{"id":1497,"date":"2025-07-25T02:14:52","date_gmt":"2025-07-25T02:14:52","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1497"},"modified":"2026-03-24T07:07:56","modified_gmt":"2026-03-24T07:07:56","slug":"binomial-theorem-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/binomial-theorem-fresh-take\/","title":{"raw":"Binomial Theorem: Fresh Take","rendered":"Binomial Theorem: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Use the Binomial Theorem to expand a binomial.<\/li>\r\n \t<li>Use the Binomial Theorem to find a specified term of a binomial expansion.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Binomial Coefficients<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Binomial Definition<\/strong>: A polynomial with two terms is called a binomial.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Binomial Theorem<\/strong>: A powerful tool for expanding expressions raised to a power, useful in combination problems.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Binomial Coefficients<\/strong>: The coefficients in the expansion of [latex](x+y)^n[\/latex], following a pattern visible in Pascal's Triangle.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Binomial Coefficient Formula<\/strong>: For integers [latex]n \\geq r \\geq 0[\/latex], [latex]\\binom{n}{r} = C(n,r) = \\frac{n!}{r!(n-r)!}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Symmetry Property<\/strong>: [latex]\\binom{n}{r} = \\binom{n}{n-r}[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Find each binomial coefficient.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]\\left(\\begin{gathered}7\\\\ 3\\end{gathered}\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\left(\\begin{gathered}11\\\\ 4\\end{gathered}\\right)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"808348\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"808348\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>35<\/li>\r\n \t<li>330<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2>The Binomial Theorem<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Binomial Expansion<\/strong>: The result of expanding [latex](x+y)^n[\/latex].<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Pascal's Triangle<\/strong>: A triangular array of binomial coefficients that follows a simple rule of construction.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Binomial Theorem<\/strong>: A formula for expanding any binomial to any power without performing repeated multiplication.<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Patterns in Binomial Expansions<\/strong><\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">When expanding [latex](x+y)^n[\/latex]:<\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">There are [latex]n+1[\/latex] terms in the expansion.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The degree (sum of exponents) for each term is [latex]n[\/latex].<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Powers of [latex]x[\/latex] decrease from [latex]n[\/latex] to [latex]0[\/latex].<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Powers of [latex]y[\/latex] increase from [latex]0[\/latex] to [latex]n[\/latex].<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Coefficients are symmetric.<\/li>\r\n<\/ol>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Pascal's Triangle and Binomial Coefficients<\/strong><\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Pascal's Triangle is formed by:<\/p>\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Starting with 1 at the top<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Each number below is the sum of the two numbers above it<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Rows are symmetric<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-pre-wrap break-words\">The numbers in Pascal's Triangle correspond to the coefficients in binomial expansions.<\/p>\r\n<p class=\"font-600 text-xl font-bold\"><strong>The Binomial Theorem<\/strong><\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[latex](x+y)^n = \\sum_{k=0}^n \\binom{n}{k} x^{n-k} y^k[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Or written out:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[latex](x+y)^n = \\binom{n}{0}x^n + \\binom{n}{1}x^{n-1}y + \\binom{n}{2}x^{n-2}y^2 + ... + \\binom{n}{n-1}xy^{n-1} + \\binom{n}{n}y^n[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Where [latex]\\binom{n}{k}[\/latex] is the binomial coefficient.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Write in expanded form.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]{\\left(x-y\\right)}^{5}[\/latex]<\/li>\r\n \t<li>[latex]{\\left(2x+5y\\right)}^{3}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"459909\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"459909\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]{x}^{5}-5{x}^{4}y+10{x}^{3}{y}^{2}-10{x}^{2}{y}^{3}+5x{y}^{4}-{y}^{5}[\/latex]<\/li>\r\n \t<li>[latex]8{x}^{3}+60{x}^{2}y+150x{y}^{2}+125{y}^{3}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cfbfegda-0l9hJC3yKW0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/0l9hJC3yKW0?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-cfbfegda-0l9hJC3yKW0\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851413&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-cfbfegda-0l9hJC3yKW0&amp;vembed=0&amp;video_id=0l9hJC3yKW0&amp;video_target=tpm-plugin-cfbfegda-0l9hJC3yKW0\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+2+-+The+Binomial+Theorem+Using+Pascal's+Triangle_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: The Binomial Theorem Using Pascal's Triangle\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-begbggge-YxysKtqpbVI\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/YxysKtqpbVI?