{"id":571,"date":"2019-04-29T14:47:32","date_gmt":"2019-04-29T18:47:32","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/?post_type=chapter&p=571"},"modified":"2019-12-29T00:37:59","modified_gmt":"2019-12-29T05:37:59","slug":"6-9-pascals-triangle-and-binomial-expansion","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/chapter\/6-9-pascals-triangle-and-binomial-expansion\/","title":{"raw":"6.9 Pascal's Triangle and Binomial Expansion","rendered":"6.9 Pascal’s Triangle and Binomial Expansion"},"content":{"raw":"[latexpage]\r\n\r\nPascal\u2019s triangle (1653) has been found in the works of mathematicians dating back before the 2nd century BC. While Pascal's triangle is useful in many different mathematical settings, it will be applied to the expansion of binomials. In this application, Pascal\u2019s triangle will generate the leading coefficient of each term of a binomial expansion in the form of:\r\n

\\((a+b)^n\\)<\/p>\r\n\"\"<\/span>\r\n\r\n\"Yang\r\n\r\nFor example:\r\n

\\(\\begin{array}{llllllllllllllll}\r\n(a&+&b)^2&=&a^2&+&2ab&+&b^2\\hspace{0.25in}&(1&+&2&+&1)&& \\\\\r\n(a&+&b)^3&=&a^3&+&3a^2b&+&b^3\\hspace{0.25in}&(1&+&3&+&3&+&1)\r\n\\end{array}\\)<\/p>\r\n\r\n

Pascal's Triangle<\/h1>\r\n

\\(\\begin{array}{lclcl}\r\n(a+b)^0&1&2^0&1&(a-b)^0 \\\\\r\n(a+b)^1&1+1&2^1&1-1&(a-b)^1 \\\\\r\n(a+b)^2&1+2+1&2^2&1-2+1&(a-b)^2 \\\\\r\n(a+b)^3&1+3+3+1&2^3&1-3+3-1&(a-b)^3 \\\\\r\n(a+b)^4&1+4+6+4+1&2^4&1-4+6-4+1&(a-b)^4 \\\\\r\n(a+b)^5&1+5+10+10+5+1&2^5&1-5+10-10+5-1&(a-b)^5 \\\\\r\n(a+b)^6&1+6+15+20+15+6+1&2^6&1-6+15-20+15-6+1&(a-b)^6 \\\\\r\n(a+b)^7&1+7+21+35+35+21+7+1&2^7&1-7+21-35+35-21+7-1&(a-b)^7\r\n\\end{array}\\)<\/p>\r\nThe generation of each row of Pascal\u2019s triangle is done by adding the two numbers above it.\r\n

\\(\\begin{array}{cl}\r\n1&\\text{Start with 1} \\\\\r\n1+1&\\text{The outside number is always 1} \\\\\r\n1+2+1&\\text{The two 1's in the last row add to 2} \\\\\r\n1+3+3+1&1+2 \\text{ above adds to 3} \\\\\r\n1+4+6+4+1& \\\\\r\n1+5+10+10+5+1& \\\\\r\n1+6+15+20+15+6+1& \\\\\r\n1+7+21+35+35+21+7+1 & \\text{We can extend Pascal's triangle using this} \\\\\r\n1+8+28+56+70+56+28+8+1&(a+b)^8 \\\\\r\n1+9+36+84+126+126+84+36+9+1&(a+b)^9 \\\\\r\n\\end{array}\\)<\/p>\r\n\r\n

\r\n

Example 6.9.1<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nUse Pascal\u2019s triangle to expand \\((a + b)^9.\\)\r\n\r\nThe variables will follow a pattern of rising and falling powers:\r\n

\\(a^9 + a^8b + a^7b^2 + a^6b^3 + a^5b^4 + a^4b^5 + a^3b^6 + a^2b^7 + ab^8 + b^9\\)<\/p>\r\nWhen we insert the coefficients found from Pascal\u2019s triangle, we create:\r\n

