{"id":89,"date":"2017-08-23T07:45:46","date_gmt":"2017-08-23T07:45:46","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=89"},"modified":"2017-09-12T10:44:41","modified_gmt":"2017-09-12T10:44:41","slug":"binomial-expansion","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/binomial-expansion\/","title":{"rendered":"Binomial Expansion"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/solving-and-graphing-quadratic-equations\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/algebra-functions-ii\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/conic-sections\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Binomial Expansion<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, you will study expanding binomials to any power using Pascal\u2019s triangle and the Binomial Theorem.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li>There are numerous ways to <em><strong>solve\u00a0quadratic equations<\/strong><\/em>:\n<ul>\n<li>graphing,<\/li>\n<li>factoring,<\/li>\n<li>completing the square, and<\/li>\n<li>the quadratic formula.<\/li>\n<\/ul>\n<\/li>\n<li>There are two forms of a\u00a0quadratic equation: <em><strong>standard form<\/strong><\/em> and\u00a0<em><strong>vertex form<\/strong><\/em>. The vertex form is used to\u00a0easily identify the coordinate location of the <strong><em>vertex<\/em><\/strong>.<\/li>\n<li>The real-number solution to a\u00a0quadratic is the point or points where <em>f<\/em>(<em>x<\/em>) = 0.<\/li>\n<li><strong><em>Roots<\/em><\/strong> of quadratic equations can\u00a0be either real or complex.<\/li>\n<\/ul>\n<section>\n<h3>Binomial Expansion<\/h3>\n<p>In our previous discussion, we combined two binomials to\u00a0produce a perfect square trinomial.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s5_p2_clip_image003.gif\" width=\"234\" height=\"17\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The process of raising a binomial to a power, and deriving the\u00a0polynomial is called binomial expansion. The binomials are of the\u00a0form\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s5_p2_clip_image006.gif\" width=\"47\" height=\"15\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>It is not practical to manually expand binomials. There are\u00a0just too many steps involved, and the chance of an error is too\u00a0great to undertake this by hand (not to mention the time\u00a0involved). Fortunately, there are methods available to help us\u00a0bypass the tedious calculations and more quickly expand binomials\u00a0of a higher degree.<\/p>\n<h3>Pascal\u2019s Triangle<\/h3>\n<p>A very simple and practical way to expand binomials is to use a\u00a0diagram called Pascal\u2019s Triangle.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s5_p2_clip_image009.gif\" width=\"407\" height=\"168\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>This is a diagram of the coefficients of the expansion. Notice\u00a0the pattern in the triangle. Each number is the sum of the two\u00a0numbers above it. For example, the central number in the\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s5_p2_clip_image012.gif\" width=\"47\" height=\"17\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/>\u00a0row is 20. This is derived from 10 + 10, the two numbers it is\u00a0centered below.<\/p>\n<p>Using this process, we can build Pascal\u2019s triangle and\u00a0use it to expand binomials to any degree.<\/p>\n<p>For example, expand\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s5_p2_clip_image015.gif\" width=\"46\" height=\"17\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p><strong>Step 1:<\/strong> From Pascal\u2019s triangle, we see\u00a0that the coefficients for a 5 th degree binomial are 1, 5, 10, 10,\u00a05, and 1.<\/p>\n<p><strong>Step 2:<\/strong> We build the polynomial by first\u00a0building each term. <span style=\"text-decoration: none;\">The\u00a0exponent on the variable <\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">starts with 5, and decreases by one each time. The exponent on the\u00a0<\/span><em><span style=\"text-decoration: none;\">a<\/span><\/em><span style=\"text-decoration: none;\">-term\u00a0starts at zero, and increases by one each time.<\/span><\/p>\n<p>1st term:\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s5_p2_clip_image018.gif\" width=\"94\" height=\"17\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<p>2nd term:\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s5_p2_clip_image021.gif\" width=\"120\" height=\"17\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<p>3 rd term:\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s5_p2_clip_image024.gif\" width=\"119\" height=\"17\" name=\"graphics10\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<p>4 th term:\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s5_p2_clip_image027.gif\" width=\"129\" height=\"17\" name=\"graphics11\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<p>5 th term:\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s5_p2_clip_image030.gif\" width=\"105\" height=\"17\" name=\"graphics12\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<p>6 th term:\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s5_p2_clip_image033.gif\" width=\"105\" height=\"17\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<p><strong>Step 3:<\/strong> Combine all terms, and write the final\u00a0answer in standard form.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s5_p2_clip_image036.gif\" width=\"285\" height=\"17\" name=\"graphics14\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<h3>The Binomial Theorem<\/h3>\n<p><span style=\"text-decoration: none;\">The Binomial Theorem is a\u00a0more formal approach to binomial expansion. The Binomial Theorem<br \/>\nstates that for positive integers <\/span><em><span style=\"text-decoration: none;\">n,<br \/>\n<\/span><\/em><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s5_p3_clip_image003.gif\" width=\"415\" height=\"19\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p>The coefficients <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s5_p3_clip_image006.gif\" width=\"22\" height=\"16\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/> are combinations, and are calculated using the formula:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s5_p3_clip_image009.gif\" width=\"103\" height=\"42\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<\/div>\n<p>For example, expand\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s5_p3_clip_image012.gif\" width=\"46\" height=\"17\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p><strong>Step 1:<\/strong> Using the Binomial Theorem, we will\u00a0first calculate the coefficients for each term.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s5_p3_clip_image015.gif\" width=\"132\" height=\"294\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p>Notice how the coefficients are identical to the 5th<br \/>\ndegree binomials in Pascal\u2019s triangle.<\/p>\n<\/div>\n<p><strong>Step 2:<\/strong> <span style=\"text-decoration: none;\">Write\u00a0the binomial expansion using <\/span><em><span style=\"text-decoration: none;\">n\u00a0<\/span><\/em><span style=\"text-decoration: none;\">= 5 and <\/span><em><span style=\"text-decoration: none;\">a\u00a0<\/span><\/em><span style=\"text-decoration: none;\">= \u20132. <\/span><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/3\/images\/s5_p3_clip_image018.gif\" width=\"500\" height=\"69\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/solving-and-graphing-quadratic-equations\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/algebra-functions-ii\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/conic-sections\">Next Lesson \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Lesson \u27a1 Binomial Expansion Objective In this lesson, you will study expanding binomials to any power using Pascal\u2019s triangle and the Binomial Theorem. Previously Covered: There are numerous ways to solve\u00a0quadratic equations: graphing, factoring, completing the square, and the quadratic formula. There are two forms of a\u00a0quadratic equation: standard form and\u00a0vertex [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-89","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/89","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=89"}],"version-history":[{"count":8,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/89\/revisions"}],"predecessor-version":[{"id":678,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/89\/revisions\/678"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=89"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}