{"id":34948,"date":"2023-10-21T13:17:42","date_gmt":"2023-10-21T07:47:42","guid":{"rendered":"https:\/\/ccssmathanswers.com\/?p=34948"},"modified":"2023-11-10T11:51:07","modified_gmt":"2023-11-10T06:21:07","slug":"factorize-by-grouping-the-terms","status":"publish","type":"post","link":"https:\/\/bigideasmathanswers.com\/factorize-by-grouping-the-terms\/","title":{"rendered":"Factorize by Grouping the Terms | How to Factor by Grouping Terms?"},"content":{"rendered":"

Factorize by grouping the terms consists of two or more product terms as resultant value. To divide the expression into product terms, we need to identify the greatest common factor which divides all the remaining terms in the expression. Step by step process on how to do Factorization<\/strong><\/a> by Grouping the Terms are clearly given in this article.<\/p>\n

Procedure to Factorize by Grouping the Terms<\/h2>\n

Refer to the below-mentioned step by step process and learn the Factoring by Grouping Terms. They are along the lines<\/p>\n

    \n
  1. Note down the given expression.<\/li>\n
  2. Group the first two terms and last two terms.<\/li>\n
  3. Factor out the greatest common factor from each group.<\/li>\n
  4. Finally, you will get two or more product terms as result.<\/li>\n<\/ol>\n

    Solved Examples on Factorization by Grouping Terms<\/h3>\n

    1. Factorize grouping the following expressions?<\/p>\n

    (i) 18x\u00b3y\u00b3 \u2013 27x\u00b2y\u00b3 + 36x\u00b3y\u00b2<\/p>\n

    Solution:
    \nThe given expression is 18x\u00b3y\u00b3 \u2013 27x\u00b2y\u00b3 + 36x\u00b3y\u00b2
    \nFactor out the greatest common factor from the above expression.
    \nThat is, 9x\u00b2y\u00b2(2xy \u2013 3y + 4x).<\/p>\n

    Therefore, the final solution for the expression 18x\u00b3y\u00b3 \u2013 27x\u00b2y\u00b3 + 36x\u00b3y\u00b2 is 9x\u00b2y\u00b2(2xy \u2013 3y + 4x).<\/p>\n

    (ii) 12a\u00b2b\u00b3 \u2013 21a\u00b3b\u00b2<\/p>\n

    Solution:
    \nThe given expression is 12a\u00b2b\u00b3 \u2013 21a\u00b3b\u00b2
    \nFactor out the greatest common factor from the above expression.
    \n3a\u00b2b\u00b2(4b \u2013 7a).<\/p>\n

    Finally, the solution for the expression 12a\u00b2b\u00b3 \u2013 21a\u00b3b\u00b2 is 3a\u00b2b\u00b2(4b \u2013 7a).<\/p>\n

    (iii) a\u00b3 – a\u00b2 + a \u2013 1.<\/p>\n

    Solution:
    \nThe given expression is a\u00b3 – a\u00b2 + a \u2013 1
    \nGroup the first two terms and last two terms.
    \nHere, first two terms are a\u00b3 – a\u00b2 and the last two terms are a \u2013 1.
    \nSo, (a\u00b3 – a\u00b2) + (a \u2013 1).
    \nNow, factor out the greatest common factor from each group.
    \nThat is, a\u00b2(a \u2013 1) + (a \u2013 1).
    \n(a \u2013 1) (a\u00b2 + 1).<\/p>\n

    Therefore, solution for the expression a\u00b3 – a\u00b2 + a \u2013 1 is (a \u2013 1) (a\u00b2 + 1).<\/p>\n

    (iv) prs + qurs \u2013 pt \u2013 qut<\/p>\n

    Solution:
    \nThe given expression is prs + qurs \u2013 pt \u2013 qut.
    \nGroup the first two terms and last two terms.
    \nHere, the first two terms are prs + qurs and last two terms are \u2013 pt \u2013 qut.
    \nThen, (prs + qurs) \u2013 (pt + qut).
    \nNow, factor out the greatest common factor from the above two groups.
    \nThat is, rs(p + qu) \u2013 t(p + qu).
    \n(p + qu) (rs \u2013 t).<\/p>\n

