Let p be a polynomial of even degree n and suppose that anao < 0, where an is th

Question

Let p be a polynomial of even degree n and suppose that anao < 0, where an is the coefficient of x^n and ao is the constant term. Prove that P has at least two roots. I know to use the fact that a0 < 0 and an > 0, the behavior of the polynomial at +/- infinity, and the Intermediate Value Theorem

Explanation / Answer

without loss of generality, a0 < 0 and an > 0. The other case of a0 > 0 and an < 0 is analogous. As, a0 < 0, p(0) < 0. Now, look at the behavior of p at +/- infinity : p(x) as x->inf is > 0 (an >0) . The sign of the polynomial is the sign of the leading coefficient. Also, p(x) as x-> - inf is > 0 (an >0) . The sign of the polynomial is the sign of the leading coefficient. this will be positive as n is even so an * x^n > 0 as x -> - infinity. so, clearly, p(0) < 0 and p becomes positive as x goes to either +/- infinity. So, by intermediate value theorem, we get that there is a positive root and a negative root. Hence proved. Message me if you have any doubt.

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