{"id":34945,"date":"2023-10-21T13:15:31","date_gmt":"2023-10-21T07:45:31","guid":{"rendered":"https:\/\/ccssmathanswers.com\/?p=34945"},"modified":"2023-11-10T11:51:19","modified_gmt":"2023-11-10T06:21:19","slug":"factorization-by-grouping","status":"publish","type":"post","link":"https:\/\/bigideasmathanswers.com\/factorization-by-grouping\/","title":{"rendered":"Factorization by Grouping | What is Factorization by Grouping?"},"content":{"rendered":"<p>Are you looking for the different problems on Factorization by Grouping? Then, you are in the right place. We have given all types of factorization problems on our website. Students can learn to Factorize by Grouping the Terms in these articles. While solving <a href=\"https:\/\/ccssmathanswers.com\/factorization\/\"><strong>Factorization<\/strong><\/a> by grouping problems, students need to group the terms with common factors before factoring.<\/p>\n<h2>How to Factor by Grouping?<\/h2>\n<p>Have a look at the Factoring by Grouping Steps and learn how to solve related problems easily. Follow the guidelines provided and perform factorization by grouping method. They are as under<\/p>\n<ul>\n<li>Note down the given expression, group the first two terms and last two terms.<\/li>\n<li>Find out the greatest common factor(GCF) from the first term and second term.<\/li>\n<li>Now, find the common factor from the above two groups.<\/li>\n<li>Finally factor out the terms in terms of product.<\/li>\n<\/ul>\n<h3>Factorization by Grouping Examples<\/h3>\n<p>1. Factor grouping the expressions?<br \/>\n1 + x + xy + x\u00b2y.<\/p>\n<p>Solution: Given Expression is 1 + x + xy + x\u00b2y.<br \/>\nGroup the first two terms and last two terms.<br \/>\nFirst two terms are 1 + x and the last two terms are xy + x\u00b2y.<br \/>\n(1+ x) + (xy + x\u00b2y).<br \/>\nFind out the common factor from the above two groups.<br \/>\n(1+x) + xy(1+x).<br \/>\nFactor out the terms in terms of product.<br \/>\n(1+x) (1+xy).<\/p>\n<p>By factor grouping the expression 1 + x + xy + x\u00b2y., we will get (1+x) (1+xy).<\/p>\n<p>2. How to factor by grouping the following algebraic expressions?<\/p>\n<p>(i) x\u00b2 \u2013 xy + xz \u2013 zy.<\/p>\n<p>Solution:<br \/>\nGiven Expression is x\u00b2 \u2013 xy + xz \u2013 zy<br \/>\nGroup the first two terms and last two terms.<br \/>\nFirst two terms are x\u00b2 \u2013 xy and the last two terms are xz \u2013 zy.<br \/>\n(x\u00b2 \u2013 xy) + (xz \u2013 zy)<br \/>\nFind out the common factor from the above two groups.<br \/>\nx(x &#8211; y) + z(x &#8211; y)<br \/>\nFactor out the terms in terms of product.<br \/>\n(x + z) (x &#8211; y)<\/p>\n<p>Factor by grouping the expression x\u00b2 &#8211; xy + xz &#8211; zy, we will get the result as (x+z) (x-y).<\/p>\n<p>(ii) x\u00b2 + 3x + xy + 3y.<\/p>\n<p>Solution:<br \/>\nGiven Expression is x\u00b2 + 3x + xy + 3y.<br \/>\nGroup the first two terms and last two terms.<br \/>\nFirst two terms are x\u00b2 + 3x and the last two terms are xy + 3y.<br \/>\n(x\u00b2 + 3x) + (xy + 3y)<br \/>\nFind out the common factor from the above two groups.<br \/>\nx(x + 3) + y(x + 3)<br \/>\nFactor out the terms in terms of product.<br \/>\n(x+y) (x+3).<\/p>\n<p>Factor by grouping the expression x\u00b2 + 3x + xy + 3y, we will get solution as (x+y) (x+3).<\/p>\n<p>3. Factorize the algebraic expressions.<\/p>\n<p>(i) 2a + ba + 2b + b\u00b2<\/p>\n<p>Solution:<br \/>\nGiven Expression is 2a + ba + 2b + b\u00b2<br \/>\nGroup the first two terms and last two terms.<br \/>\nFirst two terms are 2a + ba and the last two terms are 2b + b\u00b2.<br \/>\n(2a + ba)+ (2b + b\u00b2).<br \/>\nFind out the common factor from the above two groups.<br \/>\na(2 + b) + b(2 + b).<br \/>\nFactor out the terms in terms of product.<br \/>\n(a + b) (2 + b).<\/p>\n<p>By factorizing the expression 2a + ba + 2b + b\u00b2, we will get (a + b) (2 + b).