is the most useful.\r\n\r\nLet\u2019s see how we can use this property to combine like terms.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify [latex]3x^{2}-5x^{2}[\/latex].\r\n\r\n[reveal-answer q=\"969840\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"969840\"][latex]3x^{2}[\/latex] and [latex]5x^{2}[\/latex]<sup>\u00a0<\/sup>are like terms.\r\n<p style=\"text-align: center\">[latex]3\\left(x^{2}\\right)-5\\left(x^{2}\\right)[\/latex]<\/p>\r\nWe can rewrite the expression as the product of the difference.\r\n<p style=\"text-align: center\">[latex]\\left(3-5\\right)\\left(x^{2}\\right)[\/latex]<\/p>\r\nCalculate [latex]3\u20135[\/latex].\r\n<p style=\"text-align: center\">[latex]\\left(-2\\right)\\left(x^{2}\\right)[\/latex]<\/p>\r\nWrite the difference of [latex]3 \u2013 5[\/latex] as the new coefficient.\r\n<h4>Answer<\/h4>\r\n[latex]3x^{2}-5x^{2}=-2x^{2}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nYou may have noticed that combining like terms involves combining the coefficients to find the new coefficient of the like term. You can use this as a shortcut.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify [latex]6a^{4}+4a^{4}[\/latex].\r\n\r\n[reveal-answer q=\"840415\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"840415\"]Notice that both terms have a number multiplied by [latex]a^{4}[\/latex]. This makes them like terms.\r\n<p style=\"text-align: center\">[latex]6a^{4}+4a^{4}[\/latex]<\/p>\r\nCombine the coefficients, [latex]6[\/latex] and [latex]4[\/latex].\r\n<p style=\"text-align: center\">[latex]\\left(6+4\\right)\\left(a^{4}\\right)[\/latex]<\/p>\r\nCalculate the sum.\r\n<p style=\"text-align: center\">[latex]\\left(10\\right)\\left(a^{4}\\right)[\/latex]<\/p>\r\nWrite the sum as the new coefficient.\r\n<h4>Answer<\/h4>\r\n[latex]6a^{4}+4a^{4}=10a^{4}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWhen you have a polynomial with more terms, you have to be careful that you combine <i>only<\/i> like terms<i>.<\/i> If two terms are not like terms, you can\u2019t combine them.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify [latex]3x^{2}+3x+x+1+5x[\/latex]\r\n\r\n[reveal-answer q=\"731804\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"731804\"]First identify which terms are <i>like terms<\/i>: only [latex]3x[\/latex], [latex]x[\/latex], and [latex]5x[\/latex]\u00a0are like terms.\r\n\r\n[latex]3x[\/latex], [latex]x[\/latex], and [latex]5x[\/latex] are like terms.\r\n\r\nUse the commutative and associative properties to group the like terms together.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}3x^{2}+3x+x+1+5x\\\\3x^{2}+\\left(3x+x+5x\\right)+1\\end{array}[\/latex]<\/p>\r\nAdd the coefficients of the like terms. Remember that the coefficient of <em>x<\/em> is [latex]1\\left(x=1x\\right)[\/latex].\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}3x^{2}+\\left(3+1+5\\right)x+1\\\\3x^{2}+\\left(9\\right)x+1\\end{array}[\/latex]<\/p>\r\nWrite the sum as the new coefficient.\r\n<h4>Answer<\/h4>\r\n[latex]3x^{2}+3x+x+1+5x=3x^{2}+9x+1[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/1epjbVO_qU4\r\n<h2>Adding and Subtracting Monomials<\/h2>\r\nAdding and subtracting monomials is the same as combining like terms. Like terms must have the same variable with the same exponent. Recall that when combining like terms only the coefficients are combined, never the exponents.\r\n\r\nHere is a brief summary of the steps we will follow to add or subtract polynomials.\r\n<div class=\"textbox\">\r\n<h3>How To: Given multiple polynomials, add or subtract them to simplify the expressions<strong>\r\n<\/strong><\/h3>\r\n<ol>\r\n \t<li>Combine like terms.<\/li>\r\n \t<li>Simplify and write in standard form.<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nAdd: [latex]17{x}^{2}+6{x}^{2}[\/latex]\r\n\r\nSolution\r\n<table id=\"eip-id1168469711517\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]17{x}^{2}+6{x}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]23{x}^{2}[\/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]146075[\/ohm_question]\r\n\r\n<\/div>\r\nPay attention to signs when adding or subtracting monomials.\u00a0 In the example below, we are subtracting a monomial with a negative coefficient.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSubtract: [latex]11n-\\left(-8n\\right)[\/latex]\r\n[reveal-answer q=\"915884\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"915884\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468510729\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]11n-\\left(-8n\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]19n[\/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]146315[\/ohm_question]\r\n\r\n[ohm_question]146077[\/ohm_question]\r\n\r\n<\/div>\r\nWhenever we add monomials in which the variables are not the same, even if their exponents have the same value, they are not like terms and therefore cannot be added together.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]{a}^{2}+4{b}^{2}-7{a}^{2}[\/latex]\r\n[reveal-answer q=\"15776\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"15776\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468675660\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{a}^{2}+4{b}^{2}-7{a}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]-6{a}^{{}^{2}}+4{b}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nRemember, [latex]-6{a}^{2}[\/latex] and [latex]4{b}^{2}[\/latex] are not like terms. The variables are not the same.