{"id":2541,"date":"2016-09-09T04:30:14","date_gmt":"2016-09-09T04:30:14","guid":{"rendered":"https:\/\/cbselibrary.com\/?p=2541"},"modified":"2018-06-12T06:24:15","modified_gmt":"2018-06-12T06:24:15","slug":"types-of-factorization","status":"publish","type":"post","link":"https:\/\/cbselibrary.com\/types-of-factorization\/","title":{"rendered":"What Are The Types Of Factorization"},"content":{"rendered":"<h2><strong>Types Of Factorization Example Problems With Solutions<\/strong><\/h2>\n<p><strong>Type I: Factorization by taking out the common factors.\u00a0 <\/strong><\/p>\n<p><strong>Example 1: \u00a0 \u00a0<\/strong>Factorize the following expression<br \/>\n2x<sup>2<\/sup>y + 6xy<sup>2<\/sup>\u00a0+ 10x<sup>2<\/sup>y<sup>2<\/sup><br \/>\n<strong>Solution: \u00a0 \u00a0<\/strong>2x<sup>2<\/sup>y + 6xy<sup>2<\/sup>\u00a0+ 10x<sup>2<\/sup>y<sup>2<\/sup><br \/>\n=2xy(x + 3y + 5xy)<\/p>\n<p><strong>Type II: Factorization by grouping the terms.\u00a0\u00a0\u00a0 <\/strong><\/p>\n<p><strong>Example 2: \u00a0 \u00a0<\/strong>Factorize the following expression<br \/>\na<sup>2<\/sup> \u2013 b + ab \u2013 a<br \/>\n<strong>Solution: \u00a0 \u00a0<\/strong>a<sup>2<\/sup> \u2013 b + ab \u2013 a<br \/>\n= a<sup>2<\/sup> + ab \u2013 b \u2013 a = (a<sup>2<\/sup> + ab) \u2013 (b + a)<br \/>\n= a (a + b) \u2013 (a + b) = (a + b) (a \u2013 1)<\/p>\n<p><strong>Type III: Factorization by making a perfect square.\u00a0\u00a0 <\/strong><\/p>\n<p><strong>Example 3: \u00a0 \u00a0<\/strong>Factorize of the following expression<br \/>\n9x<sup>2<\/sup> + 12xy + 4y<sup>2<\/sup><br \/>\n<strong>Solution: \u00a0 \u00a0<\/strong>9x<sup>2<\/sup> + 12xy + 4y<sup>2<\/sup><br \/>\n= (3x)<sup>2<\/sup> + 2 \u00d7 (3x) \u00d7 (2y) + (2y)<sup>2<\/sup><br \/>\n= (3x + 2y)<sup>2<\/sup><\/p>\n<p><strong>Example 4: \u00a0 \u00a0<\/strong>Factorize of the following expression<br \/>\n\\(\\frac{{{x}^{2}}}{{{y}^{2}}}+2+\\frac{{{y}^{2}}}{{{x}^{2\\prime }}},x\\ne 0,y\\ne 0\\)<br \/>\n<strong>Solution: \u00a0 \u00a0<\/strong><br \/>\n<img loading=\"lazy\" src=\"https:\/\/c1.staticflickr.com\/9\/8439\/29521867876_bff8471966_o.png\" alt=\"Factorization-by-perfect-square-Example-1\" width=\"224\" height=\"204\" \/><\/p>\n<p><strong>Example 5: \u00a0 \u00a0<\/strong>Factorize of the following expression<br \/>\n\\({{\\left( 5x-\\frac{1}{x} \\right)}^{2}}+4\\left( 5x-\\frac{1}{x} \\right)+4,x\\ne 0\\)<br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" src=\"https:\/\/c1.staticflickr.com\/9\/8035\/28932861833_cd8c492db2_o.png\" alt=\"Factorization-by-perfect-square-Example-2\" width=\"288\" height=\"199\" \/><\/p>\n<p><strong>People also ask:<\/strong><\/p>\n<ul>\n<li><a href=\"https:\/\/cbselibrary.com\/solving-quadratic-equations-with-complex-roots\/\">Solving Quadratic Equations with Complex Roots<\/a><\/li>\n<li><a