{"id":2484,"date":"2022-12-29T10:00:38","date_gmt":"2022-12-29T04:30:38","guid":{"rendered":"https:\/\/www.aplustopper.com\/?p=2484"},"modified":"2022-12-30T09:43:17","modified_gmt":"2022-12-30T04:13:17","slug":"factor-theorem","status":"publish","type":"post","link":"https:\/\/www.aplustopper.com\/factor-theorem\/","title":{"rendered":"How Do You Use The Factor Theorem"},"content":{"rendered":"<h2><strong>Factor Theorem<\/strong><\/h2>\n<p><strong>Theorem:<\/strong> If p(x) is a polynomial of degree n \u2265\u00a01 and a is any real number, then<br \/>\n(i) x \u2013 a is a factor of p(x), if p(a) = 0, and (ii) p(a) = 0, if x \u2013 a is a factor of p(x).<br \/>\n<strong>Proof:<\/strong> By the Remainder Theorem,<br \/>\np(x) = (x \u2013 a) q(x) + p(a).<br \/>\n(i) If p(a) = 0, then p(x) = (x \u2013 a) q(x),<br \/>\nwhich shows that x \u2013 a is a factor of p(x).<br \/>\n(ii) Since x \u2013 a is a factor of p(x),<br \/>\np(x) = (x \u2013 a) g(x) for same polynomial g(x).<br \/>\nIn this case, p(a) = (a \u2013 a) g(a) = 0.<\/p>\n<p><strong>To use factor theorem<\/strong><\/p>\n<ul>\n<li><strong>Step 1:<\/strong> (x + a) is factor of a polynomial p(x) if p(\u2013a) = 0.<\/li>\n<li><strong>Step 2:<\/strong> (ax \u2013 b) is a factor of a polynomial p(x)\u00a0if p(b\/a) = 0<\/li>\n<li><strong>Step 3:<\/strong> ax + b is a factor of a polynomial p(x) if p(\u2013b\/a) = 0.<\/li>\n<li><strong>Step 4:<\/strong> (x \u2013 a) (x \u2013 b) is a factor of a polynomial p(x) if p(a) = 0 and p(b) = 0.<\/li>\n<\/ul>\n<h2><strong>Factor Theorem Example Problems With Solutions<\/strong><\/h2>\n<p><strong>Example 1:<\/strong> \u00a0 \u00a0Examine whether x + 2 is a factor of x<sup>3<\/sup> + 3x<sup>2<\/sup> + 5x + 6 and of 2x + 4.<br \/>\n<strong>Solution:<\/strong> \u00a0 \u00a0The zero of x + 2 is \u20132.<br \/>\nLet p(x) = x<sup>3<\/sup> + 3x<sup>2<\/sup> + 5x + 6 and s(x) = 2x + 4<br \/>\nThen,\u00a0\u00a0\u00a0 p(\u20132) = (\u20132)<sup>3<\/sup> + 3(\u20132)<sup>2<\/sup> + 5(\u20132) + 6<br \/>\n= \u20138 + 12 \u2013 10 + 6<br \/>\n= 0<br \/>\nSo, by the Factor Theorem, x + 2 is a factor of x<sup>3<\/sup> + 3x<sup>2<\/sup> + 5x + 6.<br \/>\nAgain,\u00a0\u00a0 s(\u20132) = 2(\u20132) + 4 = 0<br \/>\nSo, x + 2 is a factor of 2x + 4.<\/p>\n<p><strong>Example 2: \u00a0 \u00a0<\/strong>Use the factor theorem to determine whether x \u2013 1 is a factor of<br \/>\n(a) x<sup>3<\/sup>\u00a0+ 8x<sup>2<\/sup> \u2013 7x \u2013 2<br \/>\n(b) 2x<sup>3<\/sup> + 5x<sup>2<\/sup> \u2013 7<br \/>\n(c) 8x<sup>4<\/sup> + 12x<sup>3<\/sup> \u2013 18x + 14<br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2488 size-full\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/09\/Factor-Theorem-Example-1.png\" alt=\"factor-theorem-example-1\" width=\"551\" height=\"420\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/09\/Factor-Theorem-Example-1.png 551w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/09\/Factor-Theorem-Example-1-300x229.png 300w\" sizes=\"(max-width: 551px) 100vw, 551px\" \/><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2487 size-full\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/09\/Factor-Theorem-Example-1.1.png\" alt=\"factor-theorem-example-1-1\" width=\"540\" height=\"202\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/09\/Factor-Theorem-Example-1.1.png 540w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/09\/Factor-Theorem-Example-1.1-300x112.png 300w\" sizes=\"(max-width: 540px) 100vw, 540px\" \/><\/p>\n<p><strong>Example 3: \u00a0 \u00a0<\/strong>Factorize each of the following expression, given that x<sup>3<\/sup>\u00a0+ 13 x<sup>2<\/sup>\u00a0+ 32 x + 20. (x+2) is a factor.