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!
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Sign up to join this communityLet $p$ 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!
Hint: Take a look at $p(0)$ and the limits of $p$ as $x$ approaches $\pm\infty$.
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.