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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 )

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

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 )
 

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