Factoring Quadratic Trinomials

Form x² +bx + c

First, find two integers m and n, whose sum is b and whose product is c. The factors are:

(x + m)(x + n)

Example:

Factor:  x² + 9x + 18

Make an organized list of the factors of 18:

1,18,  sum = 19

2,9     sum = 11

3,6     sum = 9

m and n are 6 and 3 since 6+ 3 = 9, and 6 · 3 = 18

x² + 9x + 18 = (x + 6)(x + 3)

Rules for signs:

1. b and c are Positive:

Two numbers whose product and sum are both Positive.

2. b is Negative and c is Positive,

Two numbers whose product is Positive and sum is Negative. (Both factors have a negative sign,)

3. b is Positive and c is Negative.

Two numbers whose product is Negative and sum is Positive. (Factors have opposite signs. The larger factor will have a positive sign)

4. b is Negative and c is Positive

Two numbers whose product is Positive and sum is Negative. (Factors have opposite signs. The larger factor will have a negative sign)

Form ax² +bx + c

This quadratic can be written in the form: ax² +mx + nx + c, where a > 1

First, find two integers m and n, whose sum is b and whose product is a · c.

Example:

Factor:  6x² + 25x + 14

Make an organized list of the factors of a · c, or 6 · 14, or 84:

1,84,  sum = 85

2,42     sum = 44

3,28     sum = 31

4, 21    sum = 25

m and n are 4 and 21 since 4 + 21 = 25, and 4 · 21 = 84

6x² + 25x + 14 = 6x² +4x +21x + 14

(6x² +4x) + (21x + 14) = 2x(3x +2) + 7(3x + 2) =

(3x + 2)(2x + 7)

a, b, and c Have a Common Factor

First, factor out the common Factor.

Example:

6x² + 33x + 15

3(2x² + 11x + 5)

The factors of 10 are:

1,10

2,5

m and n are 1 and 10 since 1 + 10 = 11, and 1 · 10 = 10

3(2x² + 10x + x + 5) = 3(2x² + x) + (10x + 5) = 3(x(2x + 1) + 5(2x + 1)) =

3(2x + 1)(x + 5)

A Prime Polynomial

A polynomial is prime if it cannot be written as a product of two polynomials with integral coefficients; that is there is no combination of m and n where the sum of m + n = b, and the product of m ·  n = a·  c

Solving Quadratic Equations By Factoring

Solve: 8x² - 4x - 3 = 6 + 2x

8x² - 4x - 2x - 3 - 6 = 0

 8x² - 6x  - 9 = 0

Factors of -72 are: -12, 6

8x² - 12x + 6x  - 9 = 0

4x(2x - 3) + 3(2x - 3) = 0

(2x - 3)(4x + 3) = 0

2x - 3 = 0 and 4x + 3 = 0

x = 3/2 and x = - 3/4

Perfect Square Trinomials

A perfect square trinomial is a trinomial that is the square of a binomial.

The first term and the last term must be perfect squares.

The middle term must be twice the product of the square roots of the first and last terms.

Example: 9x² -24x +16 = (3x - 4)²

9x² and 16 are perfect squares (3x and 4).

24x = 2 ( 3x ·  4) = 2( 12x)