-2
$\begingroup$

I am learning about polynomials from Barbeau's book and I was hoping for a hint on a question which is as follows:

Let $p(t)$ and $q(t)$ be two quadratic polynomials with integer coefficients. Prove that, if they have a nonrational zero in common, then on must be a constant multiple of the other.

A hint is appreciated.

$\endgroup$
4
$\begingroup$

If a quadratic polynomial with integer coefficients has an irrational zero $\alpha=a+\sqrt{b}$, then the conjugate of $\alpha$, i.e. $a-\sqrt{b}$ is also a zero. Therefore $p(t)$ and $q(t)$ share common roots and thus one must be a constant multiple of another.

$\endgroup$
  • $\begingroup$ I wanted only a hint. Not a solution. $\endgroup$ – Gearboxx Nov 19 '16 at 7:40
  • $\begingroup$ @Gearbox Please begin such an answer by "Thanks but..." $\endgroup$ – Jean Marie Nov 19 '16 at 9:45
  • $\begingroup$ The response was intentionally composed that way; I was frustrated with the response. $\endgroup$ – Gearboxx Nov 19 '16 at 14:06

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.