Factoring Trinomials
A trinomial is an algebraic expression consisting of three terms connected by plus or minus signs. Trinomials can be
written in the following two forms. They can be written as x3 + a3 or they can be written as ax3 + bx2 + cx + d. There are
three ways trinomials can be factored. One way is by using the special factor formulas, another way is by using synthetic
division, and another way is by using factor by grouping. When the trinomial is written in this form x3 + a3, it is best to
use the special factor formulas. If the trinomial is written in this form ax3 + bx2 + cx + d, it is best to first try factor by
grouping then to try synthetic division.
Special Factor Formulas (Sum & Difference of Cubes):
x3 + a3 = (x + a) (x2 – ax + a2)
x3 – a3 = (x – a) (x2 + ax + a2)
Example:
Factor: 8x3 + 27
First re-write the problem, so it will look like one of the Special Factor formulas.
8x3 + 27
can be re-written as:
(2x)3 + (3)3
now that it looks like one of the Special Factor formulas , compare the equation to the factor
formula and substitute the proper values for x and a.
3
3
x + a = (x + a) (x2 – ax + a2)
(2x)3 + (3)3 = (2x + 3) ((2x)2 – (3)(2x) + (3)2)
(2x)3 + (3)3 = (2x + 3) (4x2 – 6x + 9)
Factor by Grouping:
acx3 + adx2 + bcx + bd = ax2 (cx + d) + b (cx + d) = (ax2 + b) (cx + d)
Example:
Factor: 2x3 + 2x2 + 3x + 3
The goal of factor by grouping is to split the given equation in half and factor both sides so that each half has
one factor in common.
2x2(x + 1) + 3(x + 1)
In this example the factor that both halves have in common is (x + 1). Next group the values in front of
(x +1), 2x2 and 3 together into another factor being multiplied by (x + 1).
2x3 + 2x2 + 3x + 3 = (2x2 + 3)(x + 1)
Other Special Factors:
x2 – a2 = (x – a) (x + a)
x4 – a4 = (x2 – a2) (x2 + a2)
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