Question

use the rational zero theorem to list all possible rational zeros for the given function.

(f(x) = 7x^4 - x^2 + 5)

a. (pm \frac{1}{7}, pm \frac{1}{5}, pm 1, pm 5, pm 7)
b. (pm \frac{1}{7}, pm \frac{5}{7}, pm 1, pm 5)
c. (pm \frac{1}{7}, pm \frac{5}{7}, pm 1, pm 5, pm 7)
d. (pm \frac{1}{5}, pm \frac{7}{5}, pm 1, pm 7)

Explanation:

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<reasoning>

Identify the leading coefficient and constant term

Using the Rational Root Theorem knowledge point
\[

$$\begin{aligned} &f(x) = 7x^4 - x^2 + 5\\ &\text{Leading coefficient } a_n = 7\\ &\text{Constant term } a_0 = 5 \end{aligned}$$

\]

Find all factors of the constant term

Using the Rational Root Theorem knowledge point
\[

$$\begin{aligned} &p \in \text{Factors of } 5\\ &p = \pm 1, \pm 5 \end{aligned}$$

\]

Find all factors of the leading coefficient

Using the Rational Root Theorem knowledge point
\[

$$\begin{aligned} &q \in \text{Factors of } 7\\ &q = \pm 1, \pm 7 \end{aligned}$$

\]

Form all possible rational zeros

Using the Rational Root Theorem knowledge point
\[

$$\begin{aligned} &\frac{p}{q} = \frac{\pm 1, \pm 5}{\pm 1, \pm 7}\\ &\frac{p}{q} = \pm 1, \pm 5, \pm \frac{1}{7}, \pm \frac{5}{7} \end{aligned}$$

\]

Match with the given options

Using the Rational Root Theorem knowledge point
\[

$$\begin{aligned} &\text{Possible zeros: } \pm \frac{1}{7}, \pm \frac{5}{7}, \pm 1, \pm 5\\ &\text{This matches Option B.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>A. \(\pm \frac{1}{7}, \pm \frac{1}{5}, \pm 1, \pm 5, \pm 7\)</mcq-option>
<mcq-correct>B. \(\pm \frac{1}{7}, \pm \frac{5}{7}, \pm 1, \pm 5\)</mcq-correct>
<mcq-option>C. \(\pm \frac{1}{7}, \pm \frac{5}{7}, \pm 1, \pm 5, \pm 7\)</mcq-option>
<mcq-option>D. \(\pm \frac{1}{5}, \pm \frac{7}{5}, \pm 1, \pm 7\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify the leading coefficient and constant term

Using the Rational Root Theorem knowledge point
\[

$$\begin{aligned} &f(x) = 7x^4 - x^2 + 5\\ &\text{Leading coefficient } a_n = 7\\ &\text{Constant term } a_0 = 5 \end{aligned}$$

\]

Find all factors of the constant term

Using the Rational Root Theorem knowledge point
\[

$$\begin{aligned} &p \in \text{Factors of } 5\\ &p = \pm 1, \pm 5 \end{aligned}$$

\]

Find all factors of the leading coefficient

Using the Rational Root Theorem knowledge point
\[

$$\begin{aligned} &q \in \text{Factors of } 7\\ &q = \pm 1, \pm 7 \end{aligned}$$

\]

Form all possible rational zeros

Using the Rational Root Theorem knowledge point
\[

$$\begin{aligned} &\frac{p}{q} = \frac{\pm 1, \pm 5}{\pm 1, \pm 7}\\ &\frac{p}{q} = \pm 1, \pm 5, \pm \frac{1}{7}, \pm \frac{5}{7} \end{aligned}$$

\]

Match with the given options

Using the Rational Root Theorem knowledge point
\[

$$\begin{aligned} &\text{Possible zeros: } \pm \frac{1}{7}, \pm \frac{5}{7}, \pm 1, \pm 5\\ &\text{This matches Option B.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>A. \(\pm \frac{1}{7}, \pm \frac{1}{5}, \pm 1, \pm 5, \pm 7\)</mcq-option>
<mcq-correct>B. \(\pm \frac{1}{7}, \pm \frac{5}{7}, \pm 1, \pm 5\)</mcq-correct>
<mcq-option>C. \(\pm \frac{1}{7}, \pm \frac{5}{7}, \pm 1, \pm 5, \pm 7\)</mcq-option>
<mcq-option>D. \(\pm \frac{1}{5}, \pm \frac{7}{5}, \pm 1, \pm 7\)</mcq-option>
</answer>

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