{"id":3212,"date":"2025-08-15T23:22:54","date_gmt":"2025-08-15T23:22:54","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3212"},"modified":"2025-10-07T21:32:05","modified_gmt":"2025-10-07T21:32:05","slug":"binomial-theorem-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/binomial-theorem-apply-it\/","title":{"raw":"Binomial Theorem: Apply It","rendered":"Binomial Theorem: Apply It"},"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<p class=\"whitespace-normal break-words\">The Binomial Theorem provides a powerful shortcut for expanding expressions like [latex](x + y)^n[\/latex] without multiplying them out repeatedly. In this page, you'll apply the Binomial Theorem to expand binomials and find specific terms within expansions efficiently.<\/p>\r\n\r\n<section class=\"textbox interact\" aria-label=\"Interact\">The coefficients [latex]C\\left(n,r\\right)[\/latex] can be calculated using technology.\r\n\r\n<strong>For TI-84:<\/strong>\r\n<ol>\r\n \t<li>On the main screen, type the value for [latex]n[\/latex] (e.g., [latex]6[\/latex]).<\/li>\r\n \t<li>Press MATH.<\/li>\r\n \t<li>Arrow right to PRB.<\/li>\r\n \t<li>Choose 3:nCr and press ENTER.<\/li>\r\n \t<li>Type the value for [latex]r[\/latex] (e.g., [latex]2[\/latex]).<\/li>\r\n \t<li>Press ENTER to compute [latex]\\binom{n}{r}[\/latex].<\/li>\r\n<\/ol>\r\n<strong>For Desmos:<\/strong>\r\n<ul>\r\n \t<li>Type the command nCr() and input numerical values for n, r. For example [latex]C\\left(5,4\\right)[\/latex] can be typed as nCr(5,3)<\/li>\r\n<\/ul>\r\n<\/section>Let's expand an example with a lot of terms. How long do you think this would take without the binomial expansion theorem?\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]313352[\/ohm_question]<\/section>\r\n<div class=\"flex-1 flex flex-col gap-3 px-4 max-w-3xl mx-auto w-full pt-1\">\r\n<div data-test-render-count=\"1\">\r\n<div class=\"group relative pb-3\" data-is-streaming=\"false\">\r\n<div class=\"font-claude-response relative leading-[1.65rem] [&amp;_pre&gt;div]:bg-bg-000\/50 [&amp;_pre&gt;div]:border-0.5 [&amp;_pre&gt;div]:border-border-400 [&amp;_.ignore-pre-bg&gt;div]:bg-transparent [&amp;_.standard-markdown_:is(p,blockquote,h1,h2,h3,h4,h5,h6)]:pl-2 [&amp;_.standard-markdown_:is(p,blockquote,ul,ol,h1,h2,h3,h4,h5,h6)]:pr-8 [&amp;_.progressive-markdown_:is(p,blockquote,h1,h2,h3,h4,h5,h6)]:pl-2 [&amp;_.progressive-markdown_:is(p,blockquote,ul,ol,h1,h2,h3,h4,h5,h6)]:pr-8\">\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 standard-markdown\"><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","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<p class=\"whitespace-normal break-words\">The Binomial Theorem provides a powerful shortcut for expanding expressions like [latex](x + y)^n[\/latex] without multiplying them out repeatedly. In this page, you&#8217;ll apply the Binomial Theorem to expand binomials and find specific terms within expansions efficiently.<\/p>\n<section class=\"textbox interact\" aria-label=\"Interact\">The coefficients [latex]C\\left(n,r\\right)[\/latex] can be calculated using technology.<\/p>\n<p><strong>For TI-84:<\/strong><\/p>\n<ol>\n<li>On the main screen, type the value for [latex]n[\/latex] (e.g., [latex]6[\/latex]).<\/li>\n<li>Press MATH.<\/li>\n<li>Arrow right to PRB.<\/li>\n<li>Choose 3:nCr and press ENTER.<\/li>\n<li>Type the value for [latex]r[\/latex] (e.g., [latex]2[\/latex]).<\/li>\n<li>Press ENTER to compute [latex]\\binom{n}{r}[\/latex].<\/li>\n<\/ol>\n<p><strong>For Desmos:<\/strong><\/p>\n<ul>\n<li>Type the command nCr() and input numerical values for n, r. For example [latex]C\\left(5,4\\right)[\/latex] can be typed as nCr(5,3)<\/li>\n<\/ul>\n<\/section>\n<p>Let&#8217;s expand an example with a lot of terms. How long do you think this would take without the binomial expansion theorem?<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm313352\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313352&theme=lumen&iframe_resize_id=ohm313352&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<div class=\"flex-1 flex flex-col gap-3 px-4 max-w-3xl mx-auto w-full pt-1\">\n<div data-test-render-count=\"1\">\n<div class=\"group relative pb-3\" data-is-streaming=\"false\">\n<div class=\"font-claude-response relative leading-[1.65rem] [&amp;_pre&gt;div]:bg-bg-000\/50 [&amp;_pre&gt;div]:border-0.5 [&amp;_pre&gt;div]:border-border-400 [&amp;_.ignore-pre-bg&gt;div]:bg-transparent [&amp;_.standard-markdown_:is(p,blockquote,h1,h2,h3,h4,h5,h6)]:pl-2 [&amp;_.standard-markdown_:is(p,blockquote,ul,ol,h1,h2,h3,h4,h5,h6)]:pr-8 [&amp;_.progressive-markdown_:is(p,blockquote,h1,h2,h3,h4,h5,h6)]:pl-2 [&amp;_.progressive-markdown_:is(p,blockquote,ul,ol,h1,h2,h3,h4,h5,h6)]:pr-8\">\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 standard-markdown\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":67,"menu_order":16,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":513,"module-header":"apply_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3212"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":6,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3212\/revisions"}],"predecessor-version":[{"id":4539,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3212\/revisions\/4539"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/513"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3212\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=3212"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=3212"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=3212"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=3212"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}