Factoring Cubic Polynomials

Algebraic Transformations

Algebraic Expressions

Basic Definitions

Polynomials

Algebraic Transformations

Outline

Factoring

Factoring Quadratic Polynomials

Factoring Cubic Polynomials

Factor Theorems

Other Transformations

Expanding

Completing Perfect Square

Rationalizing Denominators

Factoring Cubic Polynomials

Key Topics Remaining: Factor Theorens » Expanding » Completing Perfect Square » Rationalizing Denominators

Problem 1. Factor the difference between two cubes, a3 – b3.
Solution. First, subtract and add a pair of terms, a2b and a b2.
Then combine the terms by pairs and take out the common factors:

a3 – b3 = a3 – a2b + a2b – a b2 + a b2 – b3
= a2 (a – b) + a b (a – b) + b2 (a – b)
= (a – b)(a2 + a b + b2).

Therefore, we have the formula for the difference between two cubes:

a3 – b3 = (a – b)(a2 + a b + b2).

Corollary. Substituting (–b) for b, we obtain the following formula for the the sum of two cubes:

a3 + b3 = (a + b) (a2 – a b + b2)

Problem 2. Factor the following cubic polynomial,

a3 + 3 a2b + 3 ab2 + b3.

Solution. First, rewrite 3 a2b and 3 a b2, respectively, as a2b + 2 a2b and a b2 + 2 a b2.
Then combine the terms by pairs:

a3 + 3 a2b + 3 a b2 + b3 = a3 + a2b + 2 a2b + 2 a b2 + a b2 + b3
= (a3 + a2b) + (2 a2b + 2 a b2) + (a b2 + b3).

Finally, take out the common factors:

(a3 + a2b) + (2 a2b + 2 a b2) + (a b2 + b3) =
= a2(a + b) + 2 a b (a + b) + b2(a + b)
= (a + b) (a2 + 2 a b + b2)
= (a + b) (a + b)2 = (a + b)3.

Therefore,

a3 + 3 a2b + 3 a b2 + b3 = (a + b)3

Corollary. Substituting (– b) for b, we obtain the following formula for the difference cubed:

a3 – 3 a2b + 3 a b2 – b3 = (a – b)3