{"id":1753,"date":"2023-10-12T00:32:05","date_gmt":"2023-10-12T00:32:05","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/chapter\/introduction-polynomials\/"},"modified":"2023-10-12T00:32:05","modified_gmt":"2023-10-12T00:32:05","slug":"introduction-polynomials","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/chapter\/introduction-polynomials\/","title":{"raw":"Polynomial Basics","rendered":"Polynomial Basics"},"content":{"raw":"\n\n<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li>Identify the degree, leading coefficient, and leading term of a polynomial.<\/li>\n \t<li>Add and subtract polynomials.<\/li>\n \t<li>Multiply polynomials.<\/li>\n \t<li>Square a binomial.<\/li>\n \t<li>Find a difference of squares.<\/li>\n \t<li>Perform operations on polynomials with several variables.<\/li>\n<\/ul>\n<\/div>\nEarl is building a doghouse whose front is in the shape of a square topped with a triangle. There will be a rectangular door through which the dog can enter and exit the house. Earl wants to find the area of the front of the doghouse so that he can purchase the correct amount of paint. Using the measurements of the front of the house shown below, we can create an expression that combines several variable terms which allows us to solve this problem and others like it.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21205143\/CNX_CAT_Figure_01_04_001.jpg\" alt=\"Sketch of a house formed by a square and a triangle based on the top of the square. A rectangle is placed at the bottom center of the square to mark a doorway. The height of the door is labeled: x and the width of the door is labeled: 1 foot. The side of the square is labeled: 2x. The height of the triangle is labeled: 3\/2 feet.\" width=\"487\" height=\"249\"> <b>Measurements of the front of the doghouse Earl is building.<\/b>[\/caption]\n\nFirst, find the area of the square in square feet.\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill A&amp; =&amp; {s}^{2}\\hfill \\\\ &amp; =&amp; {\\left(2x\\right)}^{2}\\hfill \\\\ &amp; =&amp; 4{x}^{2}\\hfill \\end{array}[\/latex]<\/div>\nThen, find the area of the triangle in square feet.\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill A&amp; =&amp; \\frac{1}{2}bh\\hfill \\\\ &amp; =&amp; \\text{}\\frac{1}{2}\\left(2x\\right)\\left(\\frac{3}{2}\\right)\\hfill \\\\ &amp; =&amp; \\text{}\\frac{3}{2}x\\hfill \\end{array}[\/latex]<\/div>\nNext, find the area of the rectangular door in square feet.\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill A&amp; =&amp; lw\\hfill \\\\ &amp; =&amp; x\\cdot 1\\hfill \\\\ \\hfill &amp; =&amp; x\\hfill \\end{array}[\/latex]<\/div>\n<h2>Operations on Polynomials<\/h2>\nThe area of the front of the doghouse can be found by adding the areas of the square and the triangle and then subtracting the area of the rectangle. When we do this, we get [latex]4{x}^{2}+\\frac{3}{2}x-x[\/latex] ft<sup>2<\/sup>, or [latex]4{x}^{2}+\\frac{1}{2}x[\/latex] ft<sup>2<\/sup>.\n\nIn this section, we will examine expressions such as this one, which combine several variable terms.\n\nThe area of the front of the doghouse described in the introduction was&nbsp;[latex]4{x}^{2}+\\frac{1}{2}x[\/latex] ft<sup>2<\/sup>.\n\nThis is an example of a <strong>polynomial&nbsp;<\/strong>which is a sum of or difference of terms each consisting of a variable raised to a nonnegative integer power. A number multiplied by a variable raised to an exponent, such as [latex]384\\pi [\/latex], is known as a <strong>coefficient<\/strong>. Coefficients can be positive, negative, or zero, and can be whole numbers, decimals, or fractions. Each product [latex]{a}_{i}{x}^{i}[\/latex], such as [latex]384\\pi w[\/latex], is a <strong>term of a polynomial<\/strong>. If a term does not contain a variable, it is called a <em>constant<\/em>.\n\nA polynomial containing only one term, such as [latex]5{x}^{4}[\/latex], is called a <strong>monomial<\/strong>. A polynomial containing two terms, such as [latex]2x - 9[\/latex], is called a <strong>binomial<\/strong>. A polynomial containing three terms, such as [latex]-3{x}^{2}+8x - 7[\/latex], is called a <strong>trinomial<\/strong>.\n\nWe can find the <strong>degree<\/strong> of a polynomial by identifying the highest power of the variable that occurs in the polynomial. The term with the highest degree is called the <strong>leading term<\/strong> because it is usually written first. The coefficient of the leading term is called the <strong>leading coefficient<\/strong>. When a polynomial is written so that the powers are descending, we say that it is in standard form.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21205146\/CNX_CAT_Figure_01_04_002.jpg\" alt=\"A polynomial reading: a sub n times x to the nth power plus and so on plus a sub 2 times x squared plus a sub one times x plus a subzero is shown. The a in the term a sub n is labeled: leading coefficient. The n in the term x to the nth power is labeled: degree. Finally, the entire term is labeled as: Leading term.\">\n<div class=\"textbox\">\n<h3>A General Note: Polynomials<\/h3>\nA <strong>polynomial<\/strong> is an expression that can be written in the form\n<div style=\"text-align: center;\">[latex]{a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/div>\nEach real number <em>a<sub>i&nbsp;<\/sub><\/em>is called a <strong>coefficient<\/strong>. The number [latex]{a}_{0}[\/latex] that is not multiplied by a variable is called a <em>constant<\/em>. Each product [latex]{a}_{i}{x}^{i}[\/latex] is a <strong>term of a polynomial<\/strong>. The highest power of the variable that occurs in the polynomial is called the <strong>degree<\/strong> of a polynomial. The <strong>leading term<\/strong> is the term with the highest power, and its coefficient is called the <strong>leading coefficient<\/strong>.\n\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a polynomial expression, identify the degree and leading coefficient<\/h3>\n<ol>\n \t<li>Find the highest power of <em>x<\/em> to determine the degree.<\/li>\n \t<li>Identify the term containing the highest power of <em>x<\/em> to find the leading term.<\/li>\n \t<li>Identify the coefficient of the leading term.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Identifying the Degree and Leading Coefficient of a Polynomial<\/h3>\nFor the following polynomials, identify the degree, the leading term, and the leading coefficient.\n<ol>\n \t<li>[latex]3+2{x}^{2}-4{x}^{3}[\/latex]<\/li>\n \t<li>[latex]5{t}^{5}-2{t}^{3}+7t[\/latex]<\/li>\n \t<li>[latex]6p-{p}^{3}-2[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"973610\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"973610\"]\n<ol>\n \t<li>The highest power of <em>x<\/em> is 3, so the degree is 3. The leading term is the term containing that degree, [latex]-4{x}^{3}[\/latex]. The leading coefficient is the coefficient of that term, [latex]-4[\/latex].<\/li>\n \t<li>The highest power of <em>t<\/em> is [latex]5[\/latex], so the degree is [latex]5[\/latex]. The leading term is the term containing that degree, [latex]5{t}^{5}[\/latex]. The leading coefficient is the coefficient of that term, [latex]5[\/latex].<\/li>\n \t<li>The highest power of <em>p<\/em> is [latex]3[\/latex], so the degree is [latex]3[\/latex]. The leading term is the term containing that degree, [latex]-{p}^{3}[\/latex], The leading coefficient is the coefficient of that term, [latex]-1[\/latex].<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nIdentify the degree, leading term, and leading coefficient of the polynomial [latex]4{x}^{2}-{x}^{6}+2x - 6[\/latex].\n\n[reveal-answer q=\"49799\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"49799\"]\n\nThe degree is 6, the leading term is [latex]-{x}^{6}[\/latex], and the leading coefficient is [latex]-1[\/latex].\n\n[\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93531&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"350\"><\/iframe>\n\n<\/div>\n<h3>Adding and Subtracting Polynomials<\/h3>\nWe can add and subtract polynomials by combining like terms, which are terms that contain the same variables raised to the same exponents. For example, [latex]5{x}^{2}[\/latex] and [latex]-2{x}^{2}[\/latex] are like terms and can be added to get [latex]3{x}^{2}[\/latex], but [latex]3x[\/latex] and [latex]3{x}^{2}[\/latex] are not like terms and therefore cannot be added.