Factorization by grouping means that we need to group the terms with common factors before factoring.
Method of factorization by grouping the terms:
(i) From the groups of the given expression a factor can be taken out from each group.
(ii) Factorize each group
(iii) Now take out the factor common to group formed.
Now we will learn how to factor the terms by grouping.
Solved examples of factorization by grouping:
1. Factor grouping the expressions:
1 + a + ac + a^{2}c= (1 + a) + ac (1 + a)
= (1 + a) (1 + ac).
2. How to factor by grouping the following algebraic expressions?
= a(a  c) + b(a  c)
= (a  c) (a + b)
Therefore, by
factoring expressions we get (a  c)(a + b)
= a(a + 3) + c(a + 3)
= (a + 3) (a + c)
Therefore, by
factoring expressions we get (a + 3)(a + c)
3. Factorize the algebraic expressions:
= x(2 + c) + c(2 + c)
= (2 + c) (x + c)
= x(x  a) + 5(x  a)
= (x  a) (x + 5)
(iii) ax  bx  az + bz
Solution:
ax  bx  az + bz
= x(a  b)  z(a  b)
= (a  b) (x  z)
(iv) mx  2my  nx + 2ny
Solution:
mx  2my  nx + 2ny
= m(x  2y)  n(x  2y)
= (x  2y) (m  n)
= x(ax – 3by) – y(ax – 3by)
= (ax  3by) (ax  y)
4. Factor each of the following
expressions by grouping:
= x(x – 3) – y(x – 3)
= (x – 3) (x – y)
= ax(2x + 3y)  by(2x + 3y)
= (2x + 3y) (ax  by)
= mx(ax + by) – ny(ax + by)
= (ax + by) (mx – ny)
5. Factorize:
(i) (x + y) (2x +
5)  (x + y) (x + 3)
Solution:
(x + y) (2x + 5)  (x + y) (x + 3)
= (x + y) [(2x + 5)  (x + 3)]
= (x + y) [2x + 5  x 3]
= (x + y) (x + 2)
= b(6a  b) + 2c(6a  b)
= (6a  b) (b + 2c)
8th Grade Math Practice
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