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Algebra II: Factoring

Factoring Polynomials of Degree 4

Problems

Problems

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) .

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