## Factorization

1. From the given expression, first observe its form. Then use the appropriate formula and obtain the factors.
2. We derive formulae for some products. We are given the product of two factors and we need to find those factors.

Expression                                         Factors

a2 + 2ab + b2                                     ( a + b) 2

a2- 2ab + b2                                       ( a - b) 2

a2 - b2                                                 ( a + b ) ( a - b )

a3+ b3                                                 ( a + b ) ( a2 - a b + b2 )

a2 - b2                                                 ( a - b ) ( a2+ a b + b2 )

a3+ 3a2b + 3ab2 + b3                       ( a + b )3

a3 - 3a2 b + 3ab2- b3                         ( a - b )3

a2+ b2+ c2 + 2ab + 2bc + 2ca         ( a+ b + c) 2

a3+ b3 + c3 - 3abc                             ( a + b + c ) ( a2+ b2 + c2 - a b - b c - ca )

x2+ x (a + b ) + ab                             ( x + a ) ( x + b )

x3+ x2 ( a + b + c )                             ( x + a ) ( x + b ) ( x + c ) + x ( a b + b c + c a ) + abc

#### Factorize each of the following polynomials

Example 1

x2 + 6x + 9

Solution:

Using the product with a = x, b = 3, we have

x2 + 6x + 9 = ( x + 3) 2

Example 2

1 - 8x + 16x2

Solution:

Using the product with a = 1, b = 4x, we have

1 - 8x + 16x2= ( 1 - 4x) 2

Example 3

4x2 - 81y2

Solution:

Using the product with a = 2x, b = 9y, we have

4x2 - 81y2 = ( 2x) 2 - ( 9y) 2

= ( 2x - 9y ) ( 2x + 9y )

Example 4

x3y3 + 729

Solution:

Using the product with a = xy, b = 9, we have

x3y3 + 729 = (xy)3 + (9)3

= ( xy + 9 ) ( x2y2 - 9xy + 81 )

Example 5

x3 - 6x2+ 12x - 8

Solution:

Using the product with a = x, b = 2, we have

x3 - 6x2+ 12x - 8 = ( x - 2 )3

Example 6

x3 + y3 - z3+ 3xyz

Solution:

Using the product with a = x, b = y, c = -z, we have

x3 + y3 - z3+ 3xyz

= ( x + y - z ) ( x2+ y2+ z2 - xy + yz + zx )

Example 7

n4+ 4

Solution:

n4+ 4 is in the form of a2 + b2, with a = n2, b = 2

n4+ 4 = a2+ b2

= ( a2+ b2+ 2ab ) - 2ab

= ( n4+ 4+ 2n2 (2)) - 2n2(2)

= ( n2+ 22- 4n)2

= (n2 + 22 - (2n)2

which is of the form a2- b2 with a = n2 + 2, b = 2n.

Therefore, ( n2 + 2) 2 - (2n2 = ( n2+ 2n + 2 ) ( n2 - 2n + 2 ).

Therefore, n4 + 4 = ( n2 + 2n + 2 ) ( n2 - 2n + 2 ).

#### Try these questions

1. a2+ 4a + 4
2. a4+ 6a2+ 9a
3. ( 2x + 3y 2 + 2 ( 2x + 3y ) ( x + y ) + ( x + y 2
4. 4a2 - 12ab + 9b2
5. 9a2b2 - 6abc + c2
6. ( 3x + 2y 2 - 2 ( 3x + 2y ) ( x + y ) + ( x + y) 2
7. 4a2 - 9
8. 16x2 - 9y2
9. a2b2 - c2d2
10. 25a2 - 16a
11. a4- 1
12. (a + b2 - c)2
13. a2 - ( b - c) 2
14. 4a2 + 4ab + b2 - c2
15. a2 - b2 - c2 - 2bc
16. a2 - b2 - 4c2 + 4bc
17. x4 + 324
18. 4a4 + 81
19. 3x4 + 12
20. x4 + x2 + 1

