Chapter 5.5, Problem 7LC

Solution

a.

The objective is to determine whether the given statement is always, sometimes, or never true.

The given statement can never be true as 5 cannot be a root of the equation $3x^3+r{x}^2+sx+8=0$ .

Given statement:

A root of the equation $3x^3+r{x}^2+sx+8=0$ could be 5.

Concept used:

According to the Rational root theorem,

  $\text{The only possible rational roots have the form}=\frac{\text{Factor of constant term}}{\text{Factor of the leading coefficient}}$

Explanation:

Factors of the constant term $\left(8\right)$ : $\pm 1,\pm 2,\pm 4,$ and $\pm 8$

Factors the coefficient of the leading term $\left(3\right)$ : $\pm 1,$ and $\pm 3$

Here, 5 cannot divide 8, thus it cannot be a possible root of $3x^3+r{x}^2+sx+8=0$ .

Thus, the given statement can never be true.

b.

The objective is to determine whether the given statement is always, sometimes, or never true.

The given statement is always true.

Given statement:

A root of the equation $3x^3+r{x}^2+sx+8=0$ could be $-2$ .

Concept used:

According to the Rational root theorem,

  $\text{The only possible rational roots have the form}=\frac{\text{Factor of constant term}}{\text{Factor of the leading coefficient}}$

Explanation:

Factors of the constant term $\left(8\right)$ : $\pm 1,\pm 2,\pm 4,$ and $\pm 8$

Factors the coefficient of the leading term $\left(3\right)$ : $\pm 1,$ and $\pm 3$

By using Rational root theorem,

  $\text{The only possible rational roots have the form}=\pm \frac{1}{1},\pm \frac{2}{1},\pm \frac{4}{1},\pm \frac{8}{1},\pm \frac{1}{3},\pm \frac{2}{3},\pm \frac{4}{3},\text{ and }\pm \frac{8}{3}$.

Here, $-2$ is included in the list of all possible rational roots.

Thus, the given statement is always true.