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146078[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Add and Subtract Polynomials<\/h2>\r\nAdding and subtracting <b>polynomials<\/b> may sound complicated, but it\u2019s really not much different from the addition and subtraction that you do every day.\u00a0 You can add two (or more) polynomials as you have added algebraic expressions.\u00a0 Adding and subtracting polynomials can be thought of as just adding and subtracting like terms. Look for like terms\u2014those with the same variables with the same exponent. You can remove the parentheses and then use the Commutative Property to rearrange the terms to put like terms together. (It may also be helpful to underline, circle, or box like terms.)\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nAdd. [latex]\\left(3b+5\\right)+\\left(2b+4\\right)[\/latex]\r\n\r\n[reveal-answer q=\"379821\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"379821\"]Regroup\r\n<p style=\"text-align: center\">[latex]\\left(3b+2b\\right)+\\left(5+4\\right)[\/latex]<\/p>\r\nCombine like terms.\r\n<p style=\"text-align: center\">[latex]5b + 9[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(3b+5\\right)+\\left(2b+4\\right)=5b+9[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWhen you are adding polynomials that have subtraction,\u00a0it is important to remember to keep the sign on each term as you are collecting like terms.\u00a0 The next example will show you how to regroup terms that are subtracted when you are collecting like terms.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nAdd. [latex]\\left(-5x^{2}\u201310x+2\\right)+\\left(3x^{2}+7x\u20134\\right)[\/latex]\r\n\r\n[reveal-answer q=\"486380\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"486380\"]\r\n\r\nCollect like terms, making sure you keep the sign on each term. For example, when you collect\u00a0the [latex]x^2[\/latex] terms, make sure to keep the negative sign on [latex]-5x^2[\/latex].\r\n\r\nHelpful Hint: We find that it is easier to put the terms with a negative sign on the right of the terms that are positive. This would mean\u00a0that the\u00a0[latex]x^2[\/latex] terms would be grouped as\u00a0[latex]\\left(3x^{2}-5x^{2}\\right)[\/latex]. If both terms are negative, then it doesn't matter which is on the left or right.\r\n\r\nThe polynomial now looks like this, with like terms collected:\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\underbrace{\\left(3x^{2}-5x^{2}\\right)}+\\underbrace{\\left(7x-10x\\right)}+\\underbrace{\\left(2-4\\right)}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x^2\\text{ terms }\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\text{ terms}\\,\\,\\,\\,\\,\\,\\,\\,\\text{ constants }\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left\">The [latex]x^2[\/latex] terms will simplify to [latex]-2x^{2}[\/latex]<\/p>\r\n<p style=\"text-align: left\">The\u00a0[latex]x[\/latex] will simplify to [latex]-3x[\/latex]<\/p>\r\n<p style=\"text-align: left\">The constant terms will simplify to [latex]-2[\/latex]<\/p>\r\n<p style=\"text-align: left\">\u00a0Rewrite the polynomial with it's simplified terms, keeping the sign on each term.<\/p>\r\n<p style=\"text-align: center\">[latex]-2x^{2}-3x-2[\/latex]<\/p>\r\n<p style=\"text-align: left\">As a matter of convention, we write polynomials in descending order based on degree. \u00a0Notice how we put the\u00a0[latex]x^2[\/latex] term first, the\u00a0[latex]x[\/latex] term second and the constant term last.<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(-5x^{2}-10x+2\\right)+\\left(3x^{2}+7x-4\\right)=-2x^{2}-3x-2[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind the sum: [latex]\\left(4{x}^{2}-5x+1\\right)+\\left(3{x}^{2}-8x - 9\\right)[\/latex].\r\n[reveal-answer q=\"337728\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"337728\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468393629\" class=\"unnumbered unstyled\" summary=\"The top row shows parentheses 4 x squared minus 5x plus 1 plus parentheses 3 x squared minus 8x minus 9. The next row says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\left(4{x}^{2}-5x+1\\right)+\\left(3{x}^{2}-8x - 9\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Identify like terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224334\/CNX_BMath_Figure_10_01_003-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rearrange to get the like terms together.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224335\/CNX_BMath_Figure_10_01_003_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]7x^2-13x-8[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe above examples show addition of polynomials horizontally, by reading from left to right along the same line. Some people like to organize their work vertically instead, because they find it easier to be sure that they are combining like terms. The example below shows this \u201cvertical\u201d method of adding polynomials:\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nAdd. [latex]\\left(3x^{2}+2x-7\\right)+\\left(7x^{2}-4x+8\\right)[\/latex]\r\n\r\n[reveal-answer q=\"425224\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"425224\"]Write one polynomial below the other, making sure to line up like terms.