Homework Answers
Suppose that x=a/bx=a/b is a rational root, wlog in lowest terms, i.e. a,ba,b are coprime. Then
0=bnT(a/b)=an+b(cn−1an−1+⋯+c0bn−1)0=bnT(a/b)=an+b(cn−1an−1+⋯+c0bn−1)
Thus b∣an.b∣an. If |b|>1|b|>1 then some prime p∣b∣an,p∣b∣an, so p∣a,p∣a, contra a,ba,b coprime. Therefore we conclude that |b|=1,|b|=1, so a/ba/bis an integer if rational.
Remark This is essentially one half of the well-known Rational Root Test. The other half follows by reciprocation symmetry, i.e. by applying this result to the reversed polynomial (satisfied by x−1).
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