    Therefore, the solution for the expression prs + qurs \u2013 pt \u2013 qut is (p + qu) (rs \u2013 t).<\/p>\n

    (v) a\u00b2 \u2013 3a \u2013 ab + 3b
    \nSolution:<\/p>\n

    The given expression is
    \na\u00b2 \u2013 3a \u2013 ab + 3b.
    \nGroup the first two terms and last two terms.
    \nHere, the first two terms are a\u00b2-3a and the last two terms are ab + 3b.
    \nThen, (a\u00b2 – 3a) \u2013 (ab – 3b).
    \nNow, factor out the greatest common factor from the above two groups.
    \nThat is a(a – 3) – b(a – 3).
    \n(a – 3) (a – b).<\/p>\n

    Therefore, solution for the expression is a\u00b2 \u2013 3a \u2013 ab + 3b is (a – 3) (a – b).<\/p>\n

    2. How to factorize by grouping the following expressions?
    \n(i) 2q^4 \u2013 q\u00b3 + 4q – 2
    \nSolution:
    \nThe given expression is
    \n2q^4 \u2013 q\u00b3 + 4q – 2.
    \nGroup the first two terms and last two terms.
    \nHere, the first two terms are 2q^4 \u2013 q\u00b3 and the last two terms are 4q – 2.
    \nThen, (2q^4 – q^3) + (4q-2).
    \nNow, factor out the greatest common factor from the above two groups.
    \nThat is, q^3(2q – 1) +2(2q – 1).
    \n(2q – 1) (q^3 + 2).<\/p>\n

    Therefore, solution for the expression is 2q4 \u2013 q3 + 4q \u2013 2 is (2q-1) (q^3+2).<\/p>\n

    (ii) ac + bc – ad \u2013 bd.<\/p>\n

    Solution:
    \nThe given expression is ac + bc – ad \u2013 bd.
    \nGroup the first two terms and last two terms.
    \nHere, the first two terms are ac+bc and the last two terms are \u2013 ad-bd.
    \nThen, (ac + bc) – (ad + bd).
    \nNow, factor out the greatest common factor from the above two groups.
    \nThat is, c(a+b) -d(a+b).
    \n(a+b) (c-d).<\/p>\n

    Therefore, solution for the expression is ac + bc – ad \u2013 bd is (a+b) (c-d).<\/p>\n

    (iii) pa – pb – qa \u2013qb.<\/p>\n

    Solution:
    \nThe given expression is pa – pb – qa\u2013qb.
    \nGroup the first two terms and last two terms.
    \nHere, the first two terms are pa – pb, and the last two terms are qa – qb.
    \nThen, (pa – pb) – (qa -qb).
    \nNow, factor out the greatest common factor from the above two groups.
    \nThat is, p(a – b) \u2013 q(a – b).
    \n(p – q) (a – b).<\/p>\n

    Therefore, the solution for the expression is pa – pb – qa \u2013 qb is (p – q) (a – b).<\/p>\n

    3. How to factorize by grouping the algebraic expressions?
    \n(i)x^2z^2 + xzw + xyz + yw<\/p>\n

    Solution:
    \nThe given expression is
    \nx^2z^2 + xzw + xyz + yw.
    \nGroup the first two terms and last two terms.
    \nHere, the first two terms are x^2z^2 + xzw and the last two terms are xyz + yw.
    \nThen, (x^2z^2 + xzw) + (xyz + yw).
    \nNow, factor out the greatest common factor from the above two groups.
    \nThat is, xz(xz + w) + y(xz + w).
    \n(xz + y) (xz + w).<\/p>\n

    Therefore, solution for the expression is x^2z^2 + xzw + xyz + yw is (xz + y) (xz + w).<\/p>\n

    (ii) 5x + xy + 5y + y^2<\/p>\n

    Solution:
    \nThe given expression is 5x + xy + 5y + y^2
    \nGroup the first two terms and last two terms.
    \nHere, the first two terms are 5x + xy, and the last two terms are 5y + y^2.
    \nThen, (5x + xy) + (5y + y^2).
    \nNow, factor out the greatest common factor from the above two groups.
    \nThat is, x(5 + y) + y(5 + y).
    \n(5 + y) (x + y).<\/p>\n

    Therefore, solution for the expression is 5x + xy + 5y + y^2 is (5 + y) (x + y).<\/p>\n

