Chapter 2.6, Problem 36E

Solution

To describe: correct the error.

Possible zeros: $\pm 1,\pm 2,\pm 7,\pm 14$

Error description: the negative numbers were forgotten.

Given information: 

The function $f\left(x\right)=x^3+7x^2+2x+14$ .

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

  $f\left(x\right)=x^3+7x^2+2x+14$

Here, the constant term is $14$ , and the leading coefficient is $1.$

Factors of $1$ :

  $\pm 1$

Factors of $14$ :

  $\pm 1,\pm 2,\pm 7,\pm 14$

So, possible rational zeros are (write all combinations of factors of $1$ and factors of $14$ ):

  $\pm \frac{1}{1},\pm \frac{2}{1},\pm \frac{7}{1},\pm \frac{14}{1}$

Simplify fractions:

  $\pm 1,\pm 2,\pm 7,\pm 14$

In this, it is obvious that the negative numbers were forgotten.