Chapter 4.3, Problem 10AYU

In Problems 7-16, information is given about a polynomial function $f\left(x\right)$ whose coefficients are real numbers. Find the remaining zeros of $f$.

10. Degree 4; zeros: 1, 2, $2+i$

To determine

To find: The remaining zeros of the polynomial function $f\left(x\right)$ whose coefficients are real numbers.

Answer to Problem 10AYU

$\text{Complex conjugate }2–i\text{ is the remaining zero of polynomial function }f\left(x\right)$.

Explanation of Solution

Given:

Degree $=4$, zeros $=1,2,2+i$.

Since the degree $=4$ there will be 4 real zeros.

Since $2+i$ is the complex zero of the polynomial function $f\left(x\right)$ and $f\left(x\right)$ has coefficients that are real numbers Complex zero appear as conjugate pairs.

By the conjugate pair theorem $2+i$ is the complex zeros of the polynomial function $f\left(x\right)$.

The complex conjugate of $2+i=2–i$ is also the zero of polynomial function $f\left(x\right)$.