{"id":505,"date":"2022-04-13T05:23:07","date_gmt":"2022-04-13T05:23:07","guid":{"rendered":"http:\/\/8thclass.deltapublications.in\/?page_id=505"},"modified":"2025-02-23T06:58:27","modified_gmt":"2025-02-23T06:58:27","slug":"y-8-factorise-polynomials","status":"publish","type":"page","link":"https:\/\/8thclass.deltapublications.in\/index.php\/y-8-factorise-polynomials\/","title":{"rendered":"Y.8 Factorise polynomials"},"content":{"rendered":"\n<h2 class=\"wp-block-heading has-text-align-center has-text-color\" style=\"color:#00056d;text-transform:uppercase\"><strong>Factorise polynomials<\/strong><\/h2>\n\n\n\n<p class=\"has-text-color has-link-color has-huge-font-size wp-elements-4d4696560822e7cb72ec4be90d4370b1\" style=\"color:#74008b\">Key Notes:<\/p>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-bcf1677ab406ff3fe4b06094c14adfcd\" style=\"color:#000060\"><strong>Definition of Factorisation<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>Factorisation is the process of expressing a polynomial as a product of its factors.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-989bf775243dbfd79181ac12a2c6fe2e\" style=\"color:#000060\"><strong>Common Factors Method<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>Identify and take out the greatest common factor (GCF) from all terms.<\/li>\n\n\n\n<li>Example: 6x<sup>2<\/sup> + 9x = 3x(2x + 3).<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-1d4e45a97d17337603a81d402f30c44b\" style=\"color:#000060\"><strong>Grouping Method<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>Group terms to find common factors within pairs.<\/li>\n\n\n\n<li>Example: ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y).<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-639ded2e4fbbeb9d1100b84187dd4b6b\" style=\"color:#000060\"><strong>Factorising Quadratic Polynomials<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>For polynomials in the form ax<sup>2<\/sup> + bx + c, find two numbers that multiply to acacac and add to bbb.<\/li>\n\n\n\n<li>Example: x<sup>2<\/sup> + 5x + 6 = (x + 2)(x + 3).<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-cdf82e9b433fcfd394f11b033aa3bd9f\" style=\"color:#000060\"><strong>Difference of Squares<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>Use the identity a<sup>2<\/sup> \u2212 b<sup>2<\/sup> = (a \u2212 b)(a + b).<\/li>\n\n\n\n<li>Example: x<sup>2<\/sup> \u2212 9 = (x \u2212 3)(x + 3).<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-b440eb800e382580e993955a24372f89\" style=\"color:#000060\"><strong>Perfect Square Trinomials<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>Recognise and factorise expressions like a<sup>2<\/sup> + 2ab + b<sup>2 <\/sup>= (a + b)<sup>2<\/sup>.<\/li>\n\n\n\n<li>Example: x<sup>2<\/sup> + 6x + 9 = (x + 3)<sup>2<\/sup>.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-d2c6ee4aeee06c1c1ec20a32ad495230\" style=\"color:#000060\"><strong>Cubic Polynomials (Factor Theorem)<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>If f(a) = 0, then (x \u2212 a) is a factor of f(x).<\/li>\n\n\n\n<li>Use synthetic or polynomial division for further factorisation.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-4f57c484b2976bc1b5687c37429e7c85\" style=\"color:#000060\"><strong>Trial and Error Method<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>Check for integer values of xxx that make the polynomial zero and use them to factorise.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-fd03adbc95080312f20057f3dc557526\" style=\"color:#000060\"><strong>Using Algebraic Identities<\/strong><\/p>\n\n\n\n<p class=\"has-large-font-size\">Apply formulas like:<\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>(a + b)<sup>2<\/sup> = a<sup>2<\/sup> + 2ab + b<sup>2<\/sup><\/li>\n\n\n\n<li>(a \u2212 b)<sup>2<\/sup> = a<sup>2<\/sup> \u2212 2ab + b<sup>2<\/sup><\/li>\n\n\n\n<li>(a + b)(a \u2212 b) = a<sup>2<\/sup> \u2212 b<sup>2<\/sup><\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-1fc2f8fb59b3cf92a109cdc5f6119674\" style=\"color:#000060\"><strong>Checking the Factors<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>Multiply the factors to verify if they give the original polynomial.