In: Advanced Math
Suppose that f(x)=x^n+a_(n-1) x^(n-1)+⋯+a_0∈Z[x]. If r is rational and x-r divides f(x), prove that r is an integer.
solution:-
which means
Suppose
is an integer and
and
so
that the fraction is in reduced form
Therefore,
Substituting in the above
expression, we have
That is,
is a root of
If
is not an integer, then
Thus, where
is a prime number
Substituting
in
we get
Multiplying both sides by we get
Thus, we have
That is,
As
So that
Thus,
But this contradicts the fact that
Therefore, we must have
where
Which is to say that the rational root of
must be an integer