href=\"https:\/\/cbselibrary.com\/solving-a-quadratic-equation-by-factoring\/\">Factoring Quadratics<\/a><\/li>\n<li><a href=\"https:\/\/cbselibrary.com\/solving-quadratic-equations-factoring\/\">Solving Quadratic Equations by Factoring<\/a><\/li>\n<li><a href=\"https:\/\/cbselibrary.com\/solving-factorable-quadratic-equations\/\">Solving Factorable Quadratic Equations<\/a><\/li>\n<li><a href=\"https:\/\/cbselibrary.com\/solving-a-quadratic-equation-by-completing-the-square\/\">Completing the Square<\/a><\/li>\n<li><a href=\"https:\/\/cbselibrary.com\/solving-quadratic-equations-by-completing-the-square\/\">Solving Quadratic Equations by Completing the Square<\/a><\/li>\n<\/ul>\n<p><strong>Type IV: Factorizing by difference of two squares.<\/strong><\/p>\n<p><strong>Example 6: <\/strong>\u00a0 \u00a0Factorize\u00a0the following expressions<br \/>\n(a) 2x<sup>2<\/sup>y + 6 xy<sup>2<\/sup> + 10 x<sup>2<\/sup>y<sup>2<\/sup><br \/>\n(b) 2x<sup>4<\/sup> + 2x<sup>3<\/sup>y + 3xy<sup>2<\/sup> + 3y<sup>3<\/sup><br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" src=\"https:\/\/c1.staticflickr.com\/9\/8015\/28930646874_76cd9a5b12_o.png\" alt=\"Factorizing-by-difference-of-two-squares-example-1\" width=\"259\" height=\"155\" \/><\/p>\n<p><strong>Example 7: \u00a0 \u00a0<\/strong>Factorize 4x<sup>2<\/sup> + 12 xy + 9 y<sup>2<\/sup><br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" src=\"https:\/\/c1.staticflickr.com\/9\/8265\/28933056973_22613116b7_o.png\" alt=\"Factorizing-by-difference-of-two-squares-example-7\" width=\"555\" height=\"315\" \/><\/p>\n<p><strong>Example 8: \u00a0 \u00a0<\/strong>Factorize each of the following expressions<br \/>\n(i) 9x<sup>2<\/sup> \u2013 4y<sup>2<\/sup><br \/>\n(ii) x<sup>3<\/sup>\u00a0\u2013 x<br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" src=\"https:\/\/c1.staticflickr.com\/9\/8713\/28930836104_f325bd0c98_o.png\" alt=\"Factorizing-by-difference-of-two-squares-example-8\" width=\"254\" height=\"122\" \/><\/p>\n<p><strong>Example 9: \u00a0 \u00a0<\/strong>Factorize each of the following expressions<br \/>\n(i) 36x<sup>2<\/sup> \u2013 12x + 1 \u2013 25y<sup>2<\/sup><br \/>\n\\(\\text{(ii) }{{a}^{2}}-\\frac{9}{{{a}^{2}}},a\\ne 0\\)<br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" src=\"https:\/\/c1.staticflickr.com\/9\/8759\/28930836004_26a41b041d_o.png\" alt=\"Factorizing-by-difference-of-two-squares-example-9\" width=\"279\" height=\"316\" \/><\/p>\n<p><strong>Example 10: \u00a0 \u00a0<\/strong>Factorize the following algebraic expression<br \/>\nx<sup>4<\/sup>\u00a0\u2013 81y<sup>4<\/sup><br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" src=\"https:\/\/c1.staticflickr.com\/9\/8287\/29475614941_f79d3946cc_o.png\" alt=\"Factorizing-by-difference-of-two-squares-example-10\" width=\"226\" height=\"138\" \/><\/p>\n<p><strong>Example 11: \u00a0 \u00a0<\/strong>Factorize the