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-align-center has-text-color has-large-font-size\" style=\"color:#105000\"><strong>Learn with an example<\/strong><\/p>\n\n\n\n<div class=\"wp-block-group has-background\" style=\"background-color:#b2dff3\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<div class=\"wp-block-group has-large-font-size\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<div class=\"wp-block-group has-background-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p class=\"has-text-color has-link-color wp-elements-7c37302915b5fa6393b649f4f1990f01\" style=\"color:#b00012\">\ud83c\udfbb Factorise.<\/p>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-6dfd3da619c190d96c8451a6b7aee4ea\" style=\"color:#b00012\"><strong>16c<sup>2<\/sup>\u20138c+1=&#8212;&#8212;<\/strong><\/p>\n<\/div><\/div>\n\n\n\n<p>Notice&nbsp;that&nbsp;16c<sup>2<\/sup>\u20138c+1&nbsp;is a perfect square trinomial because it can be written in the form&nbsp;a<sup>2<\/sup>\u20132ab+b<sup>2<\/sup>,&nbsp;where&nbsp;a&nbsp;is&nbsp;<strong>4<\/strong>c&nbsp;and&nbsp;b&nbsp;is&nbsp;<strong>1<\/strong>.<\/p>\n\n\n\n<p>a<sup>2<\/sup>\u20132ab+b<sup>2<\/sup><\/p>\n\n\n\n<p>(<strong>4<\/strong>c)<sup>2<\/sup>\u20132 . <strong>4<\/strong>c . <strong>1<\/strong>+<strong>1<\/strong><sup>2<\/sup><\/p>\n\n\n\n<p>16c<sup>2<\/sup>\u20138c+1<\/p>\n\n\n\n<p>Now&nbsp;use the formula for factorising perfect square&nbsp;trinomials.<\/p>\n\n\n\n<p>a<sup>2<\/sup>\u20132ab+b<sup>2<\/sup>=(a\u2013b)<sup>2<\/sup><\/p>\n\n\n\n<p>(<strong>4<\/strong>c)<sup>2<\/sup>\u20132 . <strong>4<\/strong>c . <strong>1<\/strong>+<strong>1<\/strong><sup>2<\/sup>=(<strong>4<\/strong>c\u2013<strong>1<\/strong>)<sup>2<\/sup><\/p>\n\n\n\n<p>16c<sup>2<\/sup>\u20138c+1=(4c\u20131)<sup>2<\/sup><\/p>\n\n\n\n<p>The&nbsp;factorised form of&nbsp;16c<sup>2<\/sup>\u20138c+1&nbsp;is&nbsp;(4c\u20131)<sup>2<\/sup>.<\/p>\n\n\n\n<p>Finally,&nbsp;check your&nbsp;work.<\/p>\n\n\n\n<p>(4c\u20131)<sup>2<\/sup><\/p>\n\n\n\n<p>(4c\u20131)(4c\u20131)Expand<\/p>\n\n\n\n<p>16c<sup>2<\/sup>\u20134c\u20134c+1Apply&nbsp;the distributive property&nbsp;(FOIL)<\/p>\n\n\n\n<p>16c<sup>2<\/sup>\u20138c+1<\/p>\n\n\n\n<p>Yes,&nbsp;16c<sup>2<\/sup>\u20138c+1=(4c\u20131)<sup>2<\/sup>.<\/p>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background\" style=\"background-color:#c7f4cc\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<div class=\"wp-block-group has-large-font-size\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<div class=\"wp-block-group has-background-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p class=\"has-text-color has-link-color wp-elements-ca35ef8369f522de62be62da1326c310\" style=\"color:#b00012\">\ud83c\udfbb <strong>Factorise.<\/strong><\/p>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-0c35c7367eb30b8abdee751d8808292e\" style=\"color:#b00012\"><strong>3w<sup>2<\/sup>+8w+5=&#8212;&#8212;<\/strong><\/p>\n<\/div><\/div>\n\n\n\n<p>Look&nbsp;at the given&nbsp;quadratic:<\/p>\n\n\n\n<p>3w2+8w+5<\/p>\n\n\n\n<p>The&nbsp;product&nbsp;ac&nbsp;is&nbsp;15,&nbsp;so you need to find a pair of factors with a product of&nbsp;15.&nbsp;The&nbsp;b&nbsp;term is&nbsp;8,&nbsp;so you need to find a pair of factors with a sum of&nbsp;8.&nbsp;Since the product is positive&nbsp;(15)&nbsp;and the sum is positive&nbsp;(8),&nbsp;you need both factors to be&nbsp;positive.<\/p>\n\n\n\n<p>Make&nbsp;a list of the possible factor pairs with a product of&nbsp;15,&nbsp;and then find the one with a sum of&nbsp;8.<\/p>\n\n\n\n<figure id=\"yui_3_18_1_1_1675766756805_994\" class=\"wp-block-table\"><table><tbody><tr><td>Factor&nbsp;pairs of&nbsp;ac=15<\/td><td>Sum&nbsp;of factor&nbsp;pairs<\/td><\/tr><tr><td>1 . 15=15<\/td><td>1+15=16<\/td><\/tr><tr><td><strong>3 . <\/strong><strong>5<\/strong><strong>=<\/strong><strong>15<\/strong><\/td><td><strong>3<\/strong>+<strong>5<\/strong><strong>=<\/strong><strong>8<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>The&nbsp;factors&nbsp;3&nbsp;and&nbsp;5&nbsp;have a sum of&nbsp;8.