{"id":191315,"date":"2025-02-14T17:38:25","date_gmt":"2025-02-14T17:38:25","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=191315"},"modified":"2025-02-14T17:38:29","modified_gmt":"2025-02-14T17:38:29","slug":"alexa-uses-the-binomial-theorem-to-correctly-expand-x24","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/14\/alexa-uses-the-binomial-theorem-to-correctly-expand-x24\/","title":{"rendered":"Alexa uses the binomial theorem to correctly expand (x+2)^(4)"},"content":{"rendered":"\n<p>Alexa uses the binomial theorem to correctly expand (x+2)^(4). Which is one of the terms in Alexa&#8217;s answer? 2x^(4) 16 x 24x^(2) 6x^(2)<\/p>\n\n\n\n<p>2.S2 Integrals:<br>(a)&nbsp;<img loading=\"lazy\" decoding=\"async\" width=\"82\" height=\"21\" src=\"https:\/\/files.transtutors.com\/book\/qimg\/cbbcfb78-9dad-4632-9985-1075615b8173.png\"><br>b)&nbsp;<img loading=\"lazy\" decoding=\"async\" width=\"91\" height=\"21\" src=\"https:\/\/files.transtutors.com\/book\/qimg\/29302910-b1eb-4abb-a561-e7a6ca6bc01d.png\"><br>(c)&nbsp;<img loading=\"lazy\" decoding=\"async\" width=\"90\" height=\"21\" src=\"https:\/\/files.transtutors.com\/book\/qimg\/4779e35f-f14a-4e59-b5f8-b297fb127d02.png\"><br>(d)&nbsp;<img loading=\"lazy\" decoding=\"async\" width=\"116\" height=\"27\" src=\"https:\/\/files.transtutors.com\/book\/qimg\/d2790beb-f8cb-4526-83fb-3d09d33f4aca.png\"><\/p>\n\n\n\n<p>3.Solve for the unknowns in the vector equation below.<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/files.transtutors.com\/book\/qimg\/0c6f7d71-cc21-4b18-8400-3cda7bebc530.png\" alt=\"\"\/><\/figure>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\">The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>Let&#8217;s solve the first question using the <strong>binomial theorem<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Expanding (x + 2)\u2074 using the Binomial Theorem:<\/strong><\/h3>\n\n\n\n<p>The <strong>binomial theorem<\/strong> states that: (a+b)n=\u2211k=0n(nk)a(n\u2212k)bk(a + b)^n = \\sum_{k=0}^{n} \\binom{n}{k} a^{(n-k)} b^k<\/p>\n\n\n\n<p>For <strong>(x + 2)\u2074<\/strong>, we set:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>a=xa = x<\/li>\n\n\n\n<li>b=2b = 2<\/li>\n\n\n\n<li>n=4n = 4<\/li>\n<\/ul>\n\n\n\n<p>Expanding using binomial coefficients: (x+2)4=\u2211k=04(4k)x(4\u2212k)2k(x + 2)^4 = \\sum_{k=0}^{4} \\binom{4}{k} x^{(4-k)} 2^k<\/p>\n\n\n\n<p>We compute the binomial coefficients:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>(40)=1\\binom{4}{0} = 1, term: 1\u22c5x4\u22c520=x41 \\cdot x^4 \\cdot 2^0 = x^4<\/li>\n\n\n\n<li>(41)=4\\binom{4}{1} = 4, term: 4\u22c5x3\u22c521=8&#215;34 \\cdot x^3 \\cdot 2^1 = 8x^3<\/li>\n\n\n\n<li>(42)=6\\binom{4}{2} = 6, term: 6\u22c5x2\u22c522=24&#215;26 \\cdot x^2 \\cdot 2^2 = 24x^2<\/li>\n\n\n\n<li>(43)=4\\binom{4}{3} = 4, term: 4\u22c5x1\u22c523=32&#215;4 \\cdot x^1 \\cdot 2^3 = 32x<\/li>\n\n\n\n<li>(44)=1\\binom{4}{4} = 1, term: 1\u22c5x0\u22c524=161 \\cdot x^0 \\cdot 2^4 = 16<\/li>\n<\/ul>\n\n\n\n<p>Thus, the full expansion is: x4+8&#215;3+24&#215;2+32x+16x^4 + 8x^3 + 24x^2 + 32x + 16<\/p>\n\n\n\n<p>One of the terms in the expansion is <strong>24x\u00b2<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Answer:<\/strong><\/h3>\n\n\n\n<p><strong>24x\u00b2<\/strong> is one of the terms.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Understanding the Binomial Theorem<\/strong><\/h3>\n\n\n\n<p>The <strong>binomial theorem<\/strong> is a fundamental concept in algebra that allows us to expand expressions raised to a power. Instead of multiplying step by step, we use combinations to find coefficients for each term.<\/p>\n\n\n\n<p>The coefficients in the expansion follow <strong>Pascal\u2019s Triangle<\/strong>, where each coefficient is determined by the formula: (nk)=n!k!(n\u2212k)!\\binom{n}{k} = \\frac{n!}{k!(n-k)!}<\/p>\n\n\n\n<p>This method is particularly useful in algebra, probability, and calculus.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p>Now, I&#8217;ll generate an image illustrating the binomial expansion for better understanding.<\/p>\n\n\n\n<p>I have provided an image illustrating the binomial expansion of (x+2)4(x+2)^4, including Pascal&#8217;s Triangle and a breakdown of each term. Let me know if you need further explanations or modifications!<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/02\/image-535.png\" alt=\"\" class=\"wp-image-191316\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Alexa uses the binomial theorem to correctly expand (x+2)^(4). Which is one of the terms in Alexa&#8217;s answer? 2x^(4) 16 x 24x^(2) 6x^(2) 2.S2 Integrals:(a)&nbsp;b)&nbsp;(c)&nbsp;(d)&nbsp; 3.Solve for the unknowns in the vector equation below. The correct answer and explanation is: Let&#8217;s solve the first question using the binomial theorem. Expanding (x + 2)\u2074 using the [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-191315","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/191315","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=191315"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/191315\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=191315"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=191315"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=191315"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}