Factoring a4 - b4
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 x4 - y4, we treat x4 as (x2)2 and y4 as
(y2)2. Thus, x4 - y4 = (x2)2 - (y2)2 = (x2 + y2)(x2 - y2) = (x2 + y2)(x + y)(x - y). Similarly, we can treat x6 as
(x3)2 or (x2)3, and so on.
Factoring ax4 + bx2 + c
In a similar manner, we can factor some trinomials of degree 4 by
treating x4 as (x2)2m and factoring to (a1x2 + c1)(a2x2 + c2), (a1x2 - c1)(a2x2 - c2), or (a1x2 - c1)(a2x2 + c2). For example, we factor x4 +6x2 + 5 as (x2)2 +6(x2) + 5: x4 +6x2 +5 = (x2)2 +6(x2) + 5 = (x2 +5)(x2 + 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:
x4 -4x2 -45 = (x2)2 -4(x2) - 45 = (x2 -9)(x2 +5) = (x + 3)(x - 3)(x2 + 5).