Answers

1. a2 + 4a + 4 = (a2 + 2 (a) (2) + (22
= ( a + 2) 2

2. a3+ 6a2+ 9a = a ( a2 + 6a + 9 )
= a [ (a2 + 2 (a) (3) + (3) 2 ]
= a [ a + 3 ]2

3. ( 2x + 3y 2 + 2 ( 2x + 3y ) ( x + y ) + ( x + y) 2
Put 2x + 3y = a; x + y = b
Therefore given expression = a2+ 2ab + b2
= ( a + b) 2
= ( 2x + 3y + x + y) 2
= ( 3x + 4y) 2

4. 4a2- 12ab + 9b2 = ( 2a )2 + 2 ( 2a ) ( - 3b ) + ( - 3b )2
= ( 2a - 3b )2

5. 9a2b2- 6abc + c2
= ( 3ab )2 + 2 ( 3ab ) ( - c ) + ( - c )2
= ( 3ab - c )2

6. ( 3x + 2y )2 - 2 ( 3x + 2y ) ( x + y ) + ( x + y )2
This is in the form of a2 - 2ab + b2
Therefore factors = ( a - b )2
i.e., ( 3x + 2y - x - y )2 = ( 2x + y )2

7. 4a2 - 9 = ( 2a )2 - (3)2
[Therefore a2 - b2 = ( a + b ) ( a - b ) ]
= ( 2a + 3 ) ( 2a - 3 )

8. 16x2- 9y2 = ( 4x )2 - ( 3y )2
= ( 4x + 3y ) ( 4x - 3y )

9. a2b2 - c2d2= ( ab )2 - ( cd )2
= ( ab + cd ) ( ab - cd )

10. 25a2 - 16a = a ( 25a2 - 16 )
= a [ ( 5a )2- ( 4 )2]
= a ( 5a + 4 ) ( 5a - 4 )

11. a4- 1 = ( a2 ) 2- (1)2
= ( a2 + 1 ) ( a2 - 1 )
= ( a2 + 1 ) ( a + 1 ) ( a - 1 )

12. ( a + b )2 - c2 = ( a + b + c ) ( a + b - c )

13. a2- ( b - c )2 = ( a )2 - ( b - c )2
= ( a + b - c ) ( a - b + c )

14. 4a2 + 4ab + b2 - c2 = ( 2a )2 + 2 ( 2a ) ( b ) + ( b )2 - c2
= ( 2a + b )2 - ( c )2
= ( 2a + b + c ) ( 2a + b - c )

15. a2 - b2 - c2- 2bc = a2 - ( b2 + c2 + 2bc )
= ( a )2 - ( b + c )2
= ( a + b + c ) ( a - b - c )

16. a2 – b2 - 4c2 + 4bc = a2- ( b2+ 4c2 - 4bc )
= ( a )2- ( b - 2c )2
= ( a + b - 2c ) ( a - b + 2c )

17. x4 + 324 = ( x )2 + (18)2 + 2 ( x )2 (18) - 36x2
= ( x2 + 18 )2 - ( 6x )2
= ( x2 + 18 + 6x ) ( x2 + 18 - 6x )
= ( x2 + 6x + 18 ) ( x2 - 6x + 18 )

18. 4a4 + 81 = ( 2a2 )2 + (9)2 + 2( 2a2 ) (9) -36a2
= ( 2a2 + 9 )2 - ( 6a )2
= ( 2a2 + 9 + 6a ) ( 2a2 + 9 + 6a )
= ( 2a2 - 6a + 9 ) ( 2a2 + 6a + 9 )

19. 3x4 + 12 = 3 [ x4 + 4 ]
= 3 [ ( x2 )2 + (2)2 + 2 ( x2 ) (2) - 4x2 ]
= 3 [ ( x2 + 2 )2 - ( 2x )2 ]
= 3 ( x2 + 2x + 2 ) ( x2 - 2x + 2 )

20. x4 + x2 + 1 = ( x2 )2 + (1)2 + 2 ( x2 ) (1) - x2
= ( x2 + 1 )2 - (x)2
= ( x2 + x + 1 ) ( x2 - x + 1 )