following expression<br \/>\nx(x+z) \u2013 y (y+z)<br \/>\n<strong>Solution: \u00a0 \u00a0<\/strong>x(x+z) \u2013 y (y+z) = (x<sup>2<\/sup> \u2013 y<sup>2<\/sup>) + (xz\u2013yz)<br \/>\n= (x\u2013y) (x+y) + z (x\u2013y)<br \/>\n= (x\u2013y) {(x+y) + z}<br \/>\n= (x\u2013y) (x+ y + z)<\/p>\n<p><strong>Example 12: \u00a0 \u00a0<\/strong>Factorize the following expression<br \/>\nx<sup>4<\/sup>\u00a0+ x<sup>2<\/sup>\u00a0+ 1<br \/>\n<strong>Solution: \u00a0 \u00a0<\/strong>x<sup>4<\/sup> + x<sup>2<\/sup> + 1 = (x<sup>4<\/sup> + 2x<sup>2<\/sup> +1) \u2013 x<sup>2<\/sup><br \/>\n= (x<sup>2<\/sup> +1)<sup>2<\/sup> \u2013 x<sup>2<\/sup> = (x<sup>2<\/sup> + 1 \u2013 x) (x<sup>2<\/sup> + 1+x)<br \/>\n= (x<sup>2<\/sup>\u2013x + 1) (x<sup>2<\/sup> + x + 1)<\/p>\n<p><strong>Type V: Factorizing the sum and difference of cubes of two quantities.<\/strong><br \/>\n(i) (a<sup>3<\/sup> + b<sup>3<\/sup>) = (a + b) (a<sup>2<\/sup> \u2013 ab + b<sup>2<\/sup>)<br \/>\n(ii) (a<sup>3<\/sup> \u2013 b<sup>3<\/sup>) = (a \u2013 b) (a<sup>2<\/sup> + ab + b<sup>2<\/sup>)<\/p>\n<p><strong>Example 13: \u00a0 \u00a0<\/strong>Factorize the following expression<br \/>\na<sup>3<\/sup> + 27<br \/>\n<strong>Solution: \u00a0 <\/strong>a<sup>3<\/sup>\u00a0 + 27 = a<sup>3<\/sup>\u00a0 + 3<sup>3<\/sup>\u00a0= (a + 3) (a<sup>2<\/sup>\u00a0\u20133a +9)<\/p>\n<p><strong>Example 14: \u00a0 \u00a0<\/strong>Simplify : (x+ y)<sup>3<\/sup> \u2013 (x \u2013y)<sup>3<\/sup>\u00a0\u2013 6y(x<sup>2<\/sup> \u2013 y<sup>2<\/sup>)<br \/>\n<strong>Solution: \u00a0 \u00a0<\/strong><br \/>\n<img loading=\"lazy\" class=\"alignnone wp-image-2548 size-full\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2016\/09\/Factorizing-the-sum-and-difference-of-cubes.png\" alt=\"factorizing-the-sum-and-difference-of-cubes\" width=\"292\" height=\"181\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Types Of Factorization Example Problems With Solutions Type I: Factorization by taking out the common factors.\u00a0 Example 1: \u00a0 \u00a0Factorize the following expression 2x2y + 6xy2\u00a0+ 10x2y2 Solution: \u00a0 \u00a02x2y + 6xy2\u00a0+ 10x2y2 =2xy(x + 3y + 5xy) Type II: Factorization by grouping the terms.\u00a0\u00a0\u00a0 Example 2: \u00a0 \u00a0Factorize the following expression a2 \u2013 b &#8230; <a title=\"What Are The Types Of Factorization\" class=\"read-more\" href=\"https:\/\/cbselibrary.com\/types-of-factorization\/\" aria-label=\"More on What Are The Types Of Factorization\">Read more<\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[5],"tags":[1088,1102,24,1101,1103],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v17.8 (Yoast SEO v18.4.1) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>What Are The Types Of Factorization - CBSE Library<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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