Chapter 12.3, Problem 54E

To determine

To calculate:

The possible rational zeros for $f\left(x\right)=8x^3+3x-2$ .

Answer to Problem 54E

The possible rational zeros are $\pm 1,\pm 2,\pm \frac{1}{2},\pm \frac{1}{4}\pm \frac{1}{8}$ .

Explanation of Solution

Given information:

  $f\left(x\right)=8x^3+3x-2$ .

Calculation:

Consider the function $f\left(x\right)=8x^3+3x-2$ .

List all possible rational zeros of the given function.

Let $a_0{x}^n+a_1{x}^{n-1}+\mathrm{...}+a_{n-1}x+a_n=0$ represent a polynomial equation of degree $n$ with integral coefficients.

If a rational number $\frac{p}{q}$ , where $p$ and $q$ have no common factors, is a root of the equation, then $p$ is a factor of $a_n$ and $q$ is a factor of $a_0$ :

From the above rational root theorem,

If $\frac{p}{q}$ is a root of the equation then $p$ is a factor of

  $2$ and $q$ is a factor of $8$ .

Possible values of $p$ are $\pm 1,\pm 2$ .

Possible values of are $\pm 1,\pm 2,\pm 4,\pm 8$ .

Possible rational roots, $\frac{p}{q}:\pm 1,\pm 2,\pm \frac{1}{2},\pm \frac{1}{4}\pm \frac{1}{8}$ .

Hence, the possible rational zeros are $\pm 1,\pm 2,\pm \frac{1}{2},\pm \frac{1}{4}\pm \frac{1}{8}$ .