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-begbggge-YxysKtqpbVI\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851414&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-begbggge-YxysKtqpbVI&amp;vembed=0&amp;video_id=YxysKtqpbVI&amp;video_target=tpm-plugin-begbggge-YxysKtqpbVI\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/The+Binomial+Theorem+using+Combination_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cThe Binomial Theorem using Combination\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dcheebgf-2ZNmLHMZBRI\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/2ZNmLHMZBRI?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-dcheebgf-2ZNmLHMZBRI\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851415&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-dcheebgf-2ZNmLHMZBRI&amp;vembed=0&amp;video_id=2ZNmLHMZBRI&amp;video_target=tpm-plugin-dcheebgf-2ZNmLHMZBRI\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+1+-+The+Binomial+Theorem+Using+Combinations_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: The Binomial Theorem Using Combinations\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Using the Binomial Theorem to Find a Single Term<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n\r\nThe Binomial Theorem can be used to find a specific term in a binomial expansion without expanding the entire expression. This is particularly useful for binomials with high exponents.\r\n<p class=\"font-600 text-xl font-bold\"><strong>Key Formula<\/strong><\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">The [latex](r+1)[\/latex]th term of the binomial expansion of [latex](x+y)^n[\/latex]\u00a0is:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[latex]\\binom{n}{r} x^{n-r} y^r[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Where:<\/p>\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]n[\/latex] is the exponent of the binomial<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]r[\/latex] is one less than the position of the term we're looking for<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>How to Find a Specific Term<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify the exponent [latex]n[\/latex] of the binomial.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Determine which term you're looking for (let's call it the [latex]k[\/latex]th term).<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate [latex]r = k - 1[\/latex].<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Apply the formula: [latex]\\binom{n}{r} x^{n-r} y^r[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Find the sixth term of [latex]{\\left(3x-y\\right)}^{9}[\/latex] without fully expanding the binomial.[reveal-answer q=\"96932\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"96932\"][latex]-10\\text{,}206{x}^{4}{y}^{5}[\/latex][\/hidden-answer]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Use the Binomial Theorem to expand a binomial.<\/li>\n<li>Use the Binomial Theorem to find a specified term of a binomial expansion.<\/li>\n<\/ul>\n<\/section>\n<h2>Binomial Coefficients<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\"><strong>Binomial Definition<\/strong>: A polynomial with two terms is called a binomial.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Binomial Theorem<\/strong>: A powerful tool for expanding expressions raised to a power, useful in combination problems.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Binomial Coefficients<\/strong>: The coefficients in the expansion of [latex](x+y)^n[\/latex], following a pattern visible in Pascal&#8217;s Triangle.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Binomial Coefficient Formula<\/strong>: For integers [latex]n \\geq r \\geq 0[\/latex], [latex]\\binom{n}{r} = C(n,r) = \\frac{n!}{r!(n-r)!}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Symmetry Property<\/strong>: [latex]\\binom{n}{r} = \\binom{n}{n-r}[\/latex]<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Find each binomial coefficient.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\left(\\begin{gathered}7\\\\ 3\\end{gathered}\\right)[\/latex]<\/li>\n<li>[latex]\\left(\\begin{gathered}11\\\\ 4\\end{gathered}\\right)[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q808348\">Show Solution<\/button><\/p>\n<div id=\"q808348\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>35<\/li>\n<li>330<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<h2>The Binomial Theorem<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\"><strong>Binomial Expansion<\/strong>: The result of expanding [latex](x+y)^n[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Pascal&#8217;s Triangle<\/strong>: A triangular array of binomial coefficients that follows a simple rule of construction.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Binomial Theorem<\/strong>: A formula for expanding any binomial to any power without performing repeated multiplication.<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Patterns in Binomial Expansions<\/strong><\/p>\n<p class=\"whitespace-pre-wrap break-words\">When expanding [latex](x+y)^n[\/latex]:<\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">There are [latex]n+1[\/latex] terms in the expansion.<\/li>\n<li class=\"whitespace-normal break-words\">The degree (sum of exponents) for each term is [latex]n[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">Powers of [latex]x[\/latex] decrease from [latex]n[\/latex] to [latex]0[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">Powers of [latex]y[\/latex] increase from [latex]0[\/latex] to [latex]n[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">Coefficients are symmetric.