<\/p>\n<p>(ii) b\u00b2 &#8211; yb + 5b\u2013 5y.<\/p>\n<p>Solution:<br \/>\nGiven Expression is b\u00b2 &#8211; yb + 5b\u2013 5y.<br \/>\nGroup the first two terms and last two terms.<br \/>\nFirst, two terms are b\u00b2 &#8211; yb and the last two terms are 5b \u2013 5y.<br \/>\n(b\u00b2 &#8211; yb)+ (5b &#8211; 5y).<br \/>\nFind out the common factor from the above two groups.<br \/>\nb(b &#8211; y) + 5(b &#8211; y).<br \/>\nFactor out the terms in terms of product.<br \/>\n(5 + b) (b &#8211; y).<\/p>\n<p>By factorizing the expression b\u00b2 &#8211; yb + 5b\u2013 5y, we will get (5 + b) (b &#8211; y).<\/p>\n<p>(iii) pq \u2013 rq \u2013 ps + rs.<\/p>\n<p>Solution:<br \/>\nGiven Expression is pq \u2013 rq \u2013 ps + rs.<br \/>\nGroup the first two terms and last two terms.<br \/>\nFirst two terms are pq \u2013 rq and the last two terms are &#8211; ps + rs.<br \/>\n(pq &#8211; rq) &#8211; (ps &#8211; rs).<br \/>\nFind out the common factor from the above two groups.<br \/>\nq(p &#8211; r) -s(p &#8211; r).<br \/>\nFactor out the terms in terms of product.<br \/>\n(p &#8211; r) (q &#8211; s).<\/p>\n<p>By factorizing the expression pq \u2013 rq \u2013 ps + rs, we will get (p &#8211; r) (q &#8211; s).<\/p>\n<p>(iv) ab &#8211; 2ac &#8211; db + 2dc.<\/p>\n<p>Solution:<br \/>\nThe given expression is ab &#8211; 2ac &#8211; db + 2dc.<br \/>\nGroup the first two terms and last two terms.<br \/>\nFist two terms are ab \u2013 2ac and the last two terms are \u2013 db + 2dc.<br \/>\n(ab &#8211; 2ac) &#8211; (db &#8211; 2dc).<br \/>\nFind out the common factor from the above two groups.<br \/>\na(b &#8211; 2c) -d(b &#8211; 2c).<br \/>\nFactor out the terms in terms of product.<br \/>\n(b &#8211; 2c) (a &#8211; d).<\/p>\n<p>By factorizing the expression ab &#8211; 2ac &#8211; db + 2dc., we will get (b &#8211; 2c) (a &#8211; d).<\/p>\n<p>(v) pq\u00b2 \u2013 3rqs \u2013 pqs + 3rs\u00b2<\/p>\n<p>Solution:<br \/>\nThe given expression is pq\u00b2 \u2013 3rqs \u2013 pqs + 3rs\u00b2<br \/>\nGroup the first two terms and last two terms.<br \/>\nFirst two terms are pq^2 \u2013 3rqs and the last two terms are \u2013pqs + 3rs\u00b2.<br \/>\n(pq\u00b2 \u2013 3rqs) &#8211; (pqs &#8211; 3rs\u00b2).<br \/>\nFind out the common factor from the above two groups.<br \/>\nq(pq &#8211; 3rs) -s(pq &#8211; 3rs).<br \/>\nFactor out the terms in terms of product.<br \/>\n(pq &#8211; 3rs) (q &#8211; s).<\/p>\n<p>By factorizing the expression pq\u00b2 \u2013 3rqs \u2013 pqs + 3rs\u00b2, we will get (pq &#8211; 3rs) (q &#8211; s).<\/p>\n<p>4. Factor each of the following expressions by grouping<\/p>\n<p>(i) a\u00b2 \u2013 3a &#8211; ab + 3b.<\/p>\n<p>Solution:<br \/>\nGiven expression is a\u00b2 \u2013 3a &#8211; ab + 3b<br \/>\nGroup the first two terms and last two terms.<br \/>\nFirst two terms are a\u00b2 \u2013 3a and the last two terms are \u2013 ab + 3b.<br \/>\n(a\u00b2 &#8211; 3a) &#8211; (ab &#8211; 3b).<br \/>\nFind out the common factor from the above two groups.<br \/>\na(a &#8211; 3) &#8211; b(a &#8211; 3).<br \/>\nFactor out the terms in terms of product.<br \/>\n(a &#8211; 3) (a &#8211; b).<\/p>\n<p>By factorizing the expression a\u00b2 \u2013 3a &#8211; ab + 3b, we will get (a &#8211; 3) (a &#8211; b).<\/p>\n<p>(ii) pq\u00b2 + rq\u00b2 + 2p + 2r.<\/p>\n<p>Solution:<br \/>\nGiven expression is pq\u00b2 + rq\u00b2 + 2p + 2r<br \/>\nGroup the first two terms and last two terms.<br \/>\nFirst two terms are pq\u00b2 + rq\u00b2 and the last two terms are 2p + 2r.<br \/>\n(pq\u00b2 + rq\u00b2) + (2p + 2r).<br \/>\nFind out the common factor from the above two groups.<br \/>\nq\u00b2(p +r) + 2(p+r).<br \/>\nFactor out the terms in terms of product.<br \/>\n(q\u00b2+ 2) (p + r).<\/p>\n<p>By factorizing the expression pq\u00b2 + rq\u00b2 + 2p + 2r, we will get (q\u00b2+ 2) (p + r).