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.

\\1, -1, 2, -2, 3, -3, 4, -4, 6, -6, 12, -12\\
(use a comma to separate answers as needed.)

b. use synthetic division to test several possible rational zeros in order to identify one actual zero.

one rational zero of the given function is 2.
(simplify your answer.)

c. use the zero from part (b) to find all the zeros of the polynomial function.

the zeros of the function \\f(x) = x^3 - 3x^2 - 4x + 12\\ are
(simplify your answer. type an integer or a fraction. use a comma to separate answers as needed.)

Explanation:

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<reasoning>

Identify the given function and the zero from part (b)

Using the Rational Root Theorem and Synthetic Division knowledge points
\[

$$\begin{aligned} &f(x) = x^3 - 3x^2 - 4x + 12\\ &\text{Known zero from part (b): } x = 2 \end{aligned}$$

\]

Perform synthetic division to find the depressed polynomial

Using the Synthetic Division knowledge point
\[

$$\begin{array}{r|rrrr} 2 & 1 & -3 & -4 & 12 \\ & & 2 & -2 & -12 \\ \hline & 1 & -1 & -6 & 0 \end{array}$$

\]
The quotient polynomial is:
\[
q(x) = x^2 - x - 6
\]

Find the remaining zeros by factoring the quadratic quotient

Using the Quadratic Equations knowledge point
\[

$$\begin{aligned} &x^2 - x - 6 = 0\\ &(x - 3)(x + 2) = 0\\ &x = 3 \quad \text{or} \quad x = -2 \end{aligned}$$

\]

List all zeros of the polynomial function

Using the Finding Zeros of Factored Polynomials knowledge point
\[

$$\begin{aligned} &\text{Zeros: } x = 2, \quad x = 3, \quad x = -2 \end{aligned}$$

\]
</reasoning>

<answer>
The zeros of the function \(f(x) = x^3 - 3x^2 - 4x + 12\) are <blank>\(2, 3, -2\)</blank>
</answer>

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Answer:

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<reasoning>

Identify the given function and the zero from part (b)

Using the Rational Root Theorem and Synthetic Division knowledge points
\[

$$\begin{aligned} &f(x) = x^3 - 3x^2 - 4x + 12\\ &\text{Known zero from part (b): } x = 2 \end{aligned}$$

\]

Perform synthetic division to find the depressed polynomial

Using the Synthetic Division knowledge point
\[

$$\begin{array}{r|rrrr} 2 & 1 & -3 & -4 & 12 \\ & & 2 & -2 & -12 \\ \hline & 1 & -1 & -6 & 0 \end{array}$$

\]
The quotient polynomial is:
\[
q(x) = x^2 - x - 6
\]

Find the remaining zeros by factoring the quadratic quotient

Using the Quadratic Equations knowledge point
\[

$$\begin{aligned} &x^2 - x - 6 = 0\\ &(x - 3)(x + 2) = 0\\ &x = 3 \quad \text{or} \quad x = -2 \end{aligned}$$

\]

List all zeros of the polynomial function

Using the Finding Zeros of Factored Polynomials knowledge point
\[

$$\begin{aligned} &\text{Zeros: } x = 2, \quad x = 3, \quad x = -2 \end{aligned}$$

\]
</reasoning>

<answer>
The zeros of the function \(f(x) = x^3 - 3x^2 - 4x + 12\) are <blank>\(2, 3, -2\)</blank>
</answer>

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