Sum and Difference of Cubes

Sum of cubes: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

Difference of cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

Example: $x^3 - 8$ $= x^3 - 2^3 = (x - 2)(x^2 + 2x + 4)$

Factoring by Substitution

Sometimes substituting can simplify factoring.

Example: $x^4 - 5x^2 + 4$

Let $u = x^2$: $u^2 - 5u + 4 = (u - 4)(u - 1)$

Substitute back: $= (x^2 - 4)(x^2 - 1)$ $= (x + 2)(x - 2)(x + 1)(x - 1)$

Substitute back $u = x^2$: $(x^2 - 9)(x^2 - 4)$

Both are difference of squares: $= (x + 3)(x - 3)(x + 2)(x - 2)$

Answer: $(x + 3)(x - 3)(x + 2)(x - 2)$