\n<div class=\"textbox\">\n<h3>How To: Given multiple polynomials, add or subtract them to simplify the expressions<strong>\n<\/strong><\/h3>\n<ol>\n \t<li>Combine like terms.<\/li>\n \t<li>Simplify and write in standard form. Standard form means you start with the leading term, and write the rest of the terms in descending order by degree.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Adding Polynomials<\/h3>\nFind the sum.\n<p style=\"text-align: center;\">[latex]\\left(12{x}^{2}+9x - 21\\right)+\\left(4{x}^{3}+8{x}^{2}-5x+20\\right)[\/latex]<\/p>\n[reveal-answer q=\"660613\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"660613\"]\n\n[latex]\\begin{array}{cc}4{x}^{3}+\\left(12{x}^{2}+8{x}^{2}\\right)+\\left(9x - 5x\\right)+\\left(-21+20\\right) \\hfill &amp; \\text{Combine like terms}.\\hfill \\\\ 4{x}^{3}+20{x}^{2}+4x - 1\\hfill &amp; \\text{Simplify}.\\hfill \\end{array}[\/latex]\n<h4>Analysis of the Solution<\/h4>\nWe can check our answers to these types of problems using a graphing calculator. To check, graph the problem as given along with the simplified answer. The two graphs should be the same. Be sure to use the same window to compare the graphs. Using different windows can make the expressions seem equivalent when they are not.\n\n[\/hidden-answer]\n\n<\/div>\n<div><\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nFind the sum.\n\n[latex]\\left(2{x}^{3}+5{x}^{2}-x+1\\right)+\\left(2{x}^{2}-3x - 4\\right)[\/latex]\n\n[reveal-answer q=\"121561\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"121561\"]\n\n[latex]2{x}^{3}+7{x}^{2}-4x - 3[\/latex]\n\n[\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93536&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe>\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Subtracting Polynomials<\/h3>\nFind the difference.\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[reveal-answer q=\"831247\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"831247\"]\n[latex]\\begin{array}{cc}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]\n<h4>Analysis of the Solution<\/h4>\nNote that finding the difference between two polynomials is the same as adding the opposite of the second polynomial to the first.\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nFind the difference.\n\n[latex]\\left(-7{x}^{3}-7{x}^{2}+6x - 2\\right)-\\left(4{x}^{3}-6{x}^{2}-x+7\\right)[\/latex]\n\n[reveal-answer q=\"260383\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"260383\"]\n\n[latex]-11{x}^{3}-{x}^{2}+7x - 9[\/latex]\n\n[\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93537&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"350\"><\/iframe>\n\n<\/div>\n<h3>Multiplying Polynomials<\/h3>\nMultiplying polynomials is a bit more challenging than adding and subtracting polynomials. We must use the distributive property to multiply each term in the first polynomial by each term in the second polynomial. We then combine like terms. We can also use a shortcut called the FOIL method when multiplying binomials. Certain special products follow patterns that we can memorize and use instead of multiplying the polynomials by hand each time. We will look at a variety of ways to multiply polynomials.\n<h3>Multiplying Polynomials Using the Distributive Property<\/h3>\nTo multiply a number by a polynomial we use the distributive property. The number must be distributed to each term of the polynomial. In [latex]2\\left(x+7\\right)[\/latex] we can distribute [latex]2[\/latex] to obtain the expression [latex]2x+14[\/latex]. When multiplying polynomials, the distributive property allows us to multiply each term of the first polynomial by each term of the second. We then add the products together and combine like terms to simplify.\n<div class=\"textbox\">\n<h3>How To: Given the multiplication of two polynomials, use the distributive property to simplify the expression<\/h3>\n<ol>\n \t<li>Multiply each term of the first polynomial by each term of the second.<\/li>\n \t<li>Combine like terms.<\/li>\n \t<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Multiplying Polynomials Using the Distributive Property<\/h3>\nFind the product.\n<p style=\"text-align: center;\">[latex]\\left(2x+1\\right)\\left(3{x}^{2}-x+4\\right)[\/latex]<\/p>\n[reveal-answer q=\"752165\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"752165\"]\n<p style=\"text-align: center;\">[latex]\\begin{array}{cc}2x\\left(3{x}^{2}-x+4\\right)+1\\left(3{x}^{2}-x+4\\right) \\hfill &amp; \\text{Use the distributive property}.\\hfill \\\\ \\left(6{x}^{3}-2{x}^{2}+8x\\right)+\\left(3{x}^{2}-x+4\\right)\\hfill &amp; \\text{Multiply}.\\hfill \\\\ 6{x}^{3}+\\left(-2{x}^{2}+3{x}^{2}\\right)+\\left(8x-x\\right)+4\\hfill &amp; \\text{Combine like terms}.\\hfill \\\\ 6{x}^{3}+{x}^{2}+7x+4 \\hfill &amp; \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\n\n<h4>Analysis of the Solution<\/h4>\nWe can use a table to keep track of our work, as shown in the table below. Write one polynomial across the top and the other down the side. For each box in the table, multiply the term for that row by the term for that column. Then add all of the terms together, combine like terms, and simplify.\n<table style=\"height: 45px;\" summary=\"A table with 3 rows and 4 columns. The first entry of the first row is empty, the others are labeled: three times x squared, negative x, and positive four. The first entry of the second row is labeled: two times x. The second entry reads: six times x cubed. The third entry reads: negative two times x squared. The fourth entry reads: eight times x. The first entry of the third row reads: positive one. The second entry reads: three times x squared. The third entry reads: negative x. The fourth entry reads: four.\">\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"width: 115px; height: 15px;\"><\/td>\n<td style=\"width: 159px; height: 15px;\"><strong>[latex]3{x}^{2}[\/latex]<\/strong><\/td>\n<td style=\"width: 154px; height: 15px;\"><strong>[latex]-x[\/latex]<\/strong><\/td>\n<td style=\"width: 116px; height: 15px;\"><strong>[latex]+4[\/latex]<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 115px; height: 15px;\"><strong>[latex]2x[\/latex]<\/strong><\/td>\n<td style=\"width: 159px; height: 15px;\">[latex]6{x}^{3}\\\\[\/latex]<\/td>\n<td style=\"width: 154px; height: 15px;\">[latex]-2{x}^{2}[\/latex]<\/td>\n<td style=\"width: 116px; height: 15px;\">[latex]8x[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 115px; height: 15px;\"><strong>[latex]+1[\/latex]<\/strong><\/td>\n<td style=\"width: 159px; height: 15px;\">[latex]3{x}^{2}[\/latex]<\/td>\n<td style=\"width: 154px; height: 15px;\">[latex]-x[\/latex]<\/td>\n<td style=\"width: 116px; height: 15px;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nFind the product.\n<p style=\"text-align: center;\">[latex]\\left(3x+2\\right)\\left({x}^{3}-4{x}^{2}+7\\right)[\/latex]<\/p>\n[reveal-answer q=\"508646\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"508646\"]\n\n[latex]3{x}^{4}-10{x}^{3}-8{x}^{2}+21x+14[\/latex]\n<h4>Analysis of the Solution<\/h4>\nWe can use a table to keep track of our work as shown below. Write one polynomial across the top and the other down the side. For each box in the table, multiply the term for that row by the term for that column. Then add all of the terms together, combine like terms, and simplify.\n<table summary=\"A table with 3 rows and 4 columns. The first entry of the first row is empty, the others are labeled: three times x squared, negative x, and positive four. The first entry of the second row is labeled: two times x. The second entry reads: six times x cubed. The third entry reads: negative two times x squared. The fourth entry reads: eight times x. The first entry of the third row reads: positive one. The second entry reads: three times x squared. The third entry reads: negative x. The fourth entry reads: four.