Factoring the Sum of Two Cubes (a3 + b3)

Factoring the Sum of Two Cubes

How to solve factoring the sum of two cubes problems: formula, examples, and their solutions.

Formula

a^3 + b^3 = (a + b)(a^2 - ab + b^2)

a3 + b3 = (a + b)(a2 - ab + b2)

See the blue colored signs.

If the sign of b3 term in (a3 + b3) is (+),

then the sign of b term in (a + b) is (+)
and the sign of ab term in (a2 - ab + b2) is (-).

Proof

Factoring the Sum of Two Cubes: Proof of the Formula

Solve (a - b)(a2 + ab + b2).

Multiplying polynomials

Then a2b terms and ab2 terms are cancelled.

So (a - b)(a2 + ab + b2) = a3 - b3.

Switch both sides.

Then a3 - b3 = (a - b)(a2 + ab + b2).

Example 1

Factor the given polynomial. x^3 + 8

x3 + 8 = x3 + 23
= (x + 2)(x2 - x⋅2 + 22)

Example 2

Factor the given polynomial. 27y^3 + 1

27y3 + 1 = (3y)3 + 13
= (3y + 1)((3y)2 - 3y⋅1 + 12)