Chapter 4.3, Problem 5AYU

To determine

To state: True/false a polynomial function of $n$ degree with real coefficients has exactly $n$ complex zeros. At most $n$ of them are real zeros.

Answer to Problem 5AYU

$\text{False}$.

Explanation of Solution

Given:

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

If the polynomial function has $n$ degree there will $n$ zeros of the polynomial.

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

If there are exactly $n$ complex zeros there cannot be any real zeros.

And the maximum number of real zeros can be only be $n$ real zeros in the polynomial function.