\">\n<tbody>\n<tr>\n<td style=\"width: 115px;\"><\/td>\n<td style=\"width: 159px;\"><strong>[latex]3{x}^{2}[\/latex]<\/strong><\/td>\n<td style=\"width: 154px;\"><strong>[latex]-x[\/latex]<\/strong><\/td>\n<td style=\"width: 116px;\"><strong>[latex]+4[\/latex]<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 115px;\"><strong>[latex]2x[\/latex]<\/strong><\/td>\n<td style=\"width: 159px;\">[latex]6{x}^{3}\\\\[\/latex]<\/td>\n<td style=\"width: 154px;\">[latex]-2{x}^{2}[\/latex]<\/td>\n<td style=\"width: 116px;\">[latex]8x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 115px;\"><strong>[latex]+1[\/latex]<\/strong><\/td>\n<td style=\"width: 159px;\">[latex]3{x}^{2}[\/latex]<\/td>\n<td style=\"width: 154px;\">[latex]-x[\/latex]<\/td>\n<td style=\"width: 116px;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n[\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3864&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"350\"><\/iframe>\n\n<\/div>\n<h3>Using FOIL to Multiply Binomials<\/h3>\nA shortcut called FOIL is sometimes used to find the product of two binomials. It is called FOIL because we multiply the <strong>f<\/strong>irst terms, the <strong>o<\/strong>uter terms, the <strong>i<\/strong>nner terms, and then the <strong>l<\/strong>ast terms of each binomial.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21205149\/CNX_CAT_Figure_01_04_003.jpg\" alt=\"Two quantities in parentheses are being multiplied, the first being: a times x plus b and the second being: c times x plus d. This expression equals ac times x squared plus ad times x plus bc times x plus bd. The terms ax and cx are labeled: First Terms. The terms ax and d are labeled: Outer Terms. The terms b and cx are labeled: Inner Terms. The terms b and d are labeled: Last Terms.\">\n\nThe FOIL method is simply just the distributive property. We are multiplying each term of the first binomial by each term of the second binomial and then combining like terms.\n<div class=\"textbox\">\n<h3>How To: Given two binomials, Multiplying Using FOIL<\/h3>\n<ol>\n \t<li>Multiply the first terms of each binomial.<\/li>\n \t<li>Multiply the outer terms of the binomials.<\/li>\n \t<li>Multiply the inner terms of the binomials.<\/li>\n \t<li>Multiply the last terms of each binomial.<\/li>\n \t<li>Add the products.<\/li>\n \t<li>Combine like terms and simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using FOIL to Multiply Binomials<\/h3>\nUse FOIL to find the product.\n\n[latex]\\left(2x-18\\right)\\left(3x + 3\\right)[\/latex]\n\n[reveal-answer q=\"698991\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"698991\"]\nFind the product of the first terms.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21205151\/CNX_CAT_Figure_01_04_004.jpg\" alt=\"\">\n\nFind the product of the outer terms.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21205154\/CNX_CAT_Figure_01_04_005.jpg\" alt=\"\">\n\nFind the product of the inner terms.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21205156\/CNX_CAT_Figure_01_04_006.jpg\" alt=\"\">\n\nFind the product of the last terms.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21205158\/CNX_CAT_Figure_01_04_007.jpg\" alt=\"\">\n<p style=\"text-align: center;\">[latex]\\begin{array}{cc}6{x}^{2}+6x - 54x - 54\\hfill &amp; \\text{Add the products}.\\hfill \\\\ 6{x}^{2}+\\left(6x - 54x\\right)-54\\hfill &amp; \\text{Combine like terms}.\\hfill \\\\ 6{x}^{2}-48x - 54\\hfill &amp; \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nUse FOIL to find the product.\n\n[latex]\\left(x+7\\right)\\left(3x - 5\\right)[\/latex]\n\n[reveal-answer q=\"603351\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"603351\"]\n\n[latex]3{x}^{2}+16x - 35[\/latex]\n\n[\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93539&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"350\"><\/iframe>\n\n<\/div>\n<h2>Special Cases of Polynomials<\/h2>\n<h3>Perfect Square Trinomials<\/h3>\nCertain binomial products have special forms. When a binomial is squared, the result is called a <strong>perfect square trinomial<\/strong>. We can find the square by multiplying the binomial by itself. However, there is a special form that each of these perfect square trinomials takes, and memorizing the form makes squaring binomials much easier. Let\u2019s look at a few perfect square trinomials to familiarize ourselves with the form.\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill \\text{ }{\\left(x+5\\right)}^{2}&amp; =&amp; \\text{ }{x}^{2}+10x+25\\hfill \\\\ \\hfill {\\left(x - 3\\right)}^{2}&amp; =&amp; \\text{ }{x}^{2}-6x+9\\hfill \\\\ \\hfill {\\left(4x - 1\\right)}^{2}&amp; =&amp; 4{x}^{2}-8x+1\\hfill \\end{array}[\/latex]<\/div>\nNotice that the first term of each trinomial is the square of the first term of the binomial and, similarly, the last term of each trinomial is the square of the last term of the binomial. The middle term is double the product of the two terms. Lastly, we see that the first sign of the trinomial is the same as the sign of the binomial.\n<div class=\"textbox\">\n<h3>A General Note: Perfect Square Trinomials<\/h3>\nWhen a binomial is squared, the result is the first term squared added to double the product of both terms and the last term squared.\n<div style=\"text-align: center;\">[latex]{\\left(x+a\\right)}^{2}=\\left(x+a\\right)\\left(x+a\\right)={x}^{2}+2ax+{a}^{2}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a binomial, square it using the formula for perfect square trinomials<strong>\n<\/strong><\/h3>\n<ol>\n \t<li>Square the first term of the binomial.<\/li>\n \t<li>Square the last term of the binomial.<\/li>\n \t<li>For the middle term of the trinomial, double the product of the two terms.<\/li>\n \t<li>Add and simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Expanding Perfect Squares<\/h3>\nExpand [latex]{\\left(3x - 8\\right)}^{2}[\/latex].\n\n[reveal-answer q=\"733978\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"733978\"]\n\nBegin by squaring the first term and the last term. For the middle term of the trinomial, double the product of the two terms.\n<div style=\"text-align: center;\">[latex]{\\left(3x\\right)}^{2}-2\\left(3x\\right)\\left(8\\right)+{\\left(-8\\right)}^{2}[\/latex]<\/div>\n<p style=\"text-align: center;\">[latex]9{x}^{2}-48x+64[\/latex].<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nExpand [latex]{\\left(4x - 1\\right)}^{2}[\/latex].\n\n[reveal-answer q=\"278544\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"278544\"]\n\n[latex]16{x}^{2}-8x+1[\/latex][\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1825&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe>\n\n<\/div>\n<h3>Difference of Squares<\/h3>\nAnother special product is called the <strong>difference of squares&nbsp;<\/strong>which occurs when we multiply a binomial by another binomial with the same terms but the opposite sign. Let\u2019s see what happens when we multiply [latex]\\left(x+1\\right)\\left(x - 1\\right)[\/latex] using the FOIL method.\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill \\left(x+1\\right)\\left(x - 1\\right)&amp; =&amp; {x}^{2}-x+x - 1\\hfill \\\\ &amp; =&amp; {x}^{2}-1\\hfill \\end{array}[\/latex]<\/div>\nThe middle term drops out resulting in a difference of squares. Just as we did with the perfect squares, let\u2019s look at a few examples.\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill \\left(x+5\\right)\\left(x - 5\\right)&amp; =&amp; {x}^{2}-25\\hfill \\\\ \\hfill \\left(x+11\\right)\\left(x - 11\\right)&amp; =&amp; {x}^{2}-121\\hfill \\\\ \\hfill \\left(2x+3\\right)\\left(2x - 3\\right)&amp; =&amp; 4{x}^{2}-9\\hfill \\end{array}[\/latex]<\/div>\nBecause the sign changes in the second binomial, the outer and inner terms cancel each other out, and we are left only with the square of the first term minus the square of the last term.\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<strong>Is there a special form for the sum of squares?<\/strong>\n\n<em>No. The difference of squares occurs because the opposite signs of the binomials cause the middle terms to disappear. There are no two binomials that multiply to equal a sum of squares.<\/em>\n\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Difference of Squares<\/h3>\nWhen a binomial is multiplied by a binomial with the same terms separated by the opposite sign, the result is the square of the first term minus the square of the last term.