\\(a^9 + 9a^8b + 36a^7b^2 + 84a^6b^3 + 126a^5b^4 + 126a^4b^5 + 84a^3b^6 + 36a^2b^7 + 8ab^8 + b^9\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\nProblem: <\/strong>Use Pascal\u2019s triangle to expand the binomial \\((a + b)^{12}.\\)\r\n

A Visual Representation of Binomial Expansion<\/h1>\r\n\"(a+b)1=a+b.<\/span>\r\n\r\nThe fourth expansion of the binomial is generally held to represent time, with the first three expansions being width, length, and height. While we live in a four-dimensional universe (string theory suggests ten dimensions), efforts to represent the fourth dimension of time are challenging. Carl Sagan describes the fourth dimension using an analogy created by Edwin Abbot (Abbot: Flatland: A Romance of Many Dimensions<\/em>). A video clip of Sagan\u2019s \u201cTesseract, 4th Dimension Made Easy\u201d<\/a>\u00a0 can be found on YouTube.","rendered":"

Pascal\u2019s triangle (1653) has been found in the works of mathematicians dating back before the 2nd century BC. While Pascal’s triangle is useful in many different mathematical settings, it will be applied to the expansion of binomials. In this application, Pascal\u2019s triangle will generate the leading coefficient of each term of a binomial expansion in the form of:<\/p>\n

\"(a+b)^n\"<\/p>\n

\"\"<\/span><\/p>\n

\"Yang<\/p>\n

For example:<\/p>\n

\"\begin{array}{llllllllllllllll}<\/p>\n

Pascal’s Triangle<\/h1>\n

\"\begin{array}{lclcl}<\/p>\n

The generation of each row of Pascal\u2019s triangle is done by adding the two numbers above it.<\/p>\n

\"\begin{array}{cl}<\/p>\n

\n
\n

Example 6.9.1<\/p>\n<\/header>\n

\n

Use Pascal\u2019s triangle to expand \"(a + b)^9.\"<\/p>\n

The variables will follow a pattern of rising and falling powers:<\/p>\n

\"a^9 + a^8b + a^7b^2 + a^6b^3 + a^5b^4 + a^4b^5 + a^3b^6 + a^2b^7 + ab^8 + b^9\"<\/p>\n

When we insert the coefficients found from Pascal\u2019s triangle, we create:<\/p>\n

\"a^9 + 9a^8b + 36a^7b^2 + 84a^6b^3 + 126a^5b^4 + 126a^4b^5 + 84a^3b^6 + 36a^2b^7 + 8ab^8 + b^9\"<\/p>\n<\/div>\n<\/div>\n

Problem: <\/strong>Use Pascal\u2019s triangle to expand the binomial \"(a + b)^{12}.\"<\/p>\n

A Visual Representation of Binomial Expansion<\/h1>\n

\"(a+b)1=a+b.<\/span><\/p>\n

The fourth expansion of the binomial is generally held to represent time, with the first three expansions being width, length, and height. While we live in a four-dimensional universe (string theory suggests ten dimensions), efforts to represent the fourth dimension of time are challenging. Carl Sagan describes the fourth dimension using an analogy created by Edwin Abbot (Abbot: Flatland: A Romance of Many Dimensions<\/em>). A video clip of Sagan\u2019s \u201cTesseract, 4th Dimension Made Easy\u201d<\/a>\u00a0 can be found on YouTube.<\/p>\n","protected":false},"author":540,"menu_order":22,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-571","chapter","type-chapter","status-publish","hentry"],"part":376,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapters\/571","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/users\/540"}],"version-history":[{"count":13,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapters\/571\/revisions"}],"predecessor-version":[{"id":3708,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapters\/571\/revisions\/3708"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/parts\/376"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapters\/571\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/media?parent=571"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapter-type?post=571"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/contributor?post=571"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/license?post=571"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}