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}3x^{2}+2x-7\\\\+7x^{2}-4x+8\\end{array}[\/latex]<\/p>\r\nCombine like terms, paying close attention to the signs.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}3x^{2}+2x-7\\\\\\underline{+7x^{2}-4x+8}\\\\10x^{2}-2x+1\\end{array}[\/latex]<b>\u00a0<\/b><\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(3x^{2}+3x-7\\right)+\\left(7x^{2}-4x+8\\right)=10x^{2}-2x+1[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nSometimes in a vertical arrangement, you can line up every term beneath a like term, as in the example above. But sometimes it isn't so tidy. When there isn't a matching like term for every term, there will be empty places in the vertical arrangement.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nAdd. [latex]\\left(4x^{3}+5x^{2}-6x+2\\right)+\\left(-4x^{2}+10\\right)[\/latex]\r\n\r\n[reveal-answer q=\"232680\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"232680\"]Write one polynomial below the other, lining up like terms vertically.\r\n\r\nTo keep track of like terms, you can insert zeros where there aren't any shared like terms. This is optional, but some find it helpful.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}4x^{3}+5x^{2}-6x+2\\\\+0\\,\\,-4x^{2}\\,\\,+0\\,\\,+10\\end{array}[\/latex]<\/p>\r\nCombine like terms, paying close attention to the signs.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}4x^{3}+5x^{2}-6x+\\,\\,\\,2\\\\\\underline{+0\\,\\,-4x^{2}\\,\\,+0\\,\\,+10}\\\\4x^{3}\\,+\\,\\,x^{2}-6x+12\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(4x^{3}+5x^{2}-6x+2\\right)+\\left(-4x^{2}+10\\right)=4x^{3}+x^{2}-6x+12[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146084[\/ohm_question]\r\n\r\n<\/div>\r\nYou may be thinking, how is this different than combining like terms, which we did in the last section? The answer is, it's not really. We just added a layer to combining like terms by adding more terms to combine. :) Polynomials are a useful tool for describing the behavior of anything that isn't linear, and sometimes you may need to add them.\r\n\r\nIn the following video, you will see more examples of combining like terms by adding polynomials.\r\n\r\nhttps:\/\/youtu.be\/KYZR7g7QcF4\r\n\r\nIn the next section we will show how to subtract polynomials.\r\n<h2 id=\"title2\">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\/117\/2016\/06\/02202350\/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 [latex]3[\/latex] gives [latex]\u22123[\/latex], and changing the sign\u00a0of [latex]\u22123[\/latex] gives [latex]3[\/latex]. 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&nbsp;\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&nbsp;\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-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/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\n&nbsp;\r\n\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 id=\"title3\">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-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/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\nIn the following example we will show how to distribute the negative sign to each term of a polynomial that is being subtracted from another.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind the difference.\r\n<p style=\"text-align: center\">[latex]\\left(7{x}^{4}-{x}^{2}+6x+1\\right)-\\left(5{x}^{3}-2{x}^{2}+3x+2\\right)[\/latex]<\/p>\r\n<p style=\"text-align: left\">[reveal-answer q=\"279648\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"279648\"]<\/p>\r\n<p style=\"text-align: left\">[latex]\\begin{array}{ccc}7{x}^4-{x}^2+6x+1-5{x}^3+2{x}^{2}-3x-2\\text{ }\\hfill &amp; \\text{Distribute}.\\hfill \\\\ 7{x}^{4}-5{x}^{3}+\\left(-{x}^{2}+2{x}^{2}\\right)+\\left(6x - 3x\\right)+\\left(1 - 2\\right)\\text{ }\\hfill &amp; \\text{Combine like terms}.\\hfill \\\\ 7{x}^{4}-5{x}^{3}+{x}^{2}+3x - 1\\hfill &amp; \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\r\nNote that finding the difference between two polynomials is the same as adding the opposite of the second polynomial to the first.\r\n<p style=\"text-align: left\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n&nbsp;\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 examples above, try the vertical approach shown above 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\nWhen we add polynomials as we did in the last example, we can rewrite the expression without parentheses and then combine like terms. But when we subtract polynomials, we must be very careful with the signs.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the difference: [latex]\\left(7{u}^{2}-5u+3\\right)-\\left(4{u}^{2}-2\\right)[\/latex].\r\n[reveal-answer q=\"295369\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"295369\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468667646\" class=\"unnumbered unstyled\" summary=\"The top row shows parentheses 7 u squared minus 5u plus 3 minus parentheses 4 u squared minus 2. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\left(7{u}^{2}-5u+3\\right)-\\left(4{u}^{2}-2\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute and identify like terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224338\/CNX_BMath_Figure_10_01_004_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rearrange the terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224339\/CNX_BMath_Figure_10_01_004_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]3{u}^{2}-5u+5[\/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]146079[\/ohm_question]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Exercises<\/h3>\r\nSubtract [latex]\\left({m}^{2}-3m+8\\right)[\/latex] from [latex]\\left(9{m}^{2}-7m+4\\right)[\/latex].\r\n[reveal-answer q=\"661227\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"661227\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168467283477\" class=\"unnumbered unstyled\" summary=\"The top line says, \">\r\n<tbody>\r\n<tr>\r\n<td>\u00a0Subtract [latex]\\left({m}^{2}-3m+8\\right)[\/latex] from [latex]\\left(9{m}^{2}-7m+4\\right)[\/latex].<\/td>\r\n<td>[latex](9{m}^{2}-7m+4)-({m}^{2}-3m+8)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute and identify like terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224343\/CNX_BMath_Figure_10_01_005_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rearrange the terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224345\/CNX_BMath_Figure_10_01_005_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]8m^2-4m-4[\/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\u00a0IT<\/h3>\r\n[ohm_question]146085[\/ohm_question]\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\r\nIn the next video we show more examples of adding and subtracting polynomials.\r\n\r\nhttps:\/\/youtu.be\/jiq3toC7wGM\r\n\r\n&nbsp;\r\n<h4>Summary<\/h4>\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.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Add and subtract monomials<\/li>\n<li>Add and subtract polynomials<\/li>\n<\/ul>\n<\/div>\n<div id=\"attachment_4439\" style=\"width: 498px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4439\" class=\"wp-image-4439\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/07\/27223107\/Apple_and_Orange_-_they_do_not_compare-300x206.jpg\" alt=\"Apple sitting next to an Orange\" width=\"488\" height=\"335\" \/><\/p>\n<p id=\"caption-attachment-4439\" class=\"wp-caption-text\">Apple and Orange<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<h3>Combining Like Terms<\/h3>\n<p>A polynomial may need to be simplified. One way to simplify a polynomial is to combine the <b>like terms<\/b> if there are any. Two or more terms in a polynomial are like terms if they have the same variable (or variables) with the same exponent. For example, [latex]3x^{2}[\/latex] and [latex]-5x^{2}[\/latex] are like terms: They both have [latex]x[\/latex] as the variable, and the exponent is [latex]2[\/latex] for each. However, [latex]3x^{2}[\/latex]\u00a0and [latex]3x[\/latex]\u00a0are not like terms, because their exponents are different.<\/p>\n<p>Here are some examples of terms that are alike and some that are unlike.<\/p>\n<table>\n<tbody>\n<tr>\n<td>Term<\/td>\n<td>Like Terms<\/td>\n<td>UNLike Terms<\/td>\n<\/tr>\n<tr>\n<td>[latex]a[\/latex]<\/td>\n<td>[latex]3a, \\,\\,\\,-2a,\\,\\,\\, \\frac{1}{2}a[\/latex]<\/td>\n<td>[latex]a^2,\\,\\,\\,\\frac{1}{a},\\,\\,\\, \\sqrt{a}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]a^2[\/latex]<\/td>\n<td>[latex]-5a^2,\\,\\,\\,\\frac{1}{4}a^2,\\,\\,\\, 0.56a^2[\/latex]<\/td>\n<td>[latex]\\frac{1}{a^2},\\,\\,\\,\\sqrt{a^2},\\,\\,\\, a^3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]ab[\/latex]<\/td>\n<td>[latex]7ab,\\,\\,\\,0.23ab,\\,\\,\\,\\frac{2}{3}ab,\\,\\,\\,-ab[\/latex]<\/td>\n<td>[latex]a^2b,\\,\\,\\,\\frac{1}{ab},\\,\\,\\,\\sqrt{ab}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]ab^2[\/latex]<\/td>\n<td>\u00a0[latex]4ab^2,\\,\\,\\, \\frac{ab^2}{7},\\,\\,\\,0.4ab^2,\\,\\,\\, -a^2b[\/latex]<\/td>\n<td>\u00a0[latex]a^2b,\\,\\,\\, ab,\\,\\,\\,\\sqrt{ab^2},\\,\\,\\,\\frac{1}{ab^2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Which of these terms are like terms?<\/p>\n<p style=\"text-align: center\">[latex]7x^{3}, 7x, 7y, -8x^{3}, 9y, -3x^{2}, 8y^{2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q413363\">Show Solution<\/span><\/p>\n<div id=\"q413363\" class=\"hidden-answer\" style=\"display: none\">Like terms must have the same variables, so first identify which terms use the same variables.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}x:7x^{3}, 7x, -8x^{3}, -3x^{2}\\\\y:7y, 9y, 8y^{2}\\end{array}[\/latex]<\/p>\n<p>Like terms must also have the same exponents. Identify which terms with the same variables also use the same exponents.<\/p>\n<p>The <em>x<\/em> terms [latex]7x^{3}[\/latex]\u00a0and [latex]-8x^{3}[\/latex]\u00a0have the same exponent.<\/p>\n<p>The <em>y<\/em> terms [latex]7y[\/latex] and [latex]9y[\/latex] have the same exponent.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]7x^{3}[\/latex] and [latex]-8x^{3}[\/latex] are like terms.<\/p>\n<p>[latex]7y[\/latex] and [latex]9y[\/latex]\u00a0are like terms.