Factorization Any QPolynomial

Observe the following examples

1. (3x +2) (4x +3) = 3x(4x + 3) + 2(4x + 3)
= 12x2 + 9x + 8x + 6
= 12x2 + 17x + 6
The coefficient of x = 17 = 9 + 8
9 *  8 = 72 = 12 * 6

2. (2x + 3) (4x - 5) = 2x(4x - 5) + 3(4x - 5)
= 8x2 - 10x + 12x - 15
= 8x2 + 2x - 15
The coefficient of x = 2 = 12 - 10
12 *-10 = -120 = 8?? - 15

These examples suggest the following method of factorization for a general quadratic expression.

Method of factorization of quadratic expressions

1. Multiply the coefficient of x by the constant term.
2. Resolve this product into two factors such that their sum is the coefficient of x.
3. Rewrite the x term as the sum of two terms with these coefficients.
4. Group them into two parts, each containing two terms, and factorize.

Example 1

x2 - 2x - 63

Here, the coefficient of x is 1 and the constant term is -63.

So, 1 *-63 = -63

Here, -9 *  7 = -63

-2x = -9x + 7x

x2 - 2x - 63 = x2 - 9x + 7x - 63

= x ( x - 9 ) + 7 ( x - 9 )

= ( x - 9 ) ( x + 7 )

Example 2

Factorize 2x2 + 7x + 6

Here, 2 *  6 = 12

7 = 4 + 3; 4 *  3 = 12

Therefore, 2x2 + 7x + 6 = 2x2 + 4x + 3x + 6

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

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

Example 3

Factorize 3x2 - 11x + 6

3 *  6 = 18

-11x = -9x - 2x; -9 *  -2 = 18

3x2 - 11x + 6 = 3x2 - 9x - 2x + 6

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

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

Try these questions

I. Factorize the following

1. 2x2 + 7x + 6
2. 2x2 + x - 6
3. 2x2 - x - 6
4. 2x2 - 7x + 6
5. 3x2 + 17x + 20
6. 3x2 - 17x + 20
7. 3x2 - 17x - 20
8. 7x2 - 8x - 12
9. 6x2 - 5x -14
10. 3x2 - 16x + 16
11. 6 - x - 2x2
12. 6 + 7x - 3x2
13. 12 - 4x - 5x2
14. 16 + 8x - 3x2
15. 3x2 + 8xy + 4y2
16. 4x2 + 12xy + 5y2
17. 4x4 - 5x2 + 1
18. 9x4 - 40x2 + 16
19. 4x2- 25x2 + 36
20. 8x6- 65x3+ 8

1. 2x2 + 7x + 6 = 2x2 + 4x + 3x + 6
= 2x ( x + 2 ) + 3( x + 2 )
= ( x + 2 ) ( 2x + 3 )

2. 2x2 + x - 6 = 2x2 + 4x - 3x - 6
= 2x ( x + 2 ) - 3 ( x + 2 )
= ( x + 2 ) ( 2x - 3 )

3. 2x2 - x - 6 = 2x2 - 4x + 3x - 6
= 2x ( x - 2 ) + 3 ( x - 2 )
= ( x - 2 ) ( 2x + 3 )

4. 2x2 - 7x + 6 = 2x2 - 4x - 3x + 6
= 2x ( x - 2 ) - 3 ( x - 2 )
= ( x - 2 ) ( 2x - 3 )

5. 3x2 + 17x + 20 = 3x2 + 12x + 5x + 20
= 3x ( x + 4 ) + 5 ( x + 4 )
= ( x + 4 ) ( 3x + 5 )

6. 3x2 - 17x + 20 = 3x2 - 12x - 5x + 20
= 3x ( x - 4 ) - 5 ( x - 4 )
= ( x - 4 ) ( 3x - 5 )

7. 3x2 - 17x - 20 = 3x2 + 3x - 20x - 20
= 3x ( x + 1 ) - 20 ( x + 1 )
= ( x + 1 ) ( 3x - 20 )

8. 7x2 - 8x - 12 = 7x2 - 14x + 6x - 12
= 7x ( x - 2 ) + 6 ( x - 2 )
= ( x - 2 ) ( 7x + 6 )

9. 6x2 - 5x -14 = 6x2 - 12x + 7x - 14
= 6x ( x - 2 ) + 7 ( x - 2 )
= ( x - 2 ) ( 6x + 7 )

10. 3x2 - 16x + 16 = 3x2 - 12x - 4x + 16
= 3x ( x - 4 ) - 4 ( x - 4 )
= ( x - 4 ) ( 3x - 4 )

11. 6 - x - 2x2 = - ( 2x2 + x - 6)
= - [ 2x2 + 4x - 3x - 6 ]
= - [ 2x ( x + 2 ) - 3 ( x + 2 ) ]
= - [ ( x + 2 ) ( 2x - 3 ) ]
= ( x + 2 ) ( 3 - 2x )

12. 6+ 7x - 3x2 = [ 3x2 - 7x - 6]
= [ 3x2 - 9x + 2x - 6 ]
= [ 3x - 9x + 2x - 6 ]
= - [ ( x - 3 ) ( 3x + 2 ) ]
= ( 3 - x ) ( 3x + 2 )

13. 12 - 4x - 5x2 = - [ 5x2 + 4x - 12 ]
= - [ 5x2 + 10x - 6x - 12 ]
= - [ 5x ( x + 2 ) - 6 ( x + 2 ) ]
= - [ ( x + 2 ) ( 5x - 6 ) ]
= ( x + 2 ) ( 6 - 5x )

14. 16 + 8x - 3x2 = - [ 3x2 - 8x - 16 ]
= - [ 3x2 - 12x + 4x - 16 ]
= - [ 3x ( x - 4 ) + 4 ( x - 4 ) ]
= - [ ( x - 4 ) ( 3x + 4 ) ]
= ( 3x + 4 ) ( 4 - x )

15. 3x2 + 8xy + 4y2 = 3x2 + 6xy + 2xy + 4y2
= 3x ( x + 2y ) + 2y ( x + 2y )
= ( x + 2y ) ( 3x + 2y )

16. 4x2 + 12xy + 5y2 = 4x2 + 2xy + 10xy + 5y2
= 2x ( 2x + y ) + 5y ( 2x + y )
= ( 2x + y ) ( 2x + 5y )

17. 4x4 - 5x2 + 1 = 4x4- 4x2 - x2 + 1
= 4x2 ( x2 - 1 ) - 1 ( x2 - 1 )
= ( x2 - 1 ) ( 4x2 - 1 )
= ( x + 1 ) ( x - 1 ) ( 2x + 1 ) ( 2x - 1 )

18. 9x4 - 40x2 + 16 = 9x4- 36x2 - 4x2 + 16
= 9x2 ( x2 - 4 ) - 4 ( x2 - 4 )
= ( x2 - 4 ) ( 9x2 - 4 )
= ( x + 2 ) ( x - 2 ) ( 3x + 2 ) ( 3x - 2 )

19. 4x4- 25x2 + 36 = 4x4- 16x2 - 9x2 + 36
= 4x2 ( x2 - 4 ) - 9 ( x2 - 4 )
= ( x2 - 4 ) ( 4x2 - 9 )
= ( x + 2 ) ( x - 2 ) ( 2x + 3 ) ( 2x - 3 )

20. 8x6- 65x3+ 8 = 8x6- 64x3- x3+ 8
= 8x3( x3 - 8 ) - 1 ( x3 - 8 )
= ( x3- 8 ) ( 8x3- 1 )
= ( x - 2 ) ( x2 + 2x + 4 ) ( 2x - 1 ) ( 4x2 + 2x + 1 )