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.


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.

  • $\begingroup$ 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

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