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>You can use the distributive property to simplify the sum of like terms. Recall that the distributive property of addition states that the product of a number and a sum (or difference) is equal to the sum (or difference) of the products.<\/p>\n<p style=\"text-align: center\">[latex]2\\left(3+6\\right)=2\\left(3\\right)+2\\left(6\\right)[\/latex]<\/p>\n<p>Both expressions equal [latex]18[\/latex]. So you can write the expression in whichever form is the most useful.<\/p>\n<p>Let\u2019s see how we can use this property to combine like terms.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify [latex]3x^{2}-5x^{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q969840\">Show Solution<\/span><\/p>\n<div id=\"q969840\" class=\"hidden-answer\" style=\"display: none\">[latex]3x^{2}[\/latex] and [latex]5x^{2}[\/latex]<sup>\u00a0<\/sup>are like terms.<\/p>\n<p style=\"text-align: center\">[latex]3\\left(x^{2}\\right)-5\\left(x^{2}\\right)[\/latex]<\/p>\n<p>We can rewrite the expression as the product of the difference.<\/p>\n<p style=\"text-align: center\">[latex]\\left(3-5\\right)\\left(x^{2}\\right)[\/latex]<\/p>\n<p>Calculate [latex]3\u20135[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\left(-2\\right)\\left(x^{2}\\right)[\/latex]<\/p>\n<p>Write the difference of [latex]3 \u2013 5[\/latex] as the new coefficient.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]3x^{2}-5x^{2}=-2x^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>You may have noticed that combining like terms involves combining the coefficients to find the new coefficient of the like term. You can use this as a shortcut.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify [latex]6a^{4}+4a^{4}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q840415\">Show Solution<\/span><\/p>\n<div id=\"q840415\" class=\"hidden-answer\" style=\"display: none\">Notice that both terms have a number multiplied by [latex]a^{4}[\/latex]. This makes them like terms.<\/p>\n<p style=\"text-align: center\">[latex]6a^{4}+4a^{4}[\/latex]<\/p>\n<p>Combine the coefficients, [latex]6[\/latex] and [latex]4[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\left(6+4\\right)\\left(a^{4}\\right)[\/latex]<\/p>\n<p>Calculate the sum.<\/p>\n<p style=\"text-align: center\">[latex]\\left(10\\right)\\left(a^{4}\\right)[\/latex]<\/p>\n<p>Write the sum as the new coefficient.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]6a^{4}+4a^{4}=10a^{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>When you have a polynomial with more terms, you have to be careful that you combine <i>only<\/i> like terms<i>.<\/i> If two terms are not like terms, you can\u2019t combine them.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify [latex]3x^{2}+3x+x+1+5x[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q731804\">Show Solution<\/span><\/p>\n<div id=\"q731804\" class=\"hidden-answer\" style=\"display: none\">First identify which terms are <i>like terms<\/i>: only [latex]3x[\/latex], [latex]x[\/latex], and [latex]5x[\/latex]\u00a0are like terms.<\/p>\n<p>[latex]3x[\/latex], [latex]x[\/latex], and [latex]5x[\/latex] are like terms.<\/p>\n<p>Use the commutative and associative properties to group the like terms together.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}3x^{2}+3x+x+1+5x\\\\3x^{2}+\\left(3x+x+5x\\right)+1\\end{array}[\/latex]<\/p>\n<p>Add the coefficients of the like terms. Remember that the coefficient of <em>x<\/em> is [latex]1\\left(x=1x\\right)[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}3x^{2}+\\left(3+1+5\\right)x+1\\\\3x^{2}+\\left(9\\right)x+1\\end{array}[\/latex]<\/p>\n<p>Write the sum as the new coefficient.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]3x^{2}+3x+x+1+5x=3x^{2}+9x+1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Identify Like Terms and Combine Like\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/1epjbVO_qU4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Adding and Subtracting Monomials<\/h2>\n<p>Adding and subtracting monomials is the same as combining like terms. Like terms must have the same variable with the same exponent. Recall that when combining like terms only the coefficients are combined, never the exponents.<\/p>\n<p>Here is a brief summary of the steps we will follow to add or subtract polynomials.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given multiple polynomials, add or subtract them to simplify the expressions<strong><br \/>\n<\/strong><\/h3>\n<ol>\n<li>Combine like terms.<\/li>\n<li>Simplify and write in standard form.<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Add: [latex]17{x}^{2}+6{x}^{2}[\/latex]<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168469711517\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]17{x}^{2}+6{x}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]23{x}^{2}[\/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=\"ohm146075\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146075&theme=oea&iframe_resize_id=ohm146075&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Pay attention to signs when adding or subtracting monomials.\u00a0 In the example below, we are subtracting a monomial with a negative coefficient.