<\/li>\n<\/ol>\n<p class=\"font-600 text-xl font-bold\"><strong>Pascal&#8217;s Triangle and Binomial Coefficients<\/strong><\/p>\n<p class=\"whitespace-pre-wrap break-words\">Pascal&#8217;s Triangle is formed by:<\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Starting with 1 at the top<\/li>\n<li class=\"whitespace-normal break-words\">Each number below is the sum of the two numbers above it<\/li>\n<li class=\"whitespace-normal break-words\">Rows are symmetric<\/li>\n<\/ul>\n<p class=\"whitespace-pre-wrap break-words\">The numbers in Pascal&#8217;s Triangle correspond to the coefficients in binomial expansions.<\/p>\n<p class=\"font-600 text-xl font-bold\"><strong>The Binomial Theorem<\/strong><\/p>\n<p class=\"whitespace-pre-wrap break-words\">[latex](x+y)^n = \\sum_{k=0}^n \\binom{n}{k} x^{n-k} y^k[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Or written out:<\/p>\n<p class=\"whitespace-pre-wrap break-words\">[latex](x+y)^n = \\binom{n}{0}x^n + \\binom{n}{1}x^{n-1}y + \\binom{n}{2}x^{n-2}y^2 + ... + \\binom{n}{n-1}xy^{n-1} + \\binom{n}{n}y^n[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Where [latex]\\binom{n}{k}[\/latex] is the binomial coefficient.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Write in expanded form.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]{\\left(x-y\\right)}^{5}[\/latex]<\/li>\n<li>[latex]{\\left(2x+5y\\right)}^{3}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q459909\">Show Solution<\/button><\/p>\n<div id=\"q459909\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]{x}^{5}-5{x}^{4}y+10{x}^{3}{y}^{2}-10{x}^{2}{y}^{3}+5x{y}^{4}-{y}^{5}[\/latex]<\/li>\n<li>[latex]8{x}^{3}+60{x}^{2}y+150x{y}^{2}+125{y}^{3}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cfbfegda-0l9hJC3yKW0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/0l9hJC3yKW0?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-cfbfegda-0l9hJC3yKW0\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851413&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-cfbfegda-0l9hJC3yKW0&amp;vembed=0&amp;video_id=0l9hJC3yKW0&amp;video_target=tpm-plugin-cfbfegda-0l9hJC3yKW0\" type=\"text\/javascript\"><\/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+2+-+The+Binomial+Theorem+Using+Pascal's+Triangle_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: The Binomial Theorem Using Pascal&#8217;s Triangle\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-begbggge-YxysKtqpbVI\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/YxysKtqpbVI?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-begbggge-YxysKtqpbVI\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851414&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-begbggge-YxysKtqpbVI&amp;vembed=0&amp;video_id=YxysKtqpbVI&amp;video_target=tpm-plugin-begbggge-YxysKtqpbVI\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/The+Binomial+Theorem+using+Combination_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cThe Binomial Theorem using Combination\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dcheebgf-2ZNmLHMZBRI\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/2ZNmLHMZBRI?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-dcheebgf-2ZNmLHMZBRI\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851415&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-dcheebgf-2ZNmLHMZBRI&amp;vembed=0&amp;video_id=2ZNmLHMZBRI&amp;video_target=tpm-plugin-dcheebgf-2ZNmLHMZBRI\" type=\"text\/javascript\"><\/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+1+-+The+Binomial+Theorem+Using+Combinations_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: The Binomial Theorem Using Combinations\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Using the Binomial Theorem to Find a Single Term<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p>The Binomial Theorem can be used to find a specific term in a binomial expansion without expanding the entire expression. This is particularly useful for binomials with high exponents.<\/p>\n<p class=\"font-600 text-xl font-bold\"><strong>Key Formula<\/strong><\/p>\n<p class=\"whitespace-pre-wrap break-words\">The [latex](r+1)[\/latex]th term of the binomial expansion of [latex](x+y)^n[\/latex]\u00a0is:<\/p>\n<p class=\"whitespace-pre-wrap break-words\">[latex]\\binom{n}{r} x^{n-r} y^r[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Where:<\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]n[\/latex] is the exponent of the binomial<\/li>\n<li class=\"whitespace-normal break-words\">[latex]r[\/latex] is one less than the position of the term we&#8217;re looking for<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>How to Find a Specific Term<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify the exponent [latex]n[\/latex] of the binomial.<\/li>\n<li class=\"whitespace-normal break-words\">Determine which term you&#8217;re looking for (let&#8217;s call it the [latex]k[\/latex]th term).<\/li>\n<li class=\"whitespace-normal break-words\">Calculate [latex]r = k - 1[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">Apply the formula: [latex]\\binom{n}{r} x^{n-r} y^r[\/latex].<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Find the sixth term of [latex]{\\left(3x-y\\right)}^{9}[\/latex] without fully expanding the binomial.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q96932\">Show Solution<\/button><\/p>\n<div id=\"q96932\" class=\"hidden-answer\" style=\"display: none\">[latex]-10\\text{,}206{x}^{4}{y}^{5}[\/latex]<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":67,"menu_order":17,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Ex 2: The Binomial 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