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Suppose that is a polynomial of degree . Since is a polynomial, it is infinitely differentiable. Therefore, has a Taylor expansion about . Since
, the expansion terminates after the
term. Also, the
remainder of the Taylor series vanishes; i.e.,
. Thus, the function is equal to its Taylor series. Hence,
If , then
. Thus,
, where is the polynomial
. Hence, is a factor of .
Conversely, if is a factor of , then
for some polynomial . Hence,
.
It follows that is a factor of if and only if .
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