Chapter 6.7, Problem 39PPS

To determine

To find: The zeros of the function $f\left(x\right)=x^3-5x^2+17x-85$ .

Answer to Problem 39PPS

The zeros are $5$ , $\sqrt{17}i$ and $-\sqrt{17}i$ .

Explanation of Solution

Given:

The function is $f\left(x\right)=x^3-5x^2+17x-85$ .

Calculation:

Group the terms to take out the Greatest common factor as follows.

  $\begin{array}{c}f\left(x\right)=\left(x^3-5x^2\right)+\left(17x-85\right)\\ =x^2\left(x-5\right)+17\left(x-5\right)\\ =\left(x-5\right)\left(x^2+17\right)\end{array}$

Set the equation equal to zero.

  $\left(x-5\right)\left(x^2+17\right)=0$

Solve for $x$ .

  $\begin{array}{c}x-5=0\\ x=5\\ x^2+17=0\\ x=\pm \sqrt{17}i\end{array}$

Therefore, the zeros are $5$ , $\sqrt{17}i$ and $-\sqrt{17}i$ .