Chapter 4.6, Problem 46E

To determine

To find: the correct one with justification.

Answer to Problem 46E

Second claim is correct

Explanation of Solution

Given information:

First claim is $2-i$ is a complex zero of the polynomial function $f\left(x\right)=x^3-2x^2+2x+5i$ but that its conjugate is not a zero.

Second claim is both $2-i$ and its conjugate must be zeros by the complex conjugates.

Calculation:

The complex conjugate theorem states that “if $f$ is a polynomial function with real coefficient, and $a+bi$ s an imaginary zero of $f$ , then $a-bi$ is also a zero of $f$ ”.

By the complex conjugate theorem, if $2-i$ is a complex zero of function

  $f\left(x\right)=x^3-2x^2+2x+5i$ , so $2+i$ must be other zeros of function.

Therefore second claim is correct.