<\/p>\n<p>(iii) 2pq\u00b2 + 3pqr \u2013 2sqr \u2013 3sr\u00b2<\/p>\n<p>Solution:<br \/>\nGiven expression is 2pq\u00b2 + 3pqr \u2013 2sqr \u2013 3sr\u00b2<br \/>\nGroup the first two terms and last two terms.<br \/>\nFirst two terms are 2pq\u00b2 + 3pqr and the last two terms are \u2013 2sqr \u2013 3sr\u00b2.<br \/>\n(2pq\u00b2 + 3pqr) &#8211; ( 2sqr + 3sr\u00b2).<br \/>\nFind out the common factor from the above two groups.<br \/>\npq(2q + 3r) &#8211; sr(2q + 3r).<br \/>\nFactor out the terms in terms of product.<br \/>\n(2q + 3r) (pq &#8211; sr).<\/p>\n<p>By factorizing the expression 2pq\u00b2 + 3pqr \u2013 2sqr \u2013 3sr\u00b2, we will get (2q + 3r) (pq &#8211; sr).<\/p>\n<p>(iv) par\u00b2 + qars \u2013 pnrs \u2013 qns\u00b2<\/p>\n<p>Solution:<br \/>\nThe given expression is par\u00b2 + qars \u2013 pnrs \u2013 qns\u00b2<br \/>\nGroup the first two terms and last two terms.<br \/>\nFirst two terms are par\u00b2 + qars and the last two terms are \u2013 pnrs \u2013 qns\u00b2.<br \/>\n(par\u00b2 + qars) &#8211; (pnrs \u2013 qns\u00b2).<br \/>\nFind out the common factor from the above two groups.<br \/>\nar(pr + qs) &#8211; ns(pr + qs).<br \/>\nFactor out the terms in terms of product.<br \/>\n(pr + qs) (ar &#8211; ns).<\/p>\n<p>By factorizing the expression par\u00b2 + qars \u2013 pnrs \u2013 qns\u00b2, we will get (pr + qs) (ar &#8211; ns).<\/p>\n<p>5. Factorize<br \/>\n(i) (a + b) (2a + 5) &#8211; (a + b) (a + 3).<\/p>\n<p>Solution:<br \/>\nThe given expression is (a + b) (2a + 5) &#8211; (a + b) (a + 3).<br \/>\nFind out the common factor from the above expression.<br \/>\n(a + b) [(2a + 5) &#8211; (a + 3)].<br \/>\nExpand the terms.<br \/>\n(a + b)(2a + 5 &#8211; a &#8211; 3).<br \/>\nSimplify the second term.<br \/>\n(a + b) (a + 2).<\/p>\n<p>By factorizing the expression (a + b) (2a + 5) &#8211; (a + b) (a + 3), we will get (a + b)(a + 2).<\/p>\n<p>(ii) 6xy &#8211; y\u00b2 + 12xz \u2013 2yz.<\/p>\n<p>Solution:<br \/>\nThe given expression is 6xy &#8211; y\u00b2 + 12xz \u2013 2yz.<br \/>\nGroup the first two terms and last two terms.<br \/>\nFirst two terms are 6xy &#8211; y\u00b2 and the last two terms are 12xz \u2013 2yz.<br \/>\n(6xy &#8211; y\u00b2) + (12xz &#8211; 2yz).<br \/>\nFind out the common factor from the above two groups.<br \/>\ny(6x &#8211; y)+ 2z(6x &#8211; y).<br \/>\nFactor out the terms in terms of product.<br \/>\n(6x &#8211; y) (y + 2z).<\/p>\n<p>By factorizing the expression 6xy &#8211; y\u00b2 + 12xz \u2013 2yz, we will get (6x &#8211; y) (y + 2z).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Are you looking for the different problems on Factorization by Grouping? Then, you are in the right place. We have given all types of factorization problems on our website. Students can learn to Factorize by Grouping the Terms in these articles. While solving Factorization by grouping problems, students need to group the terms with common &#8230; <a title=\"Factorization by Grouping | What is Factorization by Grouping?\" class=\"read-more\" href=\"https:\/\/bigideasmathanswers.com\/factorization-by-grouping\/\" aria-label=\"More on Factorization by Grouping | What is Factorization by Grouping?\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[5],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/34945"}],"collection":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/comments?post=34945"}],"version-history":[{"count":1,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/34945\/revisions"}],"predecessor-version":[{"id":92677,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/34945\/revisions\/92677"}],"wp:attachment":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/media?parent=34945"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/categories?post=34945"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/tags?post=34945"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}