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Subtract: [latex]11n-\\left(-8n\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q915884\">Show Solution<\/span><\/p>\n<div id=\"q915884\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468510729\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]11n-\\left(-8n\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]19n[\/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=\"ohm146315\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146315&theme=oea&iframe_resize_id=ohm146315&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146077\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146077&theme=oea&iframe_resize_id=ohm146077&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Whenever we add monomials in which the variables are not the same, even if their exponents have the same value, they are not like terms and therefore cannot be added together.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]{a}^{2}+4{b}^{2}-7{a}^{2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q15776\">Show Solution<\/span><\/p>\n<div id=\"q15776\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468675660\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{a}^{2}+4{b}^{2}-7{a}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]-6{a}^{{}^{2}}+4{b}^{2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Remember, [latex]-6{a}^{2}[\/latex] and [latex]4{b}^{2}[\/latex] are not like terms. The variables are not the same.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146078\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146078&theme=oea&iframe_resize_id=ohm146078&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Add and Subtract Polynomials<\/h2>\n<p>Adding and subtracting <b>polynomials<\/b> may sound complicated, but it\u2019s really not much different from the addition and subtraction that you do every day.\u00a0 You can add two (or more) polynomials as you have added algebraic expressions.\u00a0 Adding and subtracting polynomials can be thought of as just adding and subtracting like terms. Look for like terms\u2014those with the same variables with the same exponent. You can remove the parentheses and then use the Commutative Property to rearrange the terms to put like terms together. (It may also be helpful to underline, circle, or box like terms.)<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Add. [latex]\\left(3b+5\\right)+\\left(2b+4\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q379821\">Show Solution<\/span><\/p>\n<div id=\"q379821\" class=\"hidden-answer\" style=\"display: none\">Regroup<\/p>\n<p style=\"text-align: center\">[latex]\\left(3b+2b\\right)+\\left(5+4\\right)[\/latex]<\/p>\n<p>Combine like terms.<\/p>\n<p style=\"text-align: center\">[latex]5b + 9[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(3b+5\\right)+\\left(2b+4\\right)=5b+9[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>When you are adding polynomials that have subtraction,\u00a0it is important to remember to keep the sign on each term as you are collecting like terms.\u00a0 The next example will show you how to regroup terms that are subtracted when you are collecting like terms.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Add. [latex]\\left(-5x^{2}\u201310x+2\\right)+\\left(3x^{2}+7x\u20134\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q486380\">Show Solution<\/span><\/p>\n<div id=\"q486380\" class=\"hidden-answer\" style=\"display: none\">\n<p>Collect like terms, making sure you keep the sign on each term. For example, when you collect\u00a0the [latex]x^2[\/latex] terms, make sure to keep the negative sign on [latex]-5x^2[\/latex].<\/p>\n<p>Helpful Hint: We find that it is easier to put the terms with a negative sign on the right of the terms that are positive. This would mean\u00a0that the\u00a0[latex]x^2[\/latex] terms would be grouped as\u00a0[latex]\\left(3x^{2}-5x^{2}\\right)[\/latex]. If both terms are negative, then it doesn&#8217;t matter which is on the left or right.<\/p>\n<p>The polynomial now looks like this, with like terms collected:<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\underbrace{\\left(3x^{2}-5x^{2}\\right)}+\\underbrace{\\left(7x-10x\\right)}+\\underbrace{\\left(2-4\\right)}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x^2\\text{ terms }\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\text{ terms}\\,\\,\\,\\,\\,\\,\\,\\,\\text{ constants }\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left\">The [latex]x^2[\/latex] terms will simplify to [latex]-2x^{2}[\/latex]<\/p>\n<p style=\"text-align: left\">The\u00a0[latex]x[\/latex] will simplify to [latex]-3x[\/latex]<\/p>\n<p style=\"text-align: left\">The constant terms will simplify to [latex]-2[\/latex]<\/p>\n<p style=\"text-align: left\">\u00a0Rewrite the polynomial with it&#8217;s simplified terms, keeping the sign on each term.<\/p>\n<p style=\"text-align: center\">[latex]-2x^{2}-3x-2[\/latex]<\/p>\n<p style=\"text-align: left\">As a matter of convention, we write polynomials in descending order based on degree. \u00a0Notice how we put the\u00a0[latex]x^2[\/latex] term first, the\u00a0[latex]x[\/latex] term second and the constant term last.