Chapter 2.4, Problem 56E

Solution

To find: All the zeros of the function $f\left(x\right)=3x^4-2x^3+3x^2+x-2$ .

  $\frac{2}{3}$

Given information:

  $f\left(x\right)=3x^4-2x^3+3x^2+x-2$

Concept Used:

Divide a polynomial function $g\left(x\right)$ by $\left(x-k\right)$ , using synthetic division and the number at last is zero, then $k$ is a real root of the equation $g\left(x\right)=0$ .

Calculation:

Setup the synthetic division for $x=\frac{2}{3}$ ,

  $\begin{array}{cccccc}\frac{2}{3}& 3& -2& 3& 1& -2\\ & \downarrow & 2& 0& 2& 2\\ & 3& 0& 3& 3& \boxed{0}\end{array}$

Since the last number of the bottom rows is 0, the $x=\frac{2}{3}$ is a real root of the function.

The quotient of the polynomial is $3x^3+3x+3$ .

The polynomial is prime and can not factor. Using graphing calculator to find the zeros.

It won’t be exact zeros.

It has only one exact real zeros and one real zero, others two are complex zeros.