Overview

Cubic identities are a common factoring checkpoint. Know them cold and look for near-miss patterns that become cubes after substitution.

Key Ideas

• $x^3 - y^3 = (x-y)(x^2 + xy + y^2)$.
• $x^3 + y^3 = (x+y)(x^2 - xy + y^2)$.
• $(x\pm y)^3 = x^3 \pm 3x^2y + 3xy^2 \pm y^3$.

Core Skills

Spot a Hidden Cube

Compare coefficients to $(x\pm y)^3$ and use substitution to match the pattern.

Factor Completely

After factoring the cubic, check if the quadratic factor can be simplified or factored further over reals or integers.

Use the Remainder Theorem

If a rational root is suspected, test $x=r$ to confirm a linear factor.

Worked Example

Factor $x^3 - 8$.

Write $8 = 2^3$ and apply difference of cubes: $x^3 - 2^3 = (x-2)(x^2 + 2x + 4)$.

More Examples

Example 1: Sum of Cubes

Factor $27a^3 + b^3$.

$(3a+b)(9a^2-3ab+b^2)$.

Example 2: Use Substitution

Factor $x^3+3x^2y+3xy^2+y^3$.

This is $(x+y)^3$.

Example 3: Rational Root

Factor $x^3-4x^2-7x+10$.

Try $x=1$; it works, so factor $(x-1)$ and continue.

Strategy Checklist

• Check for sum/difference of cubes first.
• Compare to $(x\pm y)^3$ for binomial cubes.
• Test small integer roots if needed.

Common Pitfalls

• Flipping the sign in the quadratic factor.
• Forgetting that $x^3 + y^3$ uses a minus in the middle term.
• Missing a common factor before applying the cube formulas.

Practice Problems