<br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2491 size-full\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/09\/Factor-Theorem-Example-2.png\" alt=\"factor-theorem-example-2\" width=\"265\" height=\"166\" \/><\/p>\n<p><strong>Example 4: \u00a0 \u00a0<\/strong>Factorize x3\u00a0\u2013 23 x<sup>2<\/sup>\u00a0+ 142 x \u2013 120<br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2492 size-full\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/09\/Factor-Theorem-Example-3.png\" alt=\"factor-theorem-example-3\" width=\"596\" height=\"307\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/09\/Factor-Theorem-Example-3.png 596w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/09\/Factor-Theorem-Example-3-300x155.png 300w\" sizes=\"(max-width: 596px) 100vw, 596px\" \/><\/p>\n<p><strong>Example 5: \u00a0 \u00a0<\/strong>Show that (x \u2013 3) is a factor of the polynomial x<sup>3<\/sup>\u00a0\u2013 3x<sup>2<\/sup> + 4x \u2013 12<br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2493 size-full\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/09\/Factor-Theorem-Example-5.png\" alt=\"factor-theorem-example-5\" width=\"621\" height=\"279\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/09\/Factor-Theorem-Example-5.png 621w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/09\/Factor-Theorem-Example-5-300x135.png 300w\" sizes=\"(max-width: 621px) 100vw, 621px\" \/><\/p>\n<p><strong>Example 6: \u00a0 \u00a0<\/strong>Show that (x \u2013 1) is a factor of x<sup>10<\/sup>\u00a0\u2013 1 and also of x<sup>11<\/sup> \u2013 1.<br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2495 size-full\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/09\/Factor-Theorem-Example-6-1.png\" alt=\"factor-theorem-example-6\" width=\"653\" height=\"184\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/09\/Factor-Theorem-Example-6-1.png 653w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/09\/Factor-Theorem-Example-6-1-300x85.png 300w\" sizes=\"(max-width: 653px) 100vw, 653px\" \/><\/p>\n<p><strong>Example 7: \u00a0 \u00a0<\/strong>Show that x + 1 and 2x \u2013 3 are factors of 2x<sup>3<\/sup>\u00a0\u2013 9x<sup>2<\/sup>\u00a0+ x + 12.<br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/c1.staticflickr.com\/9\/8224\/29504606076_a7e77ab086_o.png\" width=\"498\" height=\"514\" alt=\"Factor-Theorem-Example-7\"><\/p>\n<p><strong>Example 8: \u00a0 \u00a0<\/strong>Find the value of k, if x + 3 is a factor of 3x<sup>2<\/sup> + kx + 6.<br \/>\n<strong>Solution: \u00a0\u00a0<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2497 size-full\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/09\/Factor-Theorem-Example-8.png\" alt=\"factor-theorem-example-8\" width=\"598\" height=\"193\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/09\/Factor-Theorem-Example-8.png 598w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/09\/Factor-Theorem-Example-8-300x97.png 300w\" sizes=\"(max-width: 598px) 100vw, 598px\" \/><\/p>\n<p><strong>Example 9: \u00a0 \u00a0<\/strong>If ax<sup>3<\/sup>\u00a0+ bx<sup>2<\/sup> + x \u2013 6 has x + 2 as a factor and leaves a remainder 4 when divided by (x \u2013 2), find the values of a and b.<br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/c1.staticflickr.com\/9\/8435\/28916284293_2a0fc57bd0_o.png\" alt=\"Factor-Theorem-Example-9\" width=\"633\" height=\"481\" \/><\/p>\n<p><strong>Example 10: \u00a0 \u00a0<\/strong>If both x \u2013 2 and x \u2013 1\/2 are factors of px<sup>2<\/sup> + 5x + r, show that p = r.<br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/c1.staticflickr.com\/9\/8131\/28913757184_1f8a0099fa_o.png\" alt=\"Factor-Theorem-Example-10\" width=\"492\" height=\"538\" \/><\/p>\n<p><strong>Example 11: \u00a0 \u00a0<\/strong>If x<sup>2<\/sup>\u00a0\u2013 1 is a factor of ax<sup>4<\/sup>\u00a0+ bx<sup>3<\/sup>\u00a0+ cx<sup>2<\/sup>\u00a0+ dx + e, show that a + c + e = b + d = 0.