Question about polynomials with rational coefficients

Asked Dec 04 '19 at 18:00

Active Dec 04 '19 at 18:00

Viewed 36 times

Let $f$ and $g$ two polynomials with rational coefficients of degree $n$, $n \geq 1$. They share a same root $r$ and $f$ is irreductible. Prove that $f | g$.

I tried writing $f=(x-r)h$ where $h$ is a polynomial of degree $n-1$ but didn’t know what to do next. Please help me with a basic proof.

asked Dec 04 '19 at 18:00

furfur

So $f,g$ have the same degree, is that correct, with $r$ non-rational? – abiessu Dec 04 '19 at 22:22
$gcd(f,g)$ cannot be $1$ since they share a root, so it must be $f$ since $f$ irreducible, so $f$ divides $g$ – Conrad Dec 05 '19 at 07:09
Conrad, I can’t understand your proof. – furfur Dec 05 '19 at 20:50

Question about polynomials with rational coefficients

0 Answers0