# FACTORING BY GROUPING

Fully factorise :

Problem 1 :

ab + ac – 2a

Solution :

Given, ab + ac – 2a

We find a in common in all three terms, we factor "a" out

a(b + c - 2)

Problem 2 :

a2b2 – 2ab

Solution :

Given, a2b2 – 2ab

Factoring ab, we get

ab(ab - 2)

Problem 3 :

18x – 2x3

Solution :

= 18x – 2x3

= 2x(9 – x2)

= 2x(32 - x2)

= 2x(3 + x) (3 - x)

Problem 4 :

x2 + 14x + 49

Solution :

= x2 + 14x + 49

By decomposing the middle term, we get

= x2 + 7x + 7x + 49

= x(x + 7) + 7(x + 7)

= (x + 7)(x + 7)

= (x + 7)2

Problem 5 :

4a3 – 4ab2

Solution :

= 4a3 – 4ab2

= 4a(a2 – b2)

Using the algebraic identity, a2 – b2 = (a + b)(a - b)

= 4a(a + b) (a - b)

Problem 6 :

x3y – 4xy

Solution :

= x3y – 4xy

= xy(x2 - 4)

= xy(x2 - 22)

= xy(x + 2) (x - 2)

Problem 7 :

4x4 – 4x2

Solution :

= 4x4 – 4x2

= 4x2(x2 - 1)

= 4x(x + 1) (x - 1)

Problem 8 :

(x - 2)y – (x - 2)z

Solution :

= (x - 2)y – (x - 2)z

= (x - 2)(y - z)

Problem 9 :

(x + 1)a + (x + 1)b

Solution :

= (x + 1)a + (x + 1)b

= (x + 1)[a + b]

Problem 10 :

(x - y)a + (x - y)

Solution :

= (x - y)a + (x - y)

(x - y)(a + 1)

Problem 11 :

x(x + 2) + 3(x + 2)

Solution :

= x(x + 2) + 3(x + 2)

= (x + 2)(x + 3)

Problem 12 :

x3 + x2 + x + 1

Solution :

= x3 + x2 + x + 1

= x2(x + 1) + 1(x + 1)

= (x2 + 1) (x + 1)

Factorise completely :

Problem 13 :

7x - 35y

Solution :

= 7x - 35y

Factoring 7, we get

= 7(x - 5y)

Problem 14 :

= 2g2 - 8

Solution :

= 2g2 - 8

Factoring 2, we get

= 2(g2 - 4)

Problem 15 :

-5x2 - 10x

Solution :

= -5x2 - 10x

Factoring -5x, we get

= -5x(x + 2)

Problem 16 :

m2 + 3mp

Solution :

= m2 + 3mp

= m(m + 3p)

Problem 17 :

a2 + 8a + 15

Solution :

= a2 + 8a + 15

= a2 + 3a + 5a + 15

= a(a + 3) + 5(a + 3)

= (a + 3) (a + 5)

Problem 18 :

m2 - 6m  + 9

Solution :

= m2 - 6m  + 9

= m2 - 3m - 3m + 9

= m(m - 3) - 3(m - 3)

= (m - 3)2

Problem 19 :

5x2 + 5xy - 5x2y

Solution :

= 5x2 + 5xy - 5x2y

= 5x(x + y - xy)

Problem 20 :

xy + 2x + 2y + 4

Solution :

= xy + 2x + 2y + 4

= x(y + 2) + 2(y + 2)

= (x + 2) (y + 2)

Problem 21 :

y2 + 5y - 9y - 45

Solution :

= y2 + 5y - 9y - 45

= y(y + 5) - 9(y + 5)

= (y - 9) (y + 5)

Problem 22 :

2x2 + 10x + x + 5

Solution :

= 2x2 + 10x + x + 5

= 2x(x + 5) + (x + 5)

= (2x + 1) (x + 5)

Problem 23 :

3y2 - 147

Solution :

= 3y2 - 147

= 3 (y2 - 49)

= 3(y2 - 72)

= 3(y + 7) (y - 7)

Problem 24 :

3p2 - 3q2

Solution :

= 3p2 - 3q2

= 3(p2 - q2)

= 3(p + q) (p - q)

Problem 25 :

4c2 - 1

Solution :

= 4c2 - 1

= (2c)2 - 12

= (2c - 1) (2c + 1)

Problem 26 :

3x2 + 3x - 36

Solution :

= 3x2 + 3x - 36

= 3(x2 + x - 12)

= 3(x2 - 3x + 4x - 12)

= 3(x(x  - 3) + 4(x - 3))

= 3(x - 3) (x - 4)

Problem 27 :

2bx - 6b + 10x - 30

Solution :

= 2bx - 6b + 10x - 30

= 2(bx - 3b + 5x - 15)

= 2(b(x - 3) + 5(x - 3))

= 2(x - 3) (b + 5)

Fully factorise :

Problem 28 :

12 - 11x - x2

Solution :

= 12 - 11x - x2

= -x2 - 11x + 12

= -(x2 + 11x - 12)

= -(x2 - x + 12x - 12)

= -(x(x - 1) + 12(x - 1))

= -(x + 12) (x - 1)

Problem 29 :

-2x2 - 6 + 8x

Solution :

= -2x2 - 6 + 8x

= -2x2 + 8x - 6

= -2(x2 - 4x + 3)

= -2(x2 - x - 3x + 3)

= -2(x(x - 1) - 3(x - 1))

= -2(x - 1) (x - 3)

Problem 30 :

14 - x2 - 5x

Solution :

= 14 - x2 - 5x

= -x2 - 5x + 14

= -(x2 + 5x - 14)

= -(x2 - 2x + 7x - 14)

= -(x(x - 2) + 7(x - 2))

= -(x + 7) (x - 2)

Problem 31 :

4x2 - 2x3 - 2x

Solution :

= 4x2 - 2x3 - 2x

= 2x(2x - x2 - 1)

2x(-x2 + 2x - 1)

2x(-(x2 - 2x + 1))

2x(-(x - 1)2)

-2x(x - 1)2

Problem 32 :

(a + b)2 - 9

Solution :

= (a + b)2 - 9

= (a + b)2 - 32

= (a + b - 3) (a + b + 3)

Problem 33 :

(x + 2)2 - 4

Solution :

= (x + 2)2 - 4

Let t = (x + 2)

= t2 - 22

= (t + 2)(t - 2)

Applying the value of t, we get

= (x + 2 + 2) (x + 2 - 2)

= (x + 4)(x)

= x (x + 4)

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