# Factorization by Grouping

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 + a2c

Solution:

1 + a + ac + a2c

= (1 + a) + (ac + a2c)

= (1 + a) + ac (1 + a)

= (1 + a) (1 + ac).

2. How to factor by grouping the following algebraic expressions?

(i) a2 - ac + ab - bc

Solution:

a2 - ac + ab - bc

= a(a - c) + b(a - c)

= (a - c) (a + b)

Therefore, by factoring expressions we get (a - c)(a + b)

(ii) a2 + 3a + ac + 3c

Solution:

a2 + 3a + ac + 3c

= 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:

(i) 2x + cx + 2c + c2

Solution:

2x + cx + 2c + c2

= x(2 + c) + c(2 + c)

= (2 + c) (x + c)

(ii) x2 - ax + 5x - 5a

Solution:

x2 - ax + 5x - 5a

= 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)

(v) ax2 - 3bxy – axy + 3by2

Solution:

ax2 - 3bxy – axy + 3by2

= x(ax – 3by) – y(ax – 3by)

= (ax - 3by) (ax - y)

4. Factor each of the following expressions by grouping:

(i) x2 - 3x - xy + 3y

Solution:

x2 - 3x - xy + 3y

= x(x – 3) – y(x – 3)

= (x – 3) (x – y)

(ii) ax2 + bx2 + 2a + 2b

Solution:

ax2 + bx2 + 2a + 2b

= x2(a + b) + 2(a + b)

= (a + b) (x2 + 2)

(iii) 2ax2 + 3axy - 2bxy - 3by2

Solution:

2ax2 + 3axy - 2bxy - 3by2

= ax(2x + 3y) - by(2x + 3y)

= (2x + 3y) (ax - by)

(iv) amx2 + bmxy – anxy – bny2

Solution:

amx2 + bmxy – anxy – bny2

= 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)

(ii) 6ab - b2 + 12ac - 2bc

Solution:

6ab - b2 + 12ac - 2bc

= b(6a - b) + 2c(6a - b)

= (6a - b) (b + 2c)