\n<div style=\"text-align: center;\">[latex]\\left(a+b\\right)\\left(a-b\\right)={a}^{2}-{b}^{2}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a binomial multiplied by a binomial with the same terms but the opposite sign, find the difference of squares<\/h3>\n<ol>\n \t<li>Square the first term of the binomials.<\/li>\n \t<li>Square the last term of the binomials.<\/li>\n \t<li>Subtract the square of the last term from the square of the first term.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Multiplying Binomials Resulting in a Difference of Squares<\/h3>\nMultiply [latex]\\left(9x+4\\right)\\left(9x - 4\\right)[\/latex].\n\n[reveal-answer q=\"366563\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"366563\"]\n\nSquare the first term to get [latex]{\\left(9x\\right)}^{2}=81{x}^{2}[\/latex]. Square the last term to get [latex]{4}^{2}=16[\/latex]. Subtract the square of the last term from the square of the first term to find the product of [latex]81{x}^{2}-16[\/latex].\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nMultiply [latex]\\left(2x+7\\right)\\left(2x - 7\\right)[\/latex].\n\n[reveal-answer q=\"951379\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"951379\"]\n\n[latex]4{x}^{2}-49[\/latex][\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1856&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe>\n\n<\/div>\n<h2>Performing Operations with Polynomials of Several Variables<\/h2>\nWe have looked at polynomials containing only one variable. However, a polynomial can contain several variables. All of the same rules apply when working with polynomials containing several variables. Consider an example:\n<div style=\"text-align: center;\">[latex]\\begin{array}{cc}\\left(a+2b\\right)\\left(4a-b-c\\right)\\hfill &amp; \\hfill \\\\ a\\left(4a-b-c\\right)+2b\\left(4a-b-c\\right)\\hfill &amp; \\text{Use the distributive property}.\\hfill \\\\ 4{a}^{2}-ab-ac+8ab - 2{b}^{2}-2bc\\hfill &amp; \\text{Multiply}.\\hfill \\\\ 4{a}^{2}+\\left(-ab+8ab\\right)-ac - 2{b}^{2}-2bc\\hfill &amp; \\text{Combine like terms}.\\hfill \\\\ 4{a}^{2}+7ab-ac - 2bc - 2{b}^{2}\\hfill &amp; \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Multiplying Polynomials Containing Several Variables<\/h3>\nMultiply [latex]\\left(x+4\\right)\\left(3x - 2y+5\\right)[\/latex].\n\n[reveal-answer q=\"313997\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"313997\"]\n\nFollow the same steps that we used to multiply polynomials containing only one variable.\n<div style=\"text-align: center;\">[latex]\\begin{array}{cc}x\\left(3x - 2y+5\\right)+4\\left(3x - 2y+5\\right) \\hfill &amp; \\text{Use the distributive property}.\\hfill \\\\ 3{x}^{2}-2xy+5x+12x - 8y+20\\hfill &amp; \\text{Multiply}.\\hfill \\\\ 3{x}^{2}-2xy+\\left(5x+12x\\right)-8y+20\\hfill &amp; \\text{Combine like terms}.\\hfill \\\\ 3{x}^{2}-2xy+17x - 8y+20 \\hfill &amp; \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/div>\n<div>[\/hidden-answer]<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n[latex]\\left(3x - 1\\right)\\left(2x+7y - 9\\right)[\/latex].\n\n[reveal-answer q=\"383366\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"383366\"]\n\n[latex]6{x}^{2}+21xy - 29x - 7y+9[\/latex][\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=100774&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"350\"><\/iframe>\n\n<\/div>\n<h2>Key Equations<\/h2>\n<table>\n<tbody>\n<tr>\n<td><strong>perfect square trinomial<\/strong><\/td>\n<td>[latex]{\\left(x+a\\right)}^{2}=\\left(x+a\\right)\\left(x+a\\right)={x}^{2}+2ax+{a}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>difference of squares<\/strong><\/td>\n<td>[latex]\\left(a+b\\right)\\left(a-b\\right)={a}^{2}-{b}^{2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Key Concepts<\/h2>\n<ul>\n \t<li>A polynomial is a sum of terms each consisting of a variable raised to a nonnegative integer power. The degree is the highest power of the variable that occurs in the polynomial. The leading term is the term containing the highest degree, and the leading coefficient is the coefficient of that term.<\/li>\n \t<li>We can add and subtract polynomials by combining like terms.<\/li>\n \t<li>To multiply polynomials, use the distributive property to multiply each term in the first polynomial by each term in the second. Then add the products.<\/li>\n \t<li>FOIL (First, Outer, Inner, Last) is a shortcut that can be used to multiply binomials.<\/li>\n \t<li>Perfect square trinomials and difference of squares are special products.<\/li>\n \t<li>Follow the same rules to work with polynomials containing several variables.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165131990658\" class=\"definition\">\n \t<dt><strong>binomial<\/strong><\/dt>\n \t<dd id=\"fs-id1165131990661\">a polynomial containing two terms<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943522\" class=\"definition\">\n \t<dt><strong>coefficient<\/strong><\/dt>\n \t<dd id=\"fs-id1165132943525\">any real number [latex]{a}_{i}[\/latex] in a polynomial of the form [latex]{a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943528\" class=\"definition\">\n \t<dt><strong>degree<\/strong><\/dt>\n \t<dd id=\"fs-id1165134297639\">the highest power of the variable that occurs in a polynomial<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134297646\" class=\"definition\">\n \t<dt><strong>difference of squares<\/strong><\/dt>\n \t<dd id=\"fs-id1165135486042\">the binomial that results when a binomial is multiplied by a binomial with the same terms, but the opposite sign<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137644987\" class=\"definition\">\n \t<dt><strong>leading coefficient<\/strong><\/dt>\n \t<dd id=\"fs-id1165137644990\">the coefficient of the leading term<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n \t<dt>\n<dl id=\"fs-id1165132943528\" class=\"definition\">\n \t<dt>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n \t<dt><strong>leading term<\/strong><\/dt>\n \t<dd id=\"fs-id1165133085665\">&nbsp;the term containing the highest degree<\/dd>\n<\/dl>\n<\/dt>\n<\/dl>\n<dl id=\"fs-id1165134297646\" class=\"definition\">\n \t<dt><strong>monomial<\/strong><\/dt>\n \t<dd id=\"fs-id1165135486042\">a polynomial containing one term<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137644987\" class=\"definition\">\n \t<dt><strong>perfect square trinomial<\/strong><\/dt>\n \t<dd id=\"fs-id1165137644990\">the trinomial that results when a binomial is squared<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n \t<dt>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n \t<dt>\n<dl id=\"fs-id1165137644987\" class=\"definition\">\n \t<dt><strong>polynomial<\/strong><\/dt>\n \t<dd id=\"fs-id1165137644990\">a sum of terms each consisting of a variable raised to a nonnegative integer power<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n \t<dt><strong>term of a polynomial<\/strong><\/dt>\n \t<dd id=\"fs-id1165133085665\">any [latex]{a}_{i}{x}^{i}[\/latex] of a polynomial of the form [latex]{a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/dd>\n<\/dl>\n<\/dt>\n<\/dl>\n<\/dt>\n \t<dt>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n \t<dt>\n<dl id=\"fs-id1165137644987\" class=\"definition\">\n \t<dt><strong>trinomial<\/strong><\/dt>\n \t<dd id=\"fs-id1165137644990\">a polynomial containing three terms<\/dd>\n<\/dl>\n<\/dt>\n<\/dl>\n<\/dt>\n<\/dl>\n<\/dt>\n<\/dl>\n\n","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify the degree, leading coefficient, and leading term of a polynomial.<\/li>\n<li>Add and subtract polynomials.<\/li>\n<li>Multiply polynomials.<\/li>\n<li>Square a binomial.<\/li>\n<li>Find a difference of squares.<\/li>\n<li>Perform operations on polynomials with several variables.<\/li>\n<\/ul>\n<\/div>\n<p>Earl is building a doghouse whose front is in the shape of a square topped with a triangle. There will be a rectangular door through which the dog can enter and exit the house. Earl wants to find the area of the front of the doghouse so that he can purchase the correct amount of paint. Using the measurements of the front of the house shown below, we can create an expression that combines several variable terms which allows us to solve this problem and others like it.