<br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/c1.staticflickr.com\/9\/8817\/29249114630_285b3e4b33_o.png\" alt=\"Factor-Theorem-Example-11\" width=\"625\" height=\"320\" \/><\/p>\n<p><strong>Example 12: \u00a0 \u00a0<\/strong>Using factor theorem, show that a \u2013 b, b \u2013 c and c \u2013 a are the factors of\u00a0a(b<sup>2<\/sup>\u00a0\u2013 c<sup>2<\/sup>) + b(c<sup>2<\/sup>\u00a0\u2013 a<sup>2<\/sup>) + c(a<sup>2<\/sup>\u00a0\u2013 b<sup>2<\/sup>).<br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/c1.staticflickr.com\/9\/8705\/28913757084_af2ae968b6_o.png\" alt=\"Factor-Theorem-Example-12\" width=\"574\" height=\"347\" \/><\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Factor Theorem Theorem: If p(x) is a polynomial of degree n \u2265\u00a01 and a is any real number, then (i) x \u2013 a is a factor of p(x), if p(a) = 0, and (ii) p(a) = 0, if x \u2013 a is a factor of p(x). Proof: By the Remainder Theorem, p(x) = (x \u2013 [&hellip;]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","footnotes":""},"categories":[5],"tags":[1072,1073,24],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v17.8 (Yoast SEO v17.8) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>How Do You Use The Factor Theorem - A Plus Topper<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.aplustopper.com\/factor-theorem\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"How Do You Use The Factor Theorem\" \/>\n<meta property=\"og:description\" content=\"Factor Theorem Theorem: If p(x) is a polynomial of degree n \u2265\u00a01 and a is any real number, then (i) x \u2013 a is a factor of p(x), if p(a) = 0, and (ii) p(a) = 0, if x \u2013 a is a factor of p(x). Proof: By the Remainder Theorem, p(x) = (x \u2013 [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.aplustopper.com\/factor-theorem\/\" \/>\n<meta property=\"og:site_name\" content=\"A Plus Topper\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/aplustopper\/\" \/>\n<meta property=\"article:published_time\" content=\"2022-12-29T04:30:38+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2022-12-30T04:13:17+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/09\/Factor-Theorem-Example-1.png\" \/>\n<meta name=\"twitter:card\" content=\"summary\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Veerendra\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Organization\",\"@id\":\"https:\/\/www.aplustopper.com\/#organization\",\"name\":\"Aplus Topper\",\"url\":\"https:\/\/www.aplustopper.com\/\",\"sameAs\":[\"https:\/\/www.facebook.com\/aplustopper\/\"],\"logo\":{\"@type\":\"ImageObject\",\"@id\":\"https:\/\/www.aplustopper.com\/#logo\",\"inLanguage\":\"en-US\",\"url\":\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2018\/12\/Aplus_380x90-logo.jpg\",\"contentUrl\":\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2018\/12\/Aplus_380x90-logo.jpg\",\"width\":1585,\"height\":375,\"caption\":\"Aplus Topper\"},\"image\":{\"@id\":\"https:\/\/www.aplustopper.com\/#logo\"}},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/www.aplustopper.com\/#website\",\"url\":\"https:\/\/www.aplustopper.com\/\",\"name\":\"A Plus Topper\",\"description\":\"Improve your Grades\",\"publisher\":{\"@id\":\"https:\/\/www.aplustopper.com\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/www.aplustopper.com\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"en-US\"},{\"@type\":\"ImageObject\",\"@id\":\"https:\/\/www.aplustopper.com\/factor-theorem\/#primaryimage\",\"inLanguage\":\"en-US\",\"url\":\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/09\/Factor-Theorem-Example-1.png\",\"contentUrl\":\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/09\/Factor-Theorem-Example-1.png\",\"width\":551,\"height\":420},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/www.aplustopper.com\/factor-theorem\/#webpage\",\"url\":\"https:\/\/www.aplustopper.com\/factor-theorem\/\",\"name\":\"How Do You Use The Factor Theorem - 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