Question

using the x method to factor

\\(ax^2 + bx + c\\)

which statement is true of \\(x^2 + 8x - 6\\)?

(x - 2) is a factor of the polynomial.
(x + 2) is a factor of the polynomial.
(x - 4) is a factor of the polynomial.
the polynomial is prime.

Explanation:

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<reasoning>

Identify the coefficients of the quadratic polynomial

\[

$$\begin{aligned} &x^2 + 8x - 6\\ &a = 1,\quad b = 8,\quad c = -6\\ &ac = 1 \cdot (-6) = -6 \end{aligned}$$

\]

Find factors of ac that sum to b

\[

$$\begin{aligned} &\text{Factors of } -6: \\ &1 \text{ and } -6 \implies \text{sum} = -5 \\ &-1 \text{ and } 6 \implies \text{sum} = 5 \\ &2 \text{ and } -3 \implies \text{sum} = -1 \\ &-2 \text{ and } 3 \implies \text{sum} = 1 \end{aligned}$$

\]

Determine if the polynomial is factorable

\[

$$\begin{aligned} &\text{No integer factors of } -6 \text{ sum to } 8.\\ &\text{Therefore, the polynomial cannot be factored over the integers.}\\ &\text{The polynomial is prime.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(x - 2) is a factor of the polynomial.</mcq-option>
<mcq-option>(x + 2) is a factor of the polynomial.</mcq-option>
<mcq-option>(x - 4) is a factor of the polynomial.</mcq-option>
<mcq-correct>The polynomial is prime.</mcq-correct>
</answer>

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Answer:

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<reasoning>

Identify the coefficients of the quadratic polynomial

\[

$$\begin{aligned} &x^2 + 8x - 6\\ &a = 1,\quad b = 8,\quad c = -6\\ &ac = 1 \cdot (-6) = -6 \end{aligned}$$

\]

Find factors of ac that sum to b

\[

$$\begin{aligned} &\text{Factors of } -6: \\ &1 \text{ and } -6 \implies \text{sum} = -5 \\ &-1 \text{ and } 6 \implies \text{sum} = 5 \\ &2 \text{ and } -3 \implies \text{sum} = -1 \\ &-2 \text{ and } 3 \implies \text{sum} = 1 \end{aligned}$$

\]

Determine if the polynomial is factorable

\[

$$\begin{aligned} &\text{No integer factors of } -6 \text{ sum to } 8.\\ &\text{Therefore, the polynomial cannot be factored over the integers.}\\ &\text{The polynomial is prime.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(x - 2) is a factor of the polynomial.</mcq-option>
<mcq-option>(x + 2) is a factor of the polynomial.</mcq-option>
<mcq-option>(x - 4) is a factor of the polynomial.</mcq-option>
<mcq-correct>The polynomial is prime.</mcq-correct>
</answer>

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