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.

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)
.

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