&nbsp;So, replace the quadratic&#8217;s&nbsp;8w&nbsp;term with&nbsp;3w&nbsp;and&nbsp;5w,&nbsp;and then factor by&nbsp;grouping.<\/p>\n\n\n\n<p>3w<sup>2<\/sup>+8w+5<\/p>\n\n\n\n<p>3w<sup>2<\/sup>+<strong>3<\/strong>w+<strong>5<\/strong>w+5<\/p>\n\n\n\n<p>3w(w+1)+5(w+1)Factor&nbsp;by grouping; the expressions in brackets should&nbsp;match<\/p>\n\n\n\n<p>(3w+5)(w+1)<\/p>\n\n\n\n<p>Finally,&nbsp;check your&nbsp;work.<\/p>\n\n\n\n<p>(3w+5)(w+1)<\/p>\n\n\n\n<p>3w<sup>2<\/sup>+5w+3w+5Apply&nbsp;the distributive property&nbsp;(FOIL)<\/p>\n\n\n\n<p>3w<sup>2<\/sup>+8w+5<\/p>\n\n\n\n<p>Yes,&nbsp;3w<sup>2<\/sup>+8w+5=(3w+5)(w+1).<\/p>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background\" style=\"background-color:#b7e9d5\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<div class=\"wp-block-group has-large-font-size\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<div class=\"wp-block-group has-background-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p class=\"has-text-color has-link-color wp-elements-507ccdf1e7b57d5b74990edd1f1a3515\" style=\"color:#b00012\">\ud83c\udfbb Factor.<\/p>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-f5d625f6cb6cbfce1a171b720b36979f\" style=\"color:#b00012\"><strong>12fg+6f+10g+5=&#8212;&#8211;<\/strong><\/p>\n<\/div><\/div>\n\n\n\n<p>Factor&nbsp;by&nbsp;grouping.<\/p>\n\n\n\n<p>12fg+6f+10g+5<\/p>\n\n\n\n<p><strong>6<\/strong>f(<strong>2<\/strong>g+<strong>1<\/strong>)+<strong>5<\/strong>(<strong>2<\/strong>g+<strong>1<\/strong>)     Factor&nbsp;by grouping; the expressions in brackets should&nbsp;match<\/p>\n\n\n\n<p>(<strong>6<\/strong>f+<strong>5<\/strong>)(<strong>2<\/strong>g+<strong>1<\/strong>)        Apply&nbsp;the distributive&nbsp;property<\/p>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n<p class=\"has-text-color has-large-font-size\" style=\"color:#d90000\">let&#8217;s practice!<\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/wordwall.net\/play\/86522\/603\/969\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-164.png\" alt=\"\" class=\"wp-image-9019\" srcset=\"https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-164.png 500w, https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-164-300x300.png 300w, https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-164-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/wordwall.net\/play\/86302\/778\/691\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-169.png\" alt=\"\" class=\"wp-image-9020\" srcset=\"https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-169.png 500w, https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-169-300x300.png 300w, https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-169-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Factorise polynomials Key Notes: Definition of Factorisation Common Factors Method Grouping Method Factorising Quadratic Polynomials Difference of Squares Perfect Square Trinomials Cubic Polynomials (Factor Theorem) Trial and Error Method Using Algebraic Identities Apply formulas like: Checking the Factors Learn with an example \ud83c\udfbb Factorise. 16c2\u20138c+1=&#8212;&#8212; Notice&nbsp;that&nbsp;16c2\u20138c+1&nbsp;is a perfect square trinomial because it can be written<a class=\"more-link\" href=\"https:\/\/8thclass.deltapublications.in\/index.php\/y-8-factorise-polynomials\/\">Continue reading <span class=\"screen-reader-text\">&#8220;Y.8 Factorise polynomials&#8221;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"class_list":["post-505","page","type-page","status-publish","hentry","entry"],"aioseo_notices":[],"_hostinger_reach_plugin_has_subscription_block":false,"_hostinger_reach_plugin_is_elementor":false,"_links":{"self":[{"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/505","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/comments?post=505"}],"version-history":[{"count":24,"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/505\/revisions"}],"predecessor-version":[{"id":19949,"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/505\/revisions\/19949"}],"wp:attachment":[{"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/media?parent=505"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}