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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} +6*x*^{2} + 5 as (*x*^{2})^{2} +6(*x*^{2}) + 5: *x*^{4} +6*x*^{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} -4*x*^{2} -45 = (*x*^{2})^{2} -4(*x*^{2}) - 45 = (*x*^{2} -9)(*x*^{2} +5) = (*x* + 3)(*x* - 3)(*x*^{2} + 5).