Chapter 4.3, Problem 9AYU

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 $f$.

9. Degree 4; zeros: $i$, $1+i$

To determine

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

Answer to Problem 9AYU

$\text{Complex conjugate }–i\text{ and }1–i\text{ are the remaining zeros of polynomial function }f\left(x\right)$.

Explanation of Solution

Given:

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

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

Since $i$ and $1+i$ are complex zeros of the polynomial function $f\left(x\right)$ and $f\left(x\right)$ has coefficients that are real numbers Complex zeros appear as conjugate pairs.

By the conjugate pair theorem $i$ and $1+i$ are complex zeros of the polynomial function $f\left(x\right)$.

The complex conjugate of $i$ and $1+i=–i$ and $1–i$ are also the zero of polynomial function $f\left(x\right)$.