TITLE: Real roots of a polynomial
QUESTION [8 upvotes]: Let $p$ be an even degree polynomial with real coefficients such that the product of the constant term and the leading coefficient is negative. Show that $p$ has at least two real roots.
Thanks!

REPLY [2 votes]: By scaling, the polynomial can be written in the form
$p(x)=x^{2n}+...-1=0$.
Then $p(0)<0$ and $p(x) > 0$ for large negative and positive $x$, so $p(x) $ has at least one positive and negative root.