Chapter 2.6, Problem 4E

Solution

To find: the possible rational zeros of the function.

The possible rational zeros of the function $\pm 1,\pm 2,\pm 5,\pm 10$ .

Given information: 

The function $g\left(x\right)=x^3-4x^2+x-10$ .

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

  $g\left(x\right)=x^3-4x^2+x-10$

Here, the leading coefficient is $1$ , while the constant term is $10$ .

Factors of $1$ : $\pm 1$

Factor of $10$ :

  $\pm 1,\pm 2,\pm 5,\pm 10$

So, the possible rational zeros are

  $\pm \frac{1}{1},\pm \frac{2}{1},\pm \frac{5}{1},\pm \frac{10}{1}$

Simplify each fraction:

  $\pm 1,\pm 2,\pm 5,\pm 10$