50.6 Sum and Difference of Cubes

Sum and Difference of Cubes is a factoring method for polynomials involving cubes, using specific formulas to simplify complex expressions into manageable factors.

Sum and Difference of Cubes is the pair of special factoring patterns applied to a two-term polynomial in which both terms are perfect cubes, one pattern for when the terms are added and another for when they are subtracted, each rewriting the two-term expression as the product of a binomial factor and a trinomial factor. Unlike the difference of squares, which produces two binomial factors, these cube-based patterns always produce one binomial and one trinomial, reflecting the more elaborate identity that underlies cube factoring.

Both patterns are built from the same two cube bases, extracted from the original terms, arranged according to a fixed sign pattern that differs depending on whether the original two terms were added or subtracted.

Recognizing the Pattern

Two-Term Cube Pattern Requirement

The polynomial must consist of exactly two terms, connected by either addition or subtraction; a polynomial with more than two terms does not match either cube pattern regardless of whether individual terms happen to be perfect cubes.

a + b or a - b

First Perfect Cube Recognition

The first term is checked to confirm it is a perfect cube, meaning both its numerical coefficient and its variable factor can be expressed as the cube of some simpler expression.

8 x^{3} = (2 x)^{3}

Second Perfect Cube Recognition

The second term is checked the same way, confirming it too can be expressed as the cube of some simpler expression.

27 = 3^{3}

Extracting the Cube Bases

Cube Base Extraction

Once both terms are confirmed as perfect cubes, the cube root of each is extracted, producing the two base expressions, p and q, that will populate both the binomial and trinomial factors.

\sqrt[3]{8 x^{3}} = 2 x, \sqrt[3]{27} = 3

The Sum-of-Cubes Pattern

Sum-of-Cubes Binomial Factor

When the original two terms are added, the binomial factor is formed by adding the two extracted bases directly.

(p + q)

Sum-of-Cubes Trinomial Factor

The accompanying trinomial factor consists of the square of the first base, minus the product of the two bases, plus the square of the second base.

a^{3} + b^{3} = (p + q) (p^{2} - p q + q^{2})

8 x^{3} + 27 = (2 x + 3) (4 x^{2} - 6 x + 9)

The Difference-of-Cubes Pattern

Difference-of-Cubes Binomial Factor

When the original two terms are subtracted, the binomial factor is formed by subtracting the second base from the first.

(p - q)

Difference-of-Cubes Trinomial Factor

The accompanying trinomial factor consists of the square of the first base, plus the product of the two bases, plus the square of the second base, matching sign pattern that is exactly reversed compared to the sum-of-cubes trinomial.

a^{3} - b^{3} = (p - q) (p^{2} + p q + q^{2})

8 x^{3} - 27 = (2 x - 3) (4 x^{2} + 6 x + 9)

Confirming the Correct Pattern

Cube-Factor Sign Pattern Verification

Once a factorization is assembled, its sign pattern is checked: the binomial's connecting sign should match the sign connecting the two original terms, while the trinomial's middle term should carry the opposite sign, matching the fixed relationship each identity requires.