Chapter 4.3, Problem 3AYU

3. Every polynomial function of odd degree with real coefficients has at least _____ real zero(s).

To determine

To fill: Every polynomial function of odd degree with real coefficients has at least ------- real zero(s).

Answer to Problem 3AYU

Every polynomial function of odd degree with real coefficients has at least one real zero.

Explanation of Solution

Given:

Polynomial function $f\left(x\right)$ of odd degree with real coefficients.

In $f\left(x\right)$ complex zeros occurring will be in conjugate pairs in the polynomial function with real coefficients.

If the polynomial function has odd degree there will odd zeros of the polynomial.

But complex zeros occur in conjugate pairs which are even number of zeros and are not real zeros.

Therefore polynomial function with odd degree and real coefficients will have least one real zero.