Chapter 4.3, Problem 4AYU

4. If $3+4i$ is a zero of a polynomial function of degree 5 with real coefficients, then so is ________.

To determine

To fill: If $3+4i$ is a zero of a polynomial function of degree 5 with real coefficients, then so is ________.

Answer to Problem 4AYU

$\text{If 3 + 4i is a zero of a polynomial function of degree 5 with real coefficients, then so is 3 – 4i is zero of the polynomial function}$.

Explanation of Solution

Given:

Polynomial function $f\left(x\right)$ of degree 5 with real coefficients has 5 zeros.

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

If $3+4i$ is a zero of a polynomial function of degree 5 with real coefficients, then its conjugate pair $=3–4i$ .

$3–4i$ is also the zero the polynomial function of degree 5 with real coefficients.