Question

the following function is given.

\\f(x) = x^3 - 3x^2 - 4x + 12\\

a. list all rational zeros that are possible according to the rational zero theorem.

(use a comma to separate answers as needed.)

Explanation:

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<reasoning>

Identify the coefficients of the polynomial

Using the Rational Root Theorem knowledge point
\[

$$\begin{aligned} &f(x) = x^3 - 3x^2 - 4x + 12\\ &\text{Leading coefficient } a_n = 1\\ &\text{Constant term } a_0 = 12 \end{aligned}$$

\]

Find the factors of the constant term and leading coefficient

Using the Rational Root Theorem knowledge point
\[

$$\begin{aligned} &\text{Factors of } a_0 = 12 \quad (p): \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12\\ &\text{Factors of } a_n = 1 \quad (q): \pm 1 \end{aligned}$$

\]

List all possible rational zeros

Using the Rational Root Theorem knowledge point
\[

$$\begin{aligned} &\frac{p}{q} = \frac{\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12}{\pm 1}\\ &\frac{p}{q} = 1, -1, 2, -2, 3, -3, 4, -4, 6, -6, 12, -12 \end{aligned}$$

\]
</reasoning>

<answer>
a. List all rational zeros that are possible according to the Rational Zero Theorem.

<blank>\(1, -1, 2, -2, 3, -3, 4, -4, 6, -6, 12, -12\)</blank>

(Use a comma to separate answers as needed.)
</answer>

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Answer:

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<reasoning>

Identify the coefficients of the polynomial

Using the Rational Root Theorem knowledge point
\[

$$\begin{aligned} &f(x) = x^3 - 3x^2 - 4x + 12\\ &\text{Leading coefficient } a_n = 1\\ &\text{Constant term } a_0 = 12 \end{aligned}$$

\]

Find the factors of the constant term and leading coefficient

Using the Rational Root Theorem knowledge point
\[

$$\begin{aligned} &\text{Factors of } a_0 = 12 \quad (p): \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12\\ &\text{Factors of } a_n = 1 \quad (q): \pm 1 \end{aligned}$$

\]

List all possible rational zeros

Using the Rational Root Theorem knowledge point
\[

$$\begin{aligned} &\frac{p}{q} = \frac{\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12}{\pm 1}\\ &\frac{p}{q} = 1, -1, 2, -2, 3, -3, 4, -4, 6, -6, 12, -12 \end{aligned}$$

\]
</reasoning>

<answer>
a. List all rational zeros that are possible according to the Rational Zero Theorem.

<blank>\(1, -1, 2, -2, 3, -3, 4, -4, 6, -6, 12, -12\)</blank>

(Use a comma to separate answers as needed.)
</answer>

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