Factor Polynomials

FACTOR POLYNOMIALS

Factors are the terms you multiply. The answer is referred to as the product.

Example: Factor 36 completely. This directs you to find the prime factors of 36.

36 = 9 ∙ 4 which then can be factored further to 3 ∙ 3 ∙ 2 ∙ 2. Therefore, the factored answer is
3 ∙ 3 ∙ 2 ∙ 2.

When factoring polynomials you are given the "answer", such as m2 - 12m + 27, and are asked to find the "question" or factors, such as (m - 9)(m - 3).

You will be given several approaches to factor polynomials depending on the number of terms.

FACTOR TRINOMIALS, ax2 + bx + c (a,b,c are constants)

Product and Sum Method will work for all trinomials with the above format in which the degree of the polynomials are multiples of two, i.e. ax4 + bx2 + c.

PRODUCT AND SUM METHOD

1) find the product of a and c (the coefficients of the first and last term) which we refer to as P (product).
2) identify b (the coefficient of the middle term) which we refer to as S (sum)
3) Find two numbers whose product is the result in step 1 and whose sum is the result in step 2.
4) Rewrite the original trinomial, replacing the bx term by using the two terms found in step 3.
5) Factor by grouping.

EXAMPLES:

1) Factor 6m2 - 11m - 2 .

step 1: P = 6(-2) = -12
step 2: S = -11
step 3: The two numbers are -12 and 1. (-12 · 1 = -12 and -12 + 1 = -11)
step 4: 6m2 - 12m + m - 2
step 5: (6m2 - 12m) + (m - 2)
6m(m - 2) + 1(m - 2)
(m - 2)(6m + 1) is the factored answer.

2) Factor 4y2 + 20y + 25 .

step 1: P = 4(25) = 100
step 2: S = +20
step 3: The two numbers are +10 and +10. (you may have to try several combos)
step 4: 4y2 + 10y + 10y + 25
step 5: (4y2 + 10y) + (10y + 25)
2y(2y + 5) + 5(2y + 5)
(2y + 5)(2y + 5) is the factored answer.

3) Factor 6a2 + 5ab - 21b2 .

step 1: P = 6(-21) = -126
step 2: S = 5
step 3: the two numbers are -14 and 9.
step 4: 6a2 - 14ab + 9ab - 21b2
step 5: (6a2 - 14ab) + (9ab - 21b2)
2a(3a - 7b) + 3b(3a - 7b)
(3a - 7b)(2a + 3b) is the factored answer.

4) Factor 9y2 - 28y + 3 .

step 1: P = 9(3) = 27
step 2: S = -28
step 3: the two numbers are -27 and -1
step 4: 9y2 - 27y - y + 3
step 5: (9y2 - 27y) + (- y + 3)
9y(y - 3) - 1(y - 3)
(y - 3)(9y - 1) is the factored answer.

5) Factor 5x4 + 18x2 - 8 .

step 1: P = 5(-8) = -40
step 2: S = +18
step 3: the two numbers are 20 and -2
step 4: 5x4 + 20x2 - 2x2 - 8
step 5: (5x4 + 20x2) + (- 2x2 - 8)
5x2 (x2 + 4) - 2(x2 + 4)
(x2 + 4)(5x2 - 2) is the factored answer.

NOTE: There may be times when factoring a trinomial can be done by inspection rather than the product/sum method. This especially is true when the coefficient of the x2 term is one.

6) Factor y2 - 7y + 10 .

The following three steps may be done mentally by you. That is fine. It is not necessary for you to write it down. However, that will be the thought process that you will be doing to get the correct factors.

step 1: P = 10
step 2: S = -7
step 3: the two numbers are -5 and -2
(y - 5)(y - 2) is the factored answer.