{"id":833,"date":"2024-04-29T21:54:42","date_gmt":"2024-04-29T21:54:42","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=833"},"modified":"2025-01-16T19:43:11","modified_gmt":"2025-01-16T19:43:11","slug":"polynomial-basics-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/polynomial-basics-fresh-take\/","title":{"raw":"Polynomial Basics: Fresh Take","rendered":"Polynomial Basics: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li>Recognize polynomial functions, noting their degree and leading coefficient<\/li>\r\n \t<li>Add, subtract, and multiply polynomials using different methods, including the FOIL method for two-term polynomials<\/li>\r\n \t<li>Work with polynomials that have more than one variable, understanding how to combine and simplify them<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Identifying Polynomial Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n\r\nPolynomial functions are like the DNA of algebra\u2014they combine simplicity and complexity to form an incredibly diverse array of functions.\r\n\r\nAt their core, polynomials are sums of terms made up of coefficients and variables raised to whole number powers. The degree of a polynomial, given by the highest power of the variable, tells us a lot about the function's behavior and the shape of its graph.\r\n\r\nLet [latex]n[\/latex]\u00a0 be a non-negative integer. A <strong>polynomial function<\/strong> is a function that can be written in the form\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/p>\r\nThis is called the <strong>general form of a polynomial function<\/strong>.\r\n\r\nEach [latex]{a}_{i}[\/latex]\u00a0is a coefficient and can be any real number.\r\n\r\nEach product [latex]{a}_{i}{x}^{i}[\/latex]\u00a0is a <strong>term of a polynomial function<\/strong>.\r\n\r\n<strong>Quick Tips<\/strong>\r\n<ul>\r\n \t<li><strong>Linear, Quadratic, and Beyond:<\/strong> A first-degree polynomial is linear, a second-degree is quadratic, and higher degrees have their own characteristics and complexities.<\/li>\r\n \t<li><strong>Coefficient Clues:<\/strong> The coefficients in a polynomial can tell us about the steepness and direction of the graph. A positive leading coefficient means the graph opens upward, and a negative one indicates it opens downward.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Defining the Degree and Leading Coefficient of a Polynomial Function<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n\r\nThe degree of a polynomial is the highest power of the variable present. The leading coefficient is the coefficient of the term with the highest power. These two characteristics can tell us a lot about the function's behavior, especially its growth and end behavior.\r\n<ul>\r\n \t<li><strong>Degree Tells the Tale:<\/strong> The degree of a polynomial function hints at the number of roots and the possible number of turns on its graph. For instance, a second-degree polynomial, or a quadratic, will have at most two roots and one turn.<\/li>\r\n \t<li><strong>Leading Coefficient Impact:<\/strong> The leading coefficient isn't just a number\u2014it's the boss. It influences the end behavior of the polynomial's graph. If it's positive, the graph eventually rises; if negative, the graph falls.<\/li>\r\n<\/ul>\r\n<strong>Quick Tips: Identifying Degree and Leading Coefficient<\/strong>\r\n<ul>\r\n \t<li>Arrange terms in descending order of power to easily identify the leading term and coefficient.<\/li>\r\n \t<li>The degree gives a sense of the 'shape' of the graph of the polynomial function.<\/li>\r\n \t<li>For instance, [latex]f(x)=3+2x^2\u22124x^3[\/latex] has a degree of [latex]3[\/latex] and a leading coefficient of [latex]-4[\/latex], indicating a cubic function with a negative leading term.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">Identify the degree, leading term, and leading coefficient of the polynomial [latex]f\\left(x\\right)=4{x}^{2}-{x}^{6}+2x - 6[\/latex].[reveal-answer q=\"435637\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"435637\"]The degree is [latex]6[\/latex]. The leading term is [latex]-{x}^{6}[\/latex]. The leading coefficient is [latex]\u20131[\/latex].[\/hidden-answer]<\/section><section class=\"textbox watchIt\">Watch the following video for more examples of how to determine the degree, leading term, and leading coefficient of a polynomial.\r\n<script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hghafach-F_G_w82s0QA\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/F_G_w82s0QA?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hghafach-F_G_w82s0QA\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12843132&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-hghafach-F_G_w82s0QA&vembed=0&video_id=F_G_w82s0QA&video_target=tpm-plugin-hghafach-F_G_w82s0QA'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Degree%2C+Leading+Term%2C+and+Leading+Coefficient+of+a+Polynomial+Function_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDegree, Leading Term, and Leading Coefficient of a Polynomial Function\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Adding and Subtracting Polynomials<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea<\/strong>\r\n\r\n&nbsp;\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Like Terms:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Expressions with the same variables raised to the same powers<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Only coefficients are combined; exponents remain unchanged<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Adding Monomials:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Combine coefficients of like terms<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Keep the variable and its exponent the same<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Subtracting Monomials:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Similar to addition, but pay attention to signs<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Remember that subtracting a negative is the same as adding a positive<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Adding Polynomials:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Combine like terms across all polynomials<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use the Commutative Property to rearrange terms if needed<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Subtracting Polynomials:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Distribute the negative sign to all terms in the subtracted polynomial<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Then proceed as with addition<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Find the sum.