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21205143\/CNX_CAT_Figure_01_04_001.jpg\" alt=\"Sketch of a house formed by a square and a triangle based on the top of the square. A rectangle is placed at the bottom center of the square to mark a doorway. The height of the door is labeled: x and the width of the door is labeled: 1 foot. The side of the square is labeled: 2x. The height of the triangle is labeled: 3\/2 feet.\" width=\"487\" height=\"249\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Measurements of the front of the doghouse Earl is building.<\/b><\/p>\n<\/div>\n<p>First, find the area of the square in square feet.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill A& =& {s}^{2}\\hfill \\\\ & =& {\\left(2x\\right)}^{2}\\hfill \\\\ & =& 4{x}^{2}\\hfill \\end{array}[\/latex]<\/div>\n<p>Then, find the area of the triangle in square feet.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill A& =& \\frac{1}{2}bh\\hfill \\\\ & =& \\text{}\\frac{1}{2}\\left(2x\\right)\\left(\\frac{3}{2}\\right)\\hfill \\\\ & =& \\text{}\\frac{3}{2}x\\hfill \\end{array}[\/latex]<\/div>\n<p>Next, find the area of the rectangular door in square feet.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill A& =& lw\\hfill \\\\ & =& x\\cdot 1\\hfill \\\\ \\hfill & =& x\\hfill \\end{array}[\/latex]<\/div>\n<h2>Operations on Polynomials<\/h2>\n<p>The area of the front of the doghouse can be found by adding the areas of the square and the triangle and then subtracting the area of the rectangle. When we do this, we get [latex]4{x}^{2}+\\frac{3}{2}x-x[\/latex] ft<sup>2<\/sup>, or [latex]4{x}^{2}+\\frac{1}{2}x[\/latex] ft<sup>2<\/sup>.<\/p>\n<p>In this section, we will examine expressions such as this one, which combine several variable terms.<\/p>\n<p>The area of the front of the doghouse described in the introduction was&nbsp;[latex]4{x}^{2}+\\frac{1}{2}x[\/latex] ft<sup>2<\/sup>.<\/p>\n<p>This is an example of a <strong>polynomial&nbsp;<\/strong>which is a sum of or difference of terms each consisting of a variable raised to a nonnegative integer power. A number multiplied by a variable raised to an exponent, such as [latex]384\\pi[\/latex], is known as a <strong>coefficient<\/strong>. Coefficients can be positive, negative, or zero, and can be whole numbers, decimals, or fractions. Each product [latex]{a}_{i}{x}^{i}[\/latex], such as [latex]384\\pi w[\/latex], is a <strong>term of a polynomial<\/strong>. If a term does not contain a variable, it is called a <em>constant<\/em>.<\/p>\n<p>A polynomial containing only one term, such as [latex]5{x}^{4}[\/latex], is called a <strong>monomial<\/strong>. A polynomial containing two terms, such as [latex]2x - 9[\/latex], is called a <strong>binomial<\/strong>. A polynomial containing three terms, such as [latex]-3{x}^{2}+8x - 7[\/latex], is called a <strong>trinomial<\/strong>.<\/p>\n<p>We can find the <strong>degree<\/strong> of a polynomial by identifying the highest power of the variable that occurs in the polynomial. The term with the highest degree is called the <strong>leading term<\/strong> because it is usually written first. The coefficient of the leading term is called the <strong>leading coefficient<\/strong>. When a polynomial is written so that the powers are descending, we say that it is in standard form.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21205146\/CNX_CAT_Figure_01_04_002.jpg\" alt=\"A polynomial reading: a sub n times x to the nth power plus and so on plus a sub 2 times x squared plus a sub one times x plus a subzero is shown. The a in the term a sub n is labeled: leading coefficient. The n in the term x to the nth power is labeled: degree. Finally, the entire term is labeled as: Leading term.\" \/><\/p>\n<div class=\"textbox\">\n<h3>A General Note: Polynomials<\/h3>\n<p>A <strong>polynomial<\/strong> is an expression that can be written in the form<\/p>\n<div style=\"text-align: center;\">[latex]{a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/div>\n<p>Each real number <em>a<sub>i&nbsp;<\/sub><\/em>is called a <strong>coefficient<\/strong>. The number [latex]{a}_{0}[\/latex] that is not multiplied by a variable is called a <em>constant<\/em>. Each product [latex]{a}_{i}{x}^{i}[\/latex] is a <strong>term of a polynomial<\/strong>. The highest power of the variable that occurs in the polynomial is called the <strong>degree<\/strong> of a polynomial. The <strong>leading term<\/strong> is the term with the highest power, and its coefficient is called the <strong>leading coefficient<\/strong>.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a polynomial expression, identify the degree and leading coefficient<\/h3>\n<ol>\n<li>Find the highest power of <em>x<\/em> to determine the degree.<\/li>\n<li>Identify the term containing the highest power of <em>x<\/em> to find the leading term.<\/li>\n<li>Identify the coefficient of the leading term.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Identifying the Degree and Leading Coefficient of a Polynomial<\/h3>\n<p>For the following polynomials, identify the degree, the leading term, and the leading coefficient.<\/p>\n<ol>\n<li>[latex]3+2{x}^{2}-4{x}^{3}[\/latex]<\/li>\n<li>[latex]5{t}^{5}-2{t}^{3}+7t[\/latex]<\/li>\n<li>[latex]6p-{p}^{3}-2[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q973610\">Show Solution<\/span><\/p>\n<div id=\"q973610\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The highest power of <em>x<\/em> is 3, so the degree is 3. The leading term is the term containing that degree, [latex]-4{x}^{3}[\/latex]. The leading coefficient is the coefficient of that term, [latex]-4[\/latex].<\/li>\n<li>The highest power of <em>t<\/em> is [latex]5[\/latex], so the degree is [latex]5[\/latex]. The leading term is the term containing that degree, [latex]5{t}^{5}[\/latex]. The leading coefficient is the coefficient of that term, [latex]5[\/latex].<\/li>\n<li>The highest power of <em>p<\/em> is [latex]3[\/latex], so the degree is [latex]3[\/latex]. The leading term is the term containing that degree, [latex]-{p}^{3}[\/latex], The leading coefficient is the coefficient of that term, [latex]-1[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Identify the degree, leading term, and leading coefficient of the polynomial [latex]4{x}^{2}-{x}^{6}+2x - 6[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q49799\">Show Solution<\/span><\/p>\n<div id=\"q49799\" class=\"hidden-answer\" style=\"display: none\">\n<p>The degree is 6, the leading term is [latex]-{x}^{6}[\/latex], and the leading coefficient is [latex]-1[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93531&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<h3>Adding and Subtracting Polynomials<\/h3>\n<p>We can add and subtract polynomials by combining like terms, which are terms that contain the same variables raised to the same exponents. For example, [latex]5{x}^{2}[\/latex] and [latex]-2{x}^{2}[\/latex] are like terms and can be added to get [latex]3{x}^{2}[\/latex], but [latex]3x[\/latex] and [latex]3{x}^{2}[\/latex] are not like terms and therefore cannot be added.<\/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. Standard form means you start with the leading term, and write the rest of the terms in descending order by degree.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Adding Polynomials<\/h3>\n<p>Find the sum.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(12{x}^{2}+9x - 21\\right)+\\left(4{x}^{3}+8{x}^{2}-5x+20\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q660613\">Show Solution<\/span><\/p>\n<div id=\"q660613\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\begin{array}{cc}4{x}^{3}+\\left(12{x}^{2}+8{x}^{2}\\right)+\\left(9x - 5x\\right)+\\left(-21+20\\right) \\hfill & \\text{Combine like terms}.\\hfill \\\\ 4{x}^{3}+20{x}^{2}+4x - 1\\hfill & \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>We can check our answers to these types of problems using a graphing calculator. To check, graph the problem as given along with the simplified answer. The two graphs should be the same. Be sure to use the same window to compare the graphs. Using different windows can make the expressions seem equivalent when they are not.