Overview

When you see $x^n \pm y^n$, check if it factors. The factorization patterns help simplify large expressions quickly.

Key Ideas

• $x^n - y^n = (x-y)(x^{n-1} + x^{n-2}y + \cdots + y^{n-1})$.
• If $n$ is odd, $x^n + y^n$ factors as $(x+y)(x^{n-1} - x^{n-2}y + \cdots + y^{n-1})$.
• Even-power sums usually do not factor over the reals.

Core Skills

Check Parity of $n$

The sum $x^n+y^n$ only factors over the reals when $n$ is odd. The difference $x^n-y^n$ always factors.

Factor Out a Common Power

If $x^n \pm y^n$ shares a common factor, pull it out first to simplify the remaining expression.

Use Repeated Factoring

After applying a power factorization, check if the remaining factor is still factorable (for example, a difference of squares).

Worked Example

Factor $a^5 + b^5$.

Since $5$ is odd, $a^5 + b^5 = (a+b)(a^4 - a^3b + a^2b^2 - ab^3 + b^4)$.

More Examples

Example 1: Difference of Powers

Factor $x^6 - y^6$.

First, $x^6-y^6=(x^3-y^3)(x^3+y^3)$, then each cubic factors further.

Example 2: Sum with Odd Power

Factor $x^3 + 8y^3$.

$x^3 + (2y)^3 = (x+2y)(x^2-2xy+4y^2)$.

Example 3: Factor Out a Power

Factor $x^8 - x^4$.

$x^4(x^4-1)=x^4(x^2-1)(x^2+1)$.

Strategy Checklist

• Check whether the exponent is odd or even.
• Factor out a common power first.
• Look for repeated factoring opportunities.

Common Pitfalls

• Applying the sum formula when $n$ is even.
• Forgetting the alternating signs in the odd-power sum factor.
• Stopping after one factorization when more is possible.

Practice Problems