\r\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>\r\n[reveal-answer q=\"660613\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"660613\"]\r\n\r\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]\r\n\r\n<strong>Analysis of the Solution<\/strong>\r\n\r\nWe can check our answers to these types of problems using a graphing calculator. To check, graph the original 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.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Find the sum.[latex]\\left(2{x}^{3}+5{x}^{2}-x+1\\right)+\\left(2{x}^{2}-3x - 4\\right)[\/latex][reveal-answer q=\"121561\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"121561\"][latex]2{x}^{3}+7{x}^{2}-4x - 3[\/latex][\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Find 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[reveal-answer q=\"831247\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"831247\"]\r\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]\r\n\r\n<strong>Analysis of the Solution<\/strong>\r\n\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\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Find the difference.[latex]\\left(-7{x}^{3}-7{x}^{2}+6x - 2\\right)-\\left(4{x}^{3}-6{x}^{2}-x+7\\right)[\/latex][reveal-answer q=\"260383\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"260383\"][latex]-11{x}^{3}-{x}^{2}+7x - 9[\/latex][\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch this video to see more examples of adding and subtracting polynomials.\r\n<script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cghchbgf-jiq3toC7wGM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/jiq3toC7wGM?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-cghchbgf-jiq3toC7wGM\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12843133&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-cghchbgf-jiq3toC7wGM&vembed=0&video_id=jiq3toC7wGM&video_target=tpm-plugin-cghchbgf-jiq3toC7wGM'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Adding+and+Subtracting+Polynomials_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Adding and Subtracting Polynomials\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Multiplying Polynomials<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea<\/strong>\r\n\r\n&nbsp;\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Distributive Property:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Key principle for multiplying polynomials<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]a(b + c) = ab + ac[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">General Polynomial Multiplication:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Multiply each term of the first polynomial by every term of the second<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Combine like terms in the result<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">FOIL Method:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Specific technique for multiplying two binomials<\/li>\r\n \t<li class=\"whitespace-normal break-words\">FOIL stands for First, Outer, Inner, Last<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Multiply the following:\r\n<p style=\"text-align: center;\">[latex](2x + 1)(3x^2 - x + 4)[\/latex]<\/p>\r\n[reveal-answer q=\"516700\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"516700\"]\r\n<ol>\r\n \t<li class=\"whitespace-normal break-words\">Distribute [latex]2x[\/latex] and [latex]1[\/latex]: [latex]2x(3x^2 - x + 4) + 1(3x^2 - x + 4)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Multiply term by term: [latex](6x^3 - 2x^2 + 8x) + (3x^2 - x + 4)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Combine like terms: [latex]6x^3 + (-2x^2 + 3x^2) + (8x - x) + 4[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplify: [latex]6x^3 + x^2 + 7x + 4[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Multiply using the FOIL method:\r\n<p style=\"text-align: center;\">[latex](2x - 18)(3x + 3)[\/latex]<\/p>\r\n[reveal-answer q=\"67601\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"67601\"]\r\n<ol>\r\n \t<li class=\"whitespace-normal break-words\">First terms: [latex]2x \\cdot 3x = 6x^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Outer terms: [latex]2x \\cdot 3 = 6x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Inner terms: [latex]-18 \\cdot 3x = -54x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Last terms: [latex]-18 \\cdot 3 = -54[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Combine all terms: [latex]6x^2 + 6x - 54x - 54[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplify: [latex]6x^2 - 48x - 54[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch this video to see more examples of how to use the distributive property to multiply polynomials.\r\n<script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-efgfdfhd-bwTmApTV_8o\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/bwTmApTV_8o?