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div><\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find the sum.<\/p>\n<p>[latex]\\left(2{x}^{3}+5{x}^{2}-x+1\\right)+\\left(2{x}^{2}-3x - 4\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q121561\">Show Solution<\/span><\/p>\n<div id=\"q121561\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]2{x}^{3}+7{x}^{2}-4x - 3[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93536&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Subtracting Polynomials<\/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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q831247\">Show Solution<\/span><\/p>\n<div id=\"q831247\" class=\"hidden-answer\" style=\"display: none\">\n[latex]\\begin{array}{cc}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<h4>Analysis of the Solution<\/h4>\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<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find the difference.<\/p>\n<p>[latex]\\left(-7{x}^{3}-7{x}^{2}+6x - 2\\right)-\\left(4{x}^{3}-6{x}^{2}-x+7\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q260383\">Show Solution<\/span><\/p>\n<div id=\"q260383\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]-11{x}^{3}-{x}^{2}+7x - 9[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93537&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<h3>Multiplying Polynomials<\/h3>\n<p>Multiplying polynomials is a bit more challenging than adding and subtracting polynomials. We must use the distributive property to multiply each term in the first polynomial by each term in the second polynomial. We then combine like terms. We can also use a shortcut called the FOIL method when multiplying binomials. Certain special products follow patterns that we can memorize and use instead of multiplying the polynomials by hand each time. We will look at a variety of ways to multiply polynomials.<\/p>\n<h3>Multiplying Polynomials Using the Distributive Property<\/h3>\n<p>To multiply a number by a polynomial we use the distributive property. The number must be distributed to each term of the polynomial. In [latex]2\\left(x+7\\right)[\/latex] we can distribute [latex]2[\/latex] to obtain the expression [latex]2x+14[\/latex]. When multiplying polynomials, the distributive property allows us to multiply each term of the first polynomial by each term of the second. We then add the products together and combine like terms to simplify.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given the multiplication of two polynomials, use the distributive property to simplify the expression<\/h3>\n<ol>\n<li>Multiply each term of the first polynomial by each term of the second.<\/li>\n<li>Combine like terms.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Multiplying Polynomials Using the Distributive Property<\/h3>\n<p>Find the product.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(2x+1\\right)\\left(3{x}^{2}-x+4\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q752165\">Show Solution<\/span><\/p>\n<div id=\"q752165\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{array}{cc}2x\\left(3{x}^{2}-x+4\\right)+1\\left(3{x}^{2}-x+4\\right) \\hfill & \\text{Use the distributive property}.\\hfill \\\\ \\left(6{x}^{3}-2{x}^{2}+8x\\right)+\\left(3{x}^{2}-x+4\\right)\\hfill & \\text{Multiply}.\\hfill \\\\ 6{x}^{3}+\\left(-2{x}^{2}+3{x}^{2}\\right)+\\left(8x-x\\right)+4\\hfill & \\text{Combine like terms}.\\hfill \\\\ 6{x}^{3}+{x}^{2}+7x+4 \\hfill & \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>We can use a table to keep track of our work, as shown in the table below. Write one polynomial across the top and the other down the side. For each box in the table, multiply the term for that row by the term for that column. Then add all of the terms together, combine like terms, and simplify.<\/p>\n<table style=\"height: 45px;\" summary=\"A table with 3 rows and 4 columns. The first entry of the first row is empty, the others are labeled: three times x squared, negative x, and positive four. The first entry of the second row is labeled: two times x. The second entry reads: six times x cubed. The third entry reads: negative two times x squared. The fourth entry reads: eight times x. The first entry of the third row reads: positive one. The second entry reads: three times x squared. The third entry reads: negative x. The fourth entry reads: four.\">\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"width: 115px; height: 15px;\"><\/td>\n<td style=\"width: 159px; height: 15px;\"><strong>[latex]3{x}^{2}[\/latex]<\/strong><\/td>\n<td style=\"width: 154px; height: 15px;\"><strong>[latex]-x[\/latex]<\/strong><\/td>\n<td style=\"width: 116px; height: 15px;\"><strong>[latex]+4[\/latex]<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 115px; height: 15px;\"><strong>[latex]2x[\/latex]<\/strong><\/td>\n<td style=\"width: 159px; height: 15px;\">[latex]6{x}^{3}\\\\[\/latex]<\/td>\n<td style=\"width: 154px; height: 15px;\">[latex]-2{x}^{2}[\/latex]<\/td>\n<td style=\"width: 116px; height: 15px;\">[latex]8x[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 115px; height: 15px;\"><strong>[latex]+1[\/latex]<\/strong><\/td>\n<td style=\"width: 159px; height: 15px;\">[latex]3{x}^{2}[\/latex]<\/td>\n<td style=\"width: 154px; height: 15px;\">[latex]-x[\/latex]<\/td>\n<td style=\"width: 116px; height: 15px;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find the product.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(3x+2\\right)\\left({x}^{3}-4{x}^{2}+7\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q508646\">Show Solution<\/span><\/p>\n<div id=\"q508646\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]3{x}^{4}-10{x}^{3}-8{x}^{2}+21x+14[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>We can use a table to keep track of our work as shown below. Write one polynomial across the top and the other down the side. For each box in the table, multiply the term for that row by the term for that column. Then add all of the terms together, combine like terms, and simplify.<\/p>\n<table summary=\"A table with 3 rows and 4 columns. The first entry of the first row is empty, the others are labeled: three times x squared, negative x, and positive four. The first entry of the second row is labeled: two times x. The second entry reads: six times x cubed. The third entry reads: negative two times x squared. The fourth entry reads: eight times x. The first entry of the third row reads: positive one. The second entry reads: three times x squared. The third entry reads: negative x. The fourth entry reads: four.\">\n<tbody>\n<tr>\n<td style=\"width: 115px;\"><\/td>\n<td style=\"width: 159px;\"><strong>[latex]3{x}^{2}[\/latex]<\/strong><\/td>\n<td style=\"width: 154px;\"><strong>[latex]-x[\/latex]<\/strong><\/td>\n<td style=\"width: 116px;\"><strong>[latex]+4[\/latex]<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 115px;\"><strong>[latex]2x[\/latex]<\/strong><\/td>\n<td style=\"width: 159px;\">[latex]6{x}^{3}\\\\[\/latex]<\/td>\n<td style=\"width: 154px;\">[latex]-2{x}^{2}[\/latex]<\/td>\n<td style=\"width: 116px;\">[latex]8x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 115px;\"><strong>[latex]+1[\/latex]<\/strong><\/td>\n<td style=\"width: 159px;\">[latex]3{x}^{2}[\/latex]<\/td>\n<td style=\"width: 154px;\">[latex]-x[\/latex]<\/td>\n<td style=\"width: 116px;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3864&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<h3>Using FOIL to Multiply Binomials<\/h3>\n<p>A shortcut called FOIL is sometimes used to find the product of two binomials. It is called FOIL because we multiply the <strong>f<\/strong>irst terms, the <strong>o<\/strong>uter terms, the <strong>i<\/strong>nner terms, and then the <strong>l<\/strong>ast terms of each binomial.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21205149\/CNX_CAT_Figure_01_04_003.jpg\" alt=\"Two quantities in parentheses are being multiplied, the first being: a times x plus b and the second being: c times x plus d. This expression equals ac times x squared plus ad times x plus bc times x plus bd. The terms ax and cx are labeled: First Terms. The terms ax and d are labeled: Outer Terms. The terms b and cx are labeled: Inner Terms. The terms b and d are labeled: Last Terms.