Factoring a ⁴ - b ⁴

We can factor a difference of fourth powers (and higher powers) by treating each term as the square of another base, using the power to a power rule. For example, to factor x ⁴ - y ⁴ , we treat x ⁴ as (x ²)² and y ⁴ as (y ²)² . Thus, x ⁴ - y ⁴ = (x ²)² - (y ²)² = (x ² + y ²)(x ² - y ²) = (x ² + y ²)(x + y)(x - y) . Similarly, we can treat x ⁶ as (x ³)² or (x ²)³ , and so on.

Factoring ax ⁴ + bx ² + c

In a similar manner, we can factor some trinomials of degree 4 by treating x ⁴ as (x ²)² m and factoring to (a ₁ x ² + c ₁)(a ₂ x ² + c ₂) , (a ₁ x ² - c ₁)(a ₂ x ² - c ₂) , or (a ₁ x ² - c ₁)(a ₂ x ² + c ₂) . For example, we factor x ⁴ +6x ² + 5 as (x ²)² +6(x ²) + 5 : x ⁴ +6x ² +5 = (x ²)² +6(x ²) + 5 = (x ² +5)(x ² + 1) .

Some of these expressions can be factored further; if one of the resulting binomials is a difference of squares, for example, factor that binomial:

x ⁴ -4x ² -45 = (x ²)² -4(x ²) - 45 = (x ² -9)(x ² +5) = (x + 3)(x - 3)(x ² + 5) .