Section 4.5
Factoring Special Products
All Binomials
1) Sum and Difference of cubes
2) Difference of Squares
1. Difference of cubes / Sum of cubes:
𝑨𝟑 − 𝑩𝟑
OR
𝑨𝟑 + 𝑩𝟑
A Difference/Sum of cubes is a binomial where both terms are perfect cubes and there is a “-” or “+” sign in
between. For example: 8𝑥 − 1 , 64𝑥 + 27𝑦
A difference of cubes always factors using the formula
(𝑨 − 𝑩)(𝑨𝟐 + 𝑨𝑩 + 𝑩𝟐 ) where A is the cube root of the first term and B is the cube root of the second term.
A sum of cubes always factors using the formula
(𝑨 + 𝑩)(𝑨𝟐 − 𝑨𝑩 + 𝑩𝟐 ) where A is the cube root of the first term and B is the cube root of the second term.
Example 1: Factor the following polynomials.
𝑎) 8𝑥 + 27𝑦
b) 5𝑥 − 5
c) 128𝑥 𝑦 − 250𝑦
2. Difference of squares: 𝑨𝟐 − 𝑩𝟐
A Difference of Squares is a binomial where both terms are perfect squares and there is a “-“ sign in between.
For example: 4𝑥 − 9 ,
16𝑥 − 49𝑦
,
49𝑥 − 100
A difference of squares always factors as conjugates.
(𝑨 − 𝑩)(𝑨 + 𝑩) where A is the square root of the first term and B is the square root of the second term.
*NOTE: SUM of squares is PRIME. 𝑨𝟐 + 𝑩𝟐 cannot be factored down into smaller parts.
Example 2: Factor the following polynomials.
a) 6𝑥 − 24
b) 3𝑥 𝑦 − 27𝑥 𝑦
c) 16𝑥 + 25
d) 81𝑥 − 1
e) 81𝑥 − 16𝑦
f) 4𝑥 + 12𝑥 − 9𝑥 − 27