Factorization of Monic Quadratic Polynomials

Polynomials of degree two are called quadratic expressions. If the coefficient of the second degree term is one, then it is called a monic quadratic polynomial.

ax² + bx + c is the general form of a quadratic expression.

x² + ax + b is a monic quadratic expression.

Example 1

Factorize x² + 12x + 27

Solution:

Note that a quadratic monic polynomial in x is a perfect square if the constant term in it is equal to the square of half the coefficient of x. Therefore, if (12/ 2)2 = 62 = 36 is added to x² + 12x, it becomes a perfect square.

Therefore, the above can be written as

x² + 12x + 27 = ( x² + 12x + 36 ) - 36 + 27

                     = ( x + 6 )2 - 9

                     = ( x + 6 )2 - 32 (in the form of a2 - b2)

                     = ( x + 6 + 3 ) ( x + 6 - 3 )

                     = ( x + 9 ) ( x + 3 ).

Observe the following products:

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

                         = x² + 2x + 3x + 3 *2

                         = x² + ( 2 + 3 ) x + 3 *2

                         = x² + 5x + 6 _____________(1)

( x - 4 ) ( x + 3 ) = x ( x + 3 ) - 4 ( x + 3 )

                         = x² + 3x - 4x - 12

                         = x² + ( - 4 + 3 ) x + ( - 4 + 3 )

                         = x² - x - 12 ______________(2)

More generally,

( x + a ) ( x + b ) = x ( x + b ) + a ( x + b )

                         = x² + bx + ax + ab

                         = x² + ( a + b ) x + a *b __________(3)

What do you notice in the above products?

The x term in the expression (1) is

          5x = ( 2 + 3 )x = 2x + 3x

Product of the coefficient = 2, 3 = 6 the constant term.

Similarly, in the second expression, product x term is

                 = - x = - 4x + 3x

The product of the coefficients = 4 *  3 = - 12 the constant term.

Therefore, if the x term in the given quadratic monic polynomial is written as the sum of two terms so that the product of their coefficients is the constant term, it can be resolved into factors. This is another way to factorize a quadratic monic polynomial.

Example 2

x² + 7x + 12

7x is to be written as the sum of two terms such that the product of their coefficients is 12.

          = x² + ( 3x + 4x ) + 12

          = ( x² + 3x ) + ( 4x + 12 )

          = x ( x + 3 ) + 4 ( x + 3 )

          = ( x + 4 ) ( x + 3 )

7x = 3x + 4x; 3 *  4 = 12

Try these questions

I.    Resolve into factors

1. x² + 7x + 12
2. x² + x - 12
3. x² - x - 12
4. x² - 7x
5. x² - 2x - 15
6. x² - 8x + 15
7. x² - 3x - 18
8. x² - 9x + 18
9. x² - 2x - 63
10. x² + 4x - 21
11. x² - 4x - 192
12. x² - 3x - 180
13. 12 - x - x²
14. 18 - 7x - x²
15. 20 + x - x²
16. 27 - 6x - x²
17. x² + 3xy - 10y²
18. x² - 5xy - 6y²
19. x² + 2xy - 24y²
20. x² + 11xy + 24y²

Answers to Practice Problems

1. x² + 7x + 12 = x² + 4x + 3x + 12
                      = x ( x + 4 ) +3 ( x + 4 )
                      = ( x + 4 ) ( x + 3 )


2. x² + x - 12 = x² + 4x - 3x - 12
                    = x ( x + 4 ) - 3 ( x + 4 )
                    = ( x + 4 ) ( x - 3 )


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


4. x² - 7x + 12 = x² - 4x - 3x + 12
                      = x ( x - 4 ) - 3( x - 4 )
                      = ( x - 4 ) ( x - 3 )


5. x² - 2x - 15 = x² - 5x + 3x - 15
                      = x ( x - 5 ) + 3 ( x - 5 )
                      = ( x - 5 ) ( x + 3 )


6. x² - 8x + 15 = x² - 5x - 3x + 15
                      = x ( x - 5 ) - 3 ( x - 5 )
                      = ( x - 5 ) ( x - 3 )


7. x² - 3x - 18 = x² - 6x + 3x - 18
                      = x ( x - 6 ) + 3( x - 6 )
                      = ( x - 6 ) ( x + 3 )


8. x² - 9x + 18 = x² - 6x - 3x + 18
                      = x ( x - 6 ) - 3 ( x - 6 )
                      = ( x - 6 ) ( x - 3 )


9. x² - 2x - 63 = x² - 9x + 7x - 63
                      = x ( x - 9 ) - 7 ( x - 9 )
                      = ( x - 9 ) ( x - 7 )


10. x² + 4x - 21 = x² + 7x - 3x - 21
                       = x ( x + 7 ) - 3 ( x + 7 )
                       = ( x + 7 ) ( x - 3 )


11. x² - 4x - 192 = x² - 16x + 12x - 192
                        = x ( x - 16 ) + 12 ( x - 16 )
                        = ( x - 16 ) ( x + 12 )


12. x² - 3x - 180 = x² - 15x + 12x - 180
                         = x ( x - 15 ) + 12 ( x - 15 )
                         = ( x - 15 ) ( x + 12 )


13. 12 - x - x² = - ( x² + x - 12 )
                     = - [ x²+ 4x - 3x - 12]
                     = - [ ( x + 4 ) ( x - 3 ) ]
                     = ( x + 4 ) ( 3 - x )


14. 18 - 7x - x² = 18 - 9x + 2x - x²
                       = 9 ( 2 - x ) + x ( 2 - x )
                       = ( 2 - x ) ( 9 + x )


15. 20 + x - x² = 20 + 5x - 4x - x²
                     = 5 ( 4 + x ) - x ( 4 + x )
                     = ( 4 + x ) ( 5 - x )


16. 27 - 6x - x² = 27 - 9x + 3x - x²
                       = 9 ( 3 - x ) + x ( 3 - x )
                       = ( 3 - x ) ( 9 + x )


17. x² + 3xy - 10y² = x² + 5xy - 2xy - 10y²
                            = x ( x + 5y ) - 2y ( x + 5y )
                            = ( x + 5y ) ( x - 2y )


18. x² - 5xy - 6y² = x² - 6xy + xy - 6y²
                          = x ( x - 6y ) + y ( x - 6y )
                          = ( x - 6y ) ( x + y )


19. x² + 2xy - 24y² = x² + 6xy - 4xy - 24y²
                            = x ( x + 6y) - 4y ( x + 6y)
                            = (x + 6y) (x - 4y)


20. x² + 11xy + 24y² = x² + 8xy + 3xy + 24y²
                              = x (x + 8y ) + 3y (x + 8y )
                              = ( x + 8y ) ( x + 3y )