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)