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(-5x^{2}-10x+2\\right)+\\left(3x^{2}+7x-4\\right)=-2x^{2}-3x-2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find the sum: [latex]\\left(4{x}^{2}-5x+1\\right)+\\left(3{x}^{2}-8x - 9\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q337728\">Show Solution<\/span><\/p>\n<div id=\"q337728\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468393629\" class=\"unnumbered unstyled\" summary=\"The top row shows parentheses 4 x squared minus 5x plus 1 plus parentheses 3 x squared minus 8x minus 9. The next row says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\left(4{x}^{2}-5x+1\\right)+\\left(3{x}^{2}-8x - 9\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Identify 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\/24224334\/CNX_BMath_Figure_10_01_003-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Rearrange to get the like terms together.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224335\/CNX_BMath_Figure_10_01_003_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]7x^2-13x-8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>The above examples show addition of polynomials horizontally, by reading from left to right along the same line. Some people like to organize their work vertically instead, because they find it easier to be sure that they are combining like terms. The example below shows this \u201cvertical\u201d method of adding polynomials:<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Add. [latex]\\left(3x^{2}+2x-7\\right)+\\left(7x^{2}-4x+8\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q425224\">Show Solution<\/span><\/p>\n<div id=\"q425224\" class=\"hidden-answer\" style=\"display: none\">Write one polynomial below the other, making sure to line up like terms.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}3x^{2}+2x-7\\\\+7x^{2}-4x+8\\end{array}[\/latex]<\/p>\n<p>Combine like terms, paying close attention to the signs.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}3x^{2}+2x-7\\\\\\underline{+7x^{2}-4x+8}\\\\10x^{2}-2x+1\\end{array}[\/latex]<b>\u00a0<\/b><\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(3x^{2}+3x-7\\right)+\\left(7x^{2}-4x+8\\right)=10x^{2}-2x+1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Sometimes in a vertical arrangement, you can line up every term beneath a like term, as in the example above. But sometimes it isn&#8217;t so tidy. When there isn&#8217;t a matching like term for every term, there will be empty places in the vertical arrangement.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Add. [latex]\\left(4x^{3}+5x^{2}-6x+2\\right)+\\left(-4x^{2}+10\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q232680\">Show Solution<\/span><\/p>\n<div id=\"q232680\" class=\"hidden-answer\" style=\"display: none\">Write one polynomial below the other, lining up like terms vertically.<\/p>\n<p>To keep track of like terms, you can insert zeros where there aren&#8217;t any shared like terms. This is optional, but some find it helpful.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}4x^{3}+5x^{2}-6x+2\\\\+0\\,\\,-4x^{2}\\,\\,+0\\,\\,+10\\end{array}[\/latex]<\/p>\n<p>Combine like terms, paying close attention to the signs.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}4x^{3}+5x^{2}-6x+\\,\\,\\,2\\\\\\underline{+0\\,\\,-4x^{2}\\,\\,+0\\,\\,+10}\\\\4x^{3}\\,+\\,\\,x^{2}-6x+12\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(4x^{3}+5x^{2}-6x+2\\right)+\\left(-4x^{2}+10\\right)=4x^{3}+x^{2}-6x+12[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146084\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146084&theme=oea&iframe_resize_id=ohm146084&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>You may be thinking, how is this different than combining like terms, which we did in the last section? The answer is, it&#8217;s not really. We just added a layer to combining like terms by adding more terms to combine. :) Polynomials are a useful tool for describing the behavior of anything that isn&#8217;t linear, and sometimes you may need to add them.<\/p>\n<p>In the following video, you will see more examples of combining like terms by adding polynomials.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex:  Adding Polynomials\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/KYZR7g7QcF4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>In the next section we will show how to subtract polynomials.<\/p>\n<h2 id=\"title2\">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\/117\/2016\/06\/02202350\/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 [latex]3[\/latex] gives [latex]\u22123[\/latex], and changing the sign\u00a0of [latex]\u22123[\/latex] gives [latex]3[\/latex]. 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<p>&nbsp;<\/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<p>&nbsp;<\/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-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/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>&nbsp;<\/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 id=\"title3\">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-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/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<p>In the following example we will show how to distribute the negative sign to each term of a polynomial that is being subtracted from another.