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-efgfdfhd-bwTmApTV_8o\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=6405041&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-efgfdfhd-bwTmApTV_8o&vembed=0&video_id=bwTmApTV_8o&video_target=tpm-plugin-efgfdfhd-bwTmApTV_8o'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Multiplying+Using+the+Distributive+Property_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Multiplying Using the Distributive Property\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Special Cases of Polynomials<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea<\/strong>\r\n\r\n&nbsp;\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Perfect Square Trinomials:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Result from squaring a binomial<\/li>\r\n \t<li class=\"whitespace-normal break-words\">General form: [latex](a + b)^2 = a^2 + 2ab + b^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Also applies to subtraction: [latex](a - b)^2 = a^2 - 2ab + b^2[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Characteristics of Perfect Square Trinomials:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">First and last terms are perfect squares<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Middle term is twice the product of the terms in the original binomial<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Sign of the middle term matches the sign in the original binomial<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Difference of Squares:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Result from multiplying sum and difference of the same terms<\/li>\r\n \t<li class=\"whitespace-normal break-words\">General form: [latex](a + b)(a - b) = a^2 - b^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Middle terms cancel out<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">No Special Form for Sum of Squares:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex](a + b)^2 \\neq a^2 + b^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Always results in a perfect square trinomial<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Expand [latex](3x - 8)^2[\/latex][reveal-answer q=\"503148\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"503148\"]\r\n<ol>\r\n \t<li class=\"whitespace-normal break-words\">Identify [latex]a = 3x[\/latex] and [latex]b = -8[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Apply the formula [latex](a + b)^2 = a^2 + 2ab + b^2[\/latex]: [latex]\\begin{align} (3x - 8)^2 &amp;= (3x)^2 + 2(3x)(-8) + (-8)^2 \\ &amp;= 9x^2 - 48x + 64 \\end{align}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Multiply [latex]\\left(2x+7\\right)\\left(2x - 7\\right)[\/latex].[reveal-answer q=\"951379\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"951379\"][latex]4{x}^{2}-49[\/latex][\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video for more on how to factor a perfect square trinomial.\r\n<script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hadfdffc-Z3ZEPKVMXFI\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Z3ZEPKVMXFI?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hadfdffc-Z3ZEPKVMXFI\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12779050&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-hadfdffc-Z3ZEPKVMXFI&vembed=0&video_id=Z3ZEPKVMXFI&video_target=tpm-plugin-hadfdffc-Z3ZEPKVMXFI'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/How+to+factor+a+perfect+square+trinomial+and+why+is+it+important_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to factor a perfect square trinomial and why is it important\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video for more on differences of squares.\r\n<script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bgafefce-HLNSouzygw0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/HLNSouzygw0?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-bgafefce-HLNSouzygw0\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12843134&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-bgafefce-HLNSouzygw0&vembed=0&video_id=HLNSouzygw0&video_target=tpm-plugin-bgafefce-HLNSouzygw0'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Difference+of+squares+intro+%7C+Mathematics+II+%7C+High+School+Math+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDifference of squares intro | Mathematics II | High School Math | Khan Academy\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Performing Operations with Polynomials of Several Variables<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Extension of Single-Variable Operations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Rules for polynomial operations apply to multi-variable polynomials<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Operations include addition, subtraction, and multiplication<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Like Terms in Multi-Variable Polynomials:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Terms with the same variables raised to the same powers<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Coefficients of like terms can be combined<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Distributive Property:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Key principle in multiplying multi-variable polynomials<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Applied similarly to single-variable polynomials<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Combining Like Terms:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">This is the essential step after applying the distributive property<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Terms with different variables or exponents are not combined<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">[latex]\\left(3x - 1\\right)\\left(2x+7y - 9\\right)[\/latex].