Given f ^{2}=g _{2}+h _{2} where g(x) and h(x) are polynomials with real coefficients, prove that f has non-real roots

Question

myininaya · Answer

$$ f ^{2}=g _{2}+h _{2}   $$?

myininaya · Answer

are g and h suppose to be squared too?

myininaya · Answer

or did you mean for them to be subscripts?

myininaya · Answer

i think you meant
$$f^2=g^2+h^2$$

myininaya · Answer

because you gave us no info for the functions g_2 and h_2

anonymous · Answer

yeah, g and h are supposed to be square...

anonymous · Answer

The solution that you want to show is not true. Consider $$g(x)=h(x)=x$$ Then $$f(x)^2=2x^2$$ so $$f(x)=\pm\sqrt{2}x$$ which has no non-real roots.

anonymous · Answer

However, if you state that at least one of g or h has a nonzero constant term and that they are not both constant functions, then clearly $$f^2(x) > 0$$ for all real x, and since f^2 is clearly a polynomial, it must have at least one non-real root by the fundamental theorem of algebra