\" \/><\/p>\n<p>The FOIL method is simply just the distributive property. We are multiplying each term of the first binomial by each term of the second binomial and then combining like terms.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given two binomials, Multiplying Using FOIL<\/h3>\n<ol>\n<li>Multiply the first terms of each binomial.<\/li>\n<li>Multiply the outer terms of the binomials.<\/li>\n<li>Multiply the inner terms of the binomials.<\/li>\n<li>Multiply the last terms of each binomial.<\/li>\n<li>Add the products.<\/li>\n<li>Combine like terms and simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using FOIL to Multiply Binomials<\/h3>\n<p>Use FOIL to find the product.<\/p>\n<p>[latex]\\left(2x-18\\right)\\left(3x + 3\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q698991\">Show Solution<\/span><\/p>\n<div id=\"q698991\" class=\"hidden-answer\" style=\"display: none\">\nFind the product of the first terms.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21205151\/CNX_CAT_Figure_01_04_004.jpg\" alt=\"\" \/><\/p>\n<p>Find the product of the outer terms.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21205154\/CNX_CAT_Figure_01_04_005.jpg\" alt=\"\" \/><\/p>\n<p>Find the product of the inner terms.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21205156\/CNX_CAT_Figure_01_04_006.jpg\" alt=\"\" \/><\/p>\n<p>Find the product of the last terms.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21205158\/CNX_CAT_Figure_01_04_007.jpg\" alt=\"\" \/><\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{cc}6{x}^{2}+6x - 54x - 54\\hfill & \\text{Add the products}.\\hfill \\\\ 6{x}^{2}+\\left(6x - 54x\\right)-54\\hfill & \\text{Combine like terms}.\\hfill \\\\ 6{x}^{2}-48x - 54\\hfill & \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Use FOIL to find the product.<\/p>\n<p>[latex]\\left(x+7\\right)\\left(3x - 5\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q603351\">Show Solution<\/span><\/p>\n<div id=\"q603351\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]3{x}^{2}+16x - 35[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93539&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<h2>Special Cases of Polynomials<\/h2>\n<h3>Perfect Square Trinomials<\/h3>\n<p>Certain binomial products have special forms. When a binomial is squared, the result is called a <strong>perfect square trinomial<\/strong>. We can find the square by multiplying the binomial by itself. However, there is a special form that each of these perfect square trinomials takes, and memorizing the form makes squaring binomials much easier. Let\u2019s look at a few perfect square trinomials to familiarize ourselves with the form.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill \\text{ }{\\left(x+5\\right)}^{2}& =& \\text{ }{x}^{2}+10x+25\\hfill \\\\ \\hfill {\\left(x - 3\\right)}^{2}& =& \\text{ }{x}^{2}-6x+9\\hfill \\\\ \\hfill {\\left(4x - 1\\right)}^{2}& =& 4{x}^{2}-8x+1\\hfill \\end{array}[\/latex]<\/div>\n<p>Notice that the first term of each trinomial is the square of the first term of the binomial and, similarly, the last term of each trinomial is the square of the last term of the binomial. The middle term is double the product of the two terms. Lastly, we see that the first sign of the trinomial is the same as the sign of the binomial.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Perfect Square Trinomials<\/h3>\n<p>When a binomial is squared, the result is the first term squared added to double the product of both terms and the last term squared.<\/p>\n<div style=\"text-align: center;\">[latex]{\\left(x+a\\right)}^{2}=\\left(x+a\\right)\\left(x+a\\right)={x}^{2}+2ax+{a}^{2}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a binomial, square it using the formula for perfect square trinomials<strong><br \/>\n<\/strong><\/h3>\n<ol>\n<li>Square the first term of the binomial.<\/li>\n<li>Square the last term of the binomial.<\/li>\n<li>For the middle term of the trinomial, double the product of the two terms.<\/li>\n<li>Add and simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Expanding Perfect Squares<\/h3>\n<p>Expand [latex]{\\left(3x - 8\\right)}^{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q733978\">Show Solution<\/span><\/p>\n<div id=\"q733978\" class=\"hidden-answer\" style=\"display: none\">\n<p>Begin by squaring the first term and the last term. For the middle term of the trinomial, double the product of the two terms.<\/p>\n<div style=\"text-align: center;\">[latex]{\\left(3x\\right)}^{2}-2\\left(3x\\right)\\left(8\\right)+{\\left(-8\\right)}^{2}[\/latex]<\/div>\n<p style=\"text-align: center;\">[latex]9{x}^{2}-48x+64[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Expand [latex]{\\left(4x - 1\\right)}^{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q278544\">Show Solution<\/span><\/p>\n<div id=\"q278544\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]16{x}^{2}-8x+1[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1825&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<h3>Difference of Squares<\/h3>\n<p>Another special product is called the <strong>difference of squares&nbsp;<\/strong>which occurs when we multiply a binomial by another binomial with the same terms but the opposite sign. Let\u2019s see what happens when we multiply [latex]\\left(x+1\\right)\\left(x - 1\\right)[\/latex] using the FOIL method.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill \\left(x+1\\right)\\left(x - 1\\right)& =& {x}^{2}-x+x - 1\\hfill \\\\ & =& {x}^{2}-1\\hfill \\end{array}[\/latex]<\/div>\n<p>The middle term drops out resulting in a difference of squares. Just as we did with the perfect squares, let\u2019s look at a few examples.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill \\left(x+5\\right)\\left(x - 5\\right)& =& {x}^{2}-25\\hfill \\\\ \\hfill \\left(x+11\\right)\\left(x - 11\\right)& =& {x}^{2}-121\\hfill \\\\ \\hfill \\left(2x+3\\right)\\left(2x - 3\\right)& =& 4{x}^{2}-9\\hfill \\end{array}[\/latex]<\/div>\n<p>Because the sign changes in the second binomial, the outer and inner terms cancel each other out, and we are left only with the square of the first term minus the square of the last term.<\/p>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<p><strong>Is there a special form for the sum of squares?<\/strong><\/p>\n<p><em>No. The difference of squares occurs because the opposite signs of the binomials cause the middle terms to disappear. There are no two binomials that multiply to equal a sum of squares.<\/em><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Difference of Squares<\/h3>\n<p>When a binomial is multiplied by a binomial with the same terms separated by the opposite sign, the result is the square of the first term minus the square of the last term.<\/p>\n<div style=\"text-align: center;\">[latex]\\left(a+b\\right)\\left(a-b\\right)={a}^{2}-{b}^{2}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a binomial multiplied by a binomial with the same terms but the opposite sign, find the difference of squares<\/h3>\n<ol>\n<li>Square the first term of the binomials.<\/li>\n<li>Square the last term of the binomials.<\/li>\n<li>Subtract the square of the last term from the square of the first term.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Multiplying Binomials Resulting in a Difference of Squares<\/h3>\n<p>Multiply [latex]\\left(9x+4\\right)\\left(9x - 4\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q366563\">Show Solution<\/span><\/p>\n<div id=\"q366563\" class=\"hidden-answer\" style=\"display: none\">\n<p>Square the first term to get [latex]{\\left(9x\\right)}^{2}=81{x}^{2}[\/latex]. Square the last term to get [latex]{4}^{2}=16[\/latex]. Subtract the square of the last term from the square of the first term to find the product of [latex]81{x}^{2}-16[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Multiply [latex]\\left(2x+7\\right)\\left(2x - 7\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q951379\">Show Solution<\/span><\/p>\n<div id=\"q951379\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]4{x}^{2}-49[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1856&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<h2>Performing Operations with Polynomials of Several Variables<\/h2>\n<p>We have looked at polynomials containing only one variable. However, a polynomial can contain several variables. All of the same rules apply when working with polynomials containing several variables. Consider an example:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{cc}\\left(a+2b\\right)\\left(4a-b-c\\right)\\hfill & \\hfill \\\\ a\\left(4a-b-c\\right)+2b\\left(4a-b-c\\right)\\hfill & \\text{Use the distributive property}.