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find the difference.<\/p>\n<p style=\"text-align: center\">[latex]\\left(7{x}^{4}-{x}^{2}+6x+1\\right)-\\left(5{x}^{3}-2{x}^{2}+3x+2\\right)[\/latex]<\/p>\n<p style=\"text-align: left\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q279648\">Show Solution<\/span><\/p>\n<div id=\"q279648\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: left\">[latex]\\begin{array}{ccc}7{x}^4-{x}^2+6x+1-5{x}^3+2{x}^{2}-3x-2\\text{ }\\hfill & \\text{Distribute}.\\hfill \\\\ 7{x}^{4}-5{x}^{3}+\\left(-{x}^{2}+2{x}^{2}\\right)+\\left(6x - 3x\\right)+\\left(1 - 2\\right)\\text{ }\\hfill & \\text{Combine like terms}.\\hfill \\\\ 7{x}^{4}-5{x}^{3}+{x}^{2}+3x - 1\\hfill & \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\n<p>Note that finding the difference between two polynomials is the same as adding the opposite of the second polynomial to the first.<\/p>\n<p style=\"text-align: left\"><\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/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=\"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 examples above, try the vertical approach shown above 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>When we add polynomials as we did in the last example, we can rewrite the expression without parentheses and then combine like terms. But when we subtract polynomials, we must be very careful with the signs.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the difference: [latex]\\left(7{u}^{2}-5u+3\\right)-\\left(4{u}^{2}-2\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q295369\">Show Solution<\/span><\/p>\n<div id=\"q295369\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468667646\" class=\"unnumbered unstyled\" summary=\"The top row shows parentheses 7 u squared minus 5u plus 3 minus parentheses 4 u squared minus 2. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\left(7{u}^{2}-5u+3\\right)-\\left(4{u}^{2}-2\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute and identify 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\/24224338\/CNX_BMath_Figure_10_01_004_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Rearrange the terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224339\/CNX_BMath_Figure_10_01_004_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]3{u}^{2}-5u+5[\/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=\"ohm146079\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146079&theme=oea&iframe_resize_id=ohm146079&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Exercises<\/h3>\n<p>Subtract [latex]\\left({m}^{2}-3m+8\\right)[\/latex] from [latex]\\left(9{m}^{2}-7m+4\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q661227\">Show Solution<\/span><\/p>\n<div id=\"q661227\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168467283477\" class=\"unnumbered unstyled\" summary=\"The top line says,\">\n<tbody>\n<tr>\n<td>\u00a0Subtract [latex]\\left({m}^{2}-3m+8\\right)[\/latex] from [latex]\\left(9{m}^{2}-7m+4\\right)[\/latex].<\/td>\n<td>[latex](9{m}^{2}-7m+4)-({m}^{2}-3m+8)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute and identify 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\/24224343\/CNX_BMath_Figure_10_01_005_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Rearrange the terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224345\/CNX_BMath_Figure_10_01_005_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]8m^2-4m-4[\/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\u00a0IT<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146085\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146085&theme=oea&iframe_resize_id=ohm146085&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video, you will see more examples of subtracting polynomials.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" 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<p>In the next video we show more examples of adding and subtracting polynomials.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Ex:  Adding and Subtracting Polynomials\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/jiq3toC7wGM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<h4>Summary<\/h4>\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\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-10814\">\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 146085, 146084, 146078, 146070, 146073. <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>Ex: Adding Polynomials. <strong>Authored by<\/strong>: James Sousa (mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/KYZR7g7QcF4\">https:\/\/youtu.be\/KYZR7g7QcF4<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Adding and Subtracting Polynomials. <strong>Authored by<\/strong>: James Sousa (mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/jiq3toC7wGM\">https:\/\/youtu.be\/jiq3toC7wGM<\/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":17533,"menu_order":7,"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 146085, 146084, 146078, 146070, 146073\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Adding Polynomials\",\"author\":\"James Sousa 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