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## Factorization of Any Quadratic Polynomial

### 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 )

### 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

### Answers to Practice Problems

 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 )