[reveal-answer q=\"383366\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"383366\"][latex]6{x}^{2}+21xy - 29x - 7y+9[\/latex][\/hidden-answer]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Recognize polynomial functions, noting their degree and leading coefficient<\/li>\n<li>Add, subtract, and multiply polynomials using different methods, including the FOIL method for two-term polynomials<\/li>\n<li>Work with polynomials that have more than one variable, understanding how to combine and simplify them<\/li>\n<\/ul>\n<\/section>\n<h2>Identifying Polynomial Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>Polynomial functions are like the DNA of algebra\u2014they combine simplicity and complexity to form an incredibly diverse array of functions.<\/p>\n<p>At their core, polynomials are sums of terms made up of coefficients and variables raised to whole number powers. The degree of a polynomial, given by the highest power of the variable, tells us a lot about the function&#8217;s behavior and the shape of its graph.<\/p>\n<p>Let [latex]n[\/latex]\u00a0 be a non-negative integer. A <strong>polynomial function<\/strong> is a function that can be written in the form<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)={a}_{n}{x}^{n}+\\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/p>\n<p>This is called the <strong>general form of a polynomial function<\/strong>.<\/p>\n<p>Each [latex]{a}_{i}[\/latex]\u00a0is a coefficient and can be any real number.<\/p>\n<p>Each product [latex]{a}_{i}{x}^{i}[\/latex]\u00a0is a <strong>term of a polynomial function<\/strong>.<\/p>\n<p><strong>Quick Tips<\/strong><\/p>\n<ul>\n<li><strong>Linear, Quadratic, and Beyond:<\/strong> A first-degree polynomial is linear, a second-degree is quadratic, and higher degrees have their own characteristics and complexities.<\/li>\n<li><strong>Coefficient Clues:<\/strong> The coefficients in a polynomial can tell us about the steepness and direction of the graph. A positive leading coefficient means the graph opens upward, and a negative one indicates it opens downward.<\/li>\n<\/ul>\n<\/div>\n<h2>Defining the Degree and Leading Coefficient of a Polynomial Function<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>The degree of a polynomial is the highest power of the variable present. The leading coefficient is the coefficient of the term with the highest power. These two characteristics can tell us a lot about the function&#8217;s behavior, especially its growth and end behavior.<\/p>\n<ul>\n<li><strong>Degree Tells the Tale:<\/strong> The degree of a polynomial function hints at the number of roots and the possible number of turns on its graph. For instance, a second-degree polynomial, or a quadratic, will have at most two roots and one turn.<\/li>\n<li><strong>Leading Coefficient Impact:<\/strong> The leading coefficient isn&#8217;t just a number\u2014it&#8217;s the boss. It influences the end behavior of the polynomial&#8217;s graph. If it&#8217;s positive, the graph eventually rises; if negative, the graph falls.<\/li>\n<\/ul>\n<p><strong>Quick Tips: Identifying Degree and Leading Coefficient<\/strong><\/p>\n<ul>\n<li>Arrange terms in descending order of power to easily identify the leading term and coefficient.<\/li>\n<li>The degree gives a sense of the &#8216;shape&#8217; of the graph of the polynomial function.<\/li>\n<li>For instance, [latex]f(x)=3+2x^2\u22124x^3[\/latex] has a degree of [latex]3[\/latex] and a leading coefficient of [latex]-4[\/latex], indicating a cubic function with a negative leading term.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">Identify the degree, leading term, and leading coefficient of the polynomial [latex]f\\left(x\\right)=4{x}^{2}-{x}^{6}+2x - 6[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q435637\">Show Solution<\/button><\/p>\n<div id=\"q435637\" class=\"hidden-answer\" style=\"display: none\">The degree is [latex]6[\/latex]. The leading term is [latex]-{x}^{6}[\/latex]. The leading coefficient is [latex]\u20131[\/latex].<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\">Watch the following video for more examples of how to determine the degree, leading term, and leading coefficient of a polynomial.<br \/>\n<script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hghafach-F_G_w82s0QA\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/F_G_w82s0QA?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hghafach-F_G_w82s0QA\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12843132&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-hghafach-F_G_w82s0QA&#38;vembed=0&#38;video_id=F_G_w82s0QA&#38;video_target=tpm-plugin-hghafach-F_G_w82s0QA\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Degree%2C+Leading+Term%2C+and+Leading+Coefficient+of+a+Polynomial+Function_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDegree, Leading Term, and Leading Coefficient of a Polynomial Function\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Adding and Subtracting Polynomials<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Like Terms:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Expressions with the same variables raised to the same powers<\/li>\n<li class=\"whitespace-normal break-words\">Only coefficients are combined; exponents remain unchanged<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Adding Monomials:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Combine coefficients of like terms<\/li>\n<li class=\"whitespace-normal break-words\">Keep the variable and its exponent the same<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Subtracting Monomials:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Similar to addition, but pay attention to signs<\/li>\n<li class=\"whitespace-normal break-words\">Remember that subtracting a negative is the same as adding a positive<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Adding Polynomials:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Combine like terms across all polynomials<\/li>\n<li class=\"whitespace-normal break-words\">Use the Commutative Property to rearrange terms if needed<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Subtracting Polynomials:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Distribute the negative sign to all terms in the subtracted polynomial<\/li>\n<li class=\"whitespace-normal break-words\">Then proceed as with addition<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q660613\">Show Solution<\/button><\/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<p><strong>Analysis of the Solution<\/strong><\/p>\n<p>We can check our answers to these types of problems using a graphing calculator. To check, graph the original 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<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find the sum.