\\hfill \\\\ 4{a}^{2}-ab-ac+8ab - 2{b}^{2}-2bc\\hfill & \\text{Multiply}.\\hfill \\\\ 4{a}^{2}+\\left(-ab+8ab\\right)-ac - 2{b}^{2}-2bc\\hfill & \\text{Combine like terms}.\\hfill \\\\ 4{a}^{2}+7ab-ac - 2bc - 2{b}^{2}\\hfill & \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Multiplying Polynomials Containing Several Variables<\/h3>\n<p>Multiply [latex]\\left(x+4\\right)\\left(3x - 2y+5\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q313997\">Show Solution<\/span><\/p>\n<div id=\"q313997\" class=\"hidden-answer\" style=\"display: none\">\n<p>Follow the same steps that we used to multiply polynomials containing only one variable.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{cc}x\\left(3x - 2y+5\\right)+4\\left(3x - 2y+5\\right) \\hfill & \\text{Use the distributive property}.\\hfill \\\\ 3{x}^{2}-2xy+5x+12x - 8y+20\\hfill & \\text{Multiply}.\\hfill \\\\ 3{x}^{2}-2xy+\\left(5x+12x\\right)-8y+20\\hfill & \\text{Combine like terms}.\\hfill \\\\ 3{x}^{2}-2xy+17x - 8y+20 \\hfill & \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>[latex]\\left(3x - 1\\right)\\left(2x+7y - 9\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q383366\">Show Solution<\/span><\/p>\n<div id=\"q383366\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]6{x}^{2}+21xy - 29x - 7y+9[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=100774&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<h2>Key Equations<\/h2>\n<table>\n<tbody>\n<tr>\n<td><strong>perfect square trinomial<\/strong><\/td>\n<td>[latex]{\\left(x+a\\right)}^{2}=\\left(x+a\\right)\\left(x+a\\right)={x}^{2}+2ax+{a}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>difference of squares<\/strong><\/td>\n<td>[latex]\\left(a+b\\right)\\left(a-b\\right)={a}^{2}-{b}^{2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Key Concepts<\/h2>\n<ul>\n<li>A polynomial is a sum of terms each consisting of a variable raised to a nonnegative integer power. The degree is the highest power of the variable that occurs in the polynomial. The leading term is the term containing the highest degree, and the leading coefficient is the coefficient of that term.<\/li>\n<li>We can add and subtract polynomials by combining like terms.<\/li>\n<li>To multiply polynomials, use the distributive property to multiply each term in the first polynomial by each term in the second. Then add the products.<\/li>\n<li>FOIL (First, Outer, Inner, Last) is a shortcut that can be used to multiply binomials.<\/li>\n<li>Perfect square trinomials and difference of squares are special products.<\/li>\n<li>Follow the same rules to work with polynomials containing several variables.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165131990658\" class=\"definition\">\n<dt><strong>binomial<\/strong><\/dt>\n<dd id=\"fs-id1165131990661\">a polynomial containing two terms<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943522\" class=\"definition\">\n<dt><strong>coefficient<\/strong><\/dt>\n<dd id=\"fs-id1165132943525\">any real number [latex]{a}_{i}[\/latex] in a polynomial of the form [latex]{a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943528\" class=\"definition\">\n<dt><strong>degree<\/strong><\/dt>\n<dd id=\"fs-id1165134297639\">the highest power of the variable that occurs in a polynomial<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134297646\" class=\"definition\">\n<dt><strong>difference of squares<\/strong><\/dt>\n<dd id=\"fs-id1165135486042\">the binomial that results when a binomial is multiplied by a binomial with the same terms, but the opposite sign<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137644987\" class=\"definition\">\n<dt><strong>leading coefficient<\/strong><\/dt>\n<dd id=\"fs-id1165137644990\">the coefficient of the leading term<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n<dt>\n<\/dt>\n<dt>\n<\/dt>\n<dt><strong>leading term<\/strong><\/dt>\n<dd id=\"fs-id1165133085665\">&nbsp;the term containing the highest degree<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134297646\" class=\"definition\">\n<dt><strong>monomial<\/strong><\/dt>\n<dd id=\"fs-id1165135486042\">a polynomial containing one term<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137644987\" class=\"definition\">\n<dt><strong>perfect square trinomial<\/strong><\/dt>\n<dd id=\"fs-id1165137644990\">the trinomial that results when a binomial is squared<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n<dt>\n<\/dt>\n<dt>\n<\/dt>\n<dt><strong>polynomial<\/strong><\/dt>\n<dd id=\"fs-id1165137644990\">a sum of terms each consisting of a variable raised to a nonnegative integer power<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n<dt><strong>term of a polynomial<\/strong><\/dt>\n<dd id=\"fs-id1165133085665\">any [latex]{a}_{i}{x}^{i}[\/latex] of a polynomial of the form [latex]{a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n<dt>\n<\/dt>\n<dt><strong>trinomial<\/strong><\/dt>\n<dd id=\"fs-id1165137644990\">a polynomial containing three terms<\/dd>\n<\/dl>\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-1753\">\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>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><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <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\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Examples: Intro to Polynomials. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/3u16B2PN9zk\">https:\/\/youtu.be\/3u16B2PN9zk<\/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>Examples: 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><li>Examples: Multiplying Polynomials. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/bwTmApTV_8o\">https:\/\/youtu.be\/bwTmApTV_8o<\/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>Question ID 93531, 93536, 93537, 93539. <strong>Authored by<\/strong>: Michael Jenck. <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>: IMathAS Community License, CC-BY + GPL<\/li><li>Question ID 3864. <strong>Authored by<\/strong>: Tyler Wallace. <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>: IMathAS Community License, CC-BY + GPL<\/li><li>Question ID 1825. <strong>Authored by<\/strong>: David Whittaker. <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>: IMathAS Community License CC- BY + GPL<\/li><li>Question 1856. <strong>Authored by<\/strong>: Lawrence Morales. <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>: IMathAS Community License CC- BY + GPL<\/li><li>Question ID 100774. <strong>Authored by<\/strong>: Rick Rieman. <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>: IMathAS Community License CC- BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: OpenStax College Algebra. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/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":708740,"menu_order":13,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"College Algebra\",\"author\":\"OpenStax College 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Wallace\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License, CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 1825\",\"author\":\"David Whittaker\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC- BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question 1856\",\"author\":\"Lawrence Morales\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC- BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 100774\",\"author\":\"Rick Rieman\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC- BY + 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