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# Let p (x) and q (x) be two quadratic polynomials with integer coefficients. supposed they have a non rational zero in common. Show that : p (x)=rq(x) for some rational number r

Anish Singhal
3 years ago
Letx1 be the irrational root that is shared byp and q. Then, by the Irrational Conjugate Roots Theorem, the irrational conjugate ofx1x1must also be a root of bothp andq.

Explicitly, for any rationala,band irrational √c, ifx1=a+brootc
is a root of p, then
x2=a−brootc
must also be a root. The same reasoning implies that x^2is a root of q.Since pand qare quadratics whose two roots are the same, they must be proportional, up to a constant. Since we have integer coefficients, the constant of proportionality rmust be rational, so we are done.