[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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q121561\">Show Solution<\/button><\/p>\n<div id=\"q121561\" class=\"hidden-answer\" style=\"display: none\">[latex]2{x}^{3}+7{x}^{2}-4x - 3[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q831247\">Show Solution<\/button><\/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<p><strong>Analysis of the Solution<\/strong><\/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<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find the difference.[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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q260383\">Show Solution<\/button><\/p>\n<div id=\"q260383\" class=\"hidden-answer\" style=\"display: none\">[latex]-11{x}^{3}-{x}^{2}+7x - 9[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch this video to see more examples of adding and subtracting polynomials.<br \/>\n<script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cghchbgf-jiq3toC7wGM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/jiq3toC7wGM?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-cghchbgf-jiq3toC7wGM\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12843133&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-cghchbgf-jiq3toC7wGM&#38;vembed=0&#38;video_id=jiq3toC7wGM&#38;video_target=tpm-plugin-cghchbgf-jiq3toC7wGM\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Adding+and+Subtracting+Polynomials_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Adding and Subtracting Polynomials\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Multiplying Polynomials<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Distributive Property:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Key principle for multiplying polynomials<\/li>\n<li class=\"whitespace-normal break-words\">[latex]a(b + c) = ab + ac[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">General Polynomial Multiplication:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Multiply each term of the first polynomial by every term of the second<\/li>\n<li class=\"whitespace-normal break-words\">Combine like terms in the result<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">FOIL Method:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Specific technique for multiplying two binomials<\/li>\n<li class=\"whitespace-normal break-words\">FOIL stands for First, Outer, Inner, Last<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Multiply the following:<\/p>\n<p style=\"text-align: center;\">[latex](2x + 1)(3x^2 - x + 4)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q516700\">Show Answer<\/button><\/p>\n<div id=\"q516700\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li class=\"whitespace-normal break-words\">Distribute [latex]2x[\/latex] and [latex]1[\/latex]: [latex]2x(3x^2 - x + 4) + 1(3x^2 - x + 4)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Multiply term by term: [latex](6x^3 - 2x^2 + 8x) + (3x^2 - x + 4)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Combine like terms: [latex]6x^3 + (-2x^2 + 3x^2) + (8x - x) + 4[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Simplify: [latex]6x^3 + x^2 + 7x + 4[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Multiply using the FOIL method:<\/p>\n<p style=\"text-align: center;\">[latex](2x - 18)(3x + 3)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q67601\">Show Answer<\/button><\/p>\n<div id=\"q67601\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li class=\"whitespace-normal break-words\">First terms: [latex]2x \\cdot 3x = 6x^2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Outer terms: [latex]2x \\cdot 3 = 6x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Inner terms: [latex]-18 \\cdot 3x = -54x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Last terms: [latex]-18 \\cdot 3 = -54[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Combine all terms: [latex]6x^2 + 6x - 54x - 54[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Simplify: [latex]6x^2 - 48x - 54[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch this video to see more examples of how to use the distributive property to multiply polynomials.<br \/>\n<script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-efgfdfhd-bwTmApTV_8o\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/bwTmApTV_8o?