    (iii) xy – yz – xz + z^2<\/p>\n

    Solution:
    \nThe given expression is xy – yz – xz + z^2
    \nGroup the first two terms and last two terms.
    \nHere, the first two terms are xy – yz and the last two terms are \u2013 xz + z^2.
    \nThen, (xy – yz) – (xz – z^2).
    \nNow, factor out the greatest common factor from the above two groups.
    \nThat is, y(x \u2013 z) \u2013 z(x \u2013 z).
    \n(x – z) (y – z).<\/p>\n

    Therefore, solution for the expression xy – yz – xz + z^2 is (x – z) (y – z).<\/p>\n

    4. Factorize the expressions
    \n(i) a^4 + a^3 + 2a + 2<\/p>\n

    Solution:
    \nThe given expression is a^4 + a^3 + 2a + 2.
    \nGroup the first two terms and last two terms.
    \nHere, the first two terms are a^4 + a^3 and the last two terms are 2a + 2.
    \nThen, (a^4 + a^3) + (2a + 2).
    \nNow, factor out the greatest common factor from the above two groups.
    \nThat is, a^3(a + 1) + 2(a + 1).
    \n(a + 1) (a^3 + 2).<\/p>\n

    Therefore, solution for the expression a^4 + a^3 + 2a + 2 is (a + 1) (a^3 + 2).<\/p>\n

    (ii) a\u00b2b\u00b2 + d\u00b2b\u00b2 \u2013 cd\u00b2 \u2013 ca\u00b2<\/p>\n

    Solution:
    \nThe given expression is a\u00b2b\u00b2 + d\u00b2b\u00b2 \u2013 cd\u00b2 \u2013 ca\u00b2
    \nGroup the first two terms and last two terms.
    \nHere, the first two terms are a\u00b2b\u00b2 + d\u00b2b\u00b2 and the last two terms are \u2013 cd\u00b2 \u2013 ca\u00b2.
    \nThen, (a^2b^2 + d^2b^2) – (cd^2 + ca^2).
    \nNow, factor out the greatest common factor from the above two groups.
    \nThat is, b^2(a^2 + d^2) \u2013 c(d^2 + a^2).
    \n(a^2 + d^2) (b^2 \u2013 c).<\/p>\n

    Therefore, solution for the expression a2b2 + d2b2 \u2013 cd2 \u2013 ca2 is(a^2 + d^2) (b^2 \u2013 c).<\/p>\n

    5. Factorize by grouping the terms
    \n(x^2 + 3x)^2 – 2(x^2 + 3x) \u2013 y(x^2 + 3x) + 2y<\/p>\n

    Solution:
    \nThegiven expression is (x^2 + 3x)^2 – 2(x^2 + 3x) \u2013 y(x^2 + 3x) + 2y.
    \nGroup the first two terms and last two terms.
    \nhere, first two terms are (x^2 + 3x)^2 \u2013 2(x^2 + 3x) and the last two terms are \u2013y(x^2 + 3x) + 2y.
    \nThen, [(x^2 + 3x)^2 \u2013 2(x^2 + 3x)] \u2013[y(x^2 + 3x) – 2y].
    \nThat is, (x^2 + 3x)(x^2 + 3x \u2013 2) \u2013 y(x^2 + 3x \u2013 2).
    \n(x^2 + 3x \u2013 2)(x^2 + 3x \u2013 y).<\/p>\n

    Therefore, solution for the expression (x2 + 3x)2 – 2(x2 + 3x) \u2013 y(x2 + 3x) + 2y is (x^2 + 3x \u2013 2)(x^2 + 3x \u2013 y).<\/p>\n","protected":false},"excerpt":{"rendered":"

    Factorize by grouping the terms consists of two or more product terms as resultant value. To divide the expression into product terms, we need to identify the greatest common factor which divides all the remaining terms in the expression. Step by step process on how to do Factorization by Grouping the Terms are clearly given … 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\/34948"}],"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=34948"}],"version-history":[{"count":1,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/34948\/revisions"}],"predecessor-version":[{"id":92676,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/34948\/revisions\/92676"}],"wp:attachment":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/media?parent=34948"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/categories?post=34948"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/tags?post=34948"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}