Chapter 2.6, Problem 7E

Solution

To find: the possible rational zeros of the function.

The possible rational zeros of the function $\pm 1,\pm 2,\pm 7,\pm 14,\pm \frac{1}{2},\pm \frac{7}{2},\pm \frac{1}{4},\pm \frac{7}{4}$ .

Given information: 

The function $g\left(x\right)=4x^5+3x^3-2x-14$ .

Formula used:

The Rational Zero Theorem:

If $f\left(x\right)=a_n{x}^n+\cdots +a_{1}x+a_0$ consists of integer coefficients, then the shape of each rational zero of $f$ is as follows:

  $\frac{p}{q}=\frac{\text{ factor of constant term }{a}_0}{\text{ factor of leading coefficient }{a}_n}$

Calculation:

Consider the function

  $g\left(x\right)=4x^5+3x^3-2x-14$

Factors of constant term 14:

  $p=1,2,7,14$

Leading coefficient factors $4$ :

  $q=1,2,4$

All possible combinations of $\pm \frac{p}{q}$ .

Some or none may be zeros of the function.

Simplify and remove any extra duplicate values.

  $\begin{array}{c}\pm \frac{p}{q}=\pm \left(1,2,7,14,\frac{1}{2},\frac{2}{2},\frac{7}{2},\frac{14}{2},\frac{1}{4},\frac{2}{4},\frac{7}{4}\frac{14}{4}\right)\\ =\pm 1,\pm 2,\pm 7,\pm 14,\pm \frac{1}{2},\pm \frac{7}{2},\pm \frac{1}{4},\pm \frac{7}{4}\end{array}$