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-efgfdfhd-bwTmApTV_8o\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=6405041&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-efgfdfhd-bwTmApTV_8o&#38;vembed=0&#38;video_id=bwTmApTV_8o&#38;video_target=tpm-plugin-efgfdfhd-bwTmApTV_8o\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Multiplying+Using+the+Distributive+Property_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Multiplying Using the Distributive Property\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Special Cases of Polynomials<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Perfect Square Trinomials:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Result from squaring a binomial<\/li>\n<li class=\"whitespace-normal break-words\">General form: [latex](a + b)^2 = a^2 + 2ab + b^2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Also applies to subtraction: [latex](a - b)^2 = a^2 - 2ab + b^2[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Characteristics of Perfect Square Trinomials:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">First and last terms are perfect squares<\/li>\n<li class=\"whitespace-normal break-words\">Middle term is twice the product of the terms in the original binomial<\/li>\n<li class=\"whitespace-normal break-words\">Sign of the middle term matches the sign in the original binomial<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Difference of Squares:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Result from multiplying sum and difference of the same terms<\/li>\n<li class=\"whitespace-normal break-words\">General form: [latex](a + b)(a - b) = a^2 - b^2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Middle terms cancel out<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">No Special Form for Sum of Squares:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex](a + b)^2 \\neq a^2 + b^2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Always results in a perfect square trinomial<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Expand [latex](3x - 8)^2[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q503148\">Show Answer<\/button><\/p>\n<div id=\"q503148\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li class=\"whitespace-normal break-words\">Identify [latex]a = 3x[\/latex] and [latex]b = -8[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Apply the formula [latex](a + b)^2 = a^2 + 2ab + b^2[\/latex]: [latex]\\begin{align} (3x - 8)^2 &= (3x)^2 + 2(3x)(-8) + (-8)^2 \\ &= 9x^2 - 48x + 64 \\end{align}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Multiply [latex]\\left(2x+7\\right)\\left(2x - 7\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q951379\">Show Solution<\/button><\/p>\n<div id=\"q951379\" class=\"hidden-answer\" style=\"display: none\">[latex]4{x}^{2}-49[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video for more on how to factor a perfect square trinomial.<br \/>\n<script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hadfdffc-Z3ZEPKVMXFI\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Z3ZEPKVMXFI?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hadfdffc-Z3ZEPKVMXFI\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12779050&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-hadfdffc-Z3ZEPKVMXFI&#38;vembed=0&#38;video_id=Z3ZEPKVMXFI&#38;video_target=tpm-plugin-hadfdffc-Z3ZEPKVMXFI\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/How+to+factor+a+perfect+square+trinomial+and+why+is+it+important_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to factor a perfect square trinomial and why is it important\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video for more on differences of squares.<br \/>\n<script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bgafefce-HLNSouzygw0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/HLNSouzygw0?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-bgafefce-HLNSouzygw0\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12843134&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-bgafefce-HLNSouzygw0&#38;vembed=0&#38;video_id=HLNSouzygw0&#38;video_target=tpm-plugin-bgafefce-HLNSouzygw0\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Difference+of+squares+intro+%7C+Mathematics+II+%7C+High+School+Math+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDifference of squares intro | Mathematics II | High School Math | Khan Academy\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Performing Operations with Polynomials of Several Variables<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Extension of Single-Variable Operations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Rules for polynomial operations apply to multi-variable polynomials<\/li>\n<li class=\"whitespace-normal break-words\">Operations include addition, subtraction, and multiplication<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Like Terms in Multi-Variable Polynomials:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Terms with the same variables raised to the same powers<\/li>\n<li class=\"whitespace-normal break-words\">Coefficients of like terms can be combined<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Distributive Property:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Key principle in multiplying multi-variable polynomials<\/li>\n<li class=\"whitespace-normal break-words\">Applied similarly to single-variable polynomials<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Combining Like Terms:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">This is the essential step after applying the distributive property<\/li>\n<li class=\"whitespace-normal break-words\">Terms with different variables or exponents are not combined<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">[latex]\\left(3x - 1\\right)\\left(2x+7y - 9\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q383366\">Show Solution<\/button><\/p>\n<div id=\"q383366\" class=\"hidden-answer\" style=\"display: none\">[latex]6{x}^{2}+21xy - 29x - 7y+9[\/latex]<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":12,"menu_order":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Degree, Leading Term, and Leading Coefficient of a Polynomial Function \",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/www.youtube.com\/watch?v=F_G_w82s0QA\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex: Adding and Subtracting Polynomials 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