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.
GUIDE TO FACTOR POLYNOMIALS
Step 1: Factor the greatest common factor (gcf) when possible. This should always be done
first.
Step 2: Determine how many terms in the polynomial.
If there are four terms, then factor by grouping.
If there are three terms, then factor as a trinomial.
If there are two terms, then factor as a difference of squares or factor as
sum/difference of cubes.
Step 3: Factor again when possible.
EXAMPLES:
1) Factor completely: 2x3 – 4x2 + 10x
Factor a GCF: 2x (x2 – 2x + 5).
Try and factor x2 – 2x + 5 since it is a trinomial. However, it cannot be factored.
2x (x2 – 2x + 5) is the final factored answer.
2) Factor completely: 3x3y – 48xy3.
Factor a GCF: 3xy(x2 – 16y2)
Factor (x2 – 16y2) as a difference of two squares: (x – 4y)(x + 4y)
3x (x – 4y)(x + 4y) is the final factored answer.
3) Factor completely: 6x2 + 15x – 8xy – 20y
Factor a GCF: There is no GCF.
Since there are four terms, factor by grouping: (6x2 + 15x) (– 8xy – 20y)
3x(2x + 5) – 4y(2x + 5)
(2x + 5)(3x – 4y)
(2x + 5)(3x – 4y) is the final factored answer.
4) Factor completely: x5 – 18x2 + 81x
Factor a GCF: x(x4 – 18x + 81)
Factor (x4 – 18x + 81) as a trinomial: (x2 – 9)(x2 – 9)
Factor both, (x2 – 9)(x2 – 9), as difference of two squares: (x – 3)(x + 3) and (x – 3)(x + 3).
x(x – 3)(x + 3)(x – 3)(x + 3) is the final factored answer.
5) Factor completely: 2x2 – 4x + 9
Factor a GCF: there is no GCF.
Factor as a trinomial: There are no factors that work.
Therefore, the final answer is that the polynomial is prime.