Question

what is the factorization of (3x^2 - 8x + 5)?

(3x - 1)(x + 5)
(3x + 5)(x - 1)
(3x - 1)(x - 5)
(3x - 5)(x - 1)

Explanation:

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<reasoning>

Identify the coefficients of the quadratic trinomial

\[
a = 3, \quad b = -8, \quad c = 5
\]

Find two numbers that multiply to \(ac\) and add to \(b\)

\[

$$\begin{aligned} &ac = 3 \times 5 = 15\\ &b = -8\\ &\text{The two numbers are } -3 \text{ and } -5 \text{ since:}\\ &(-3) \times (-5) = 15\\ &(-3) + (-5) = -8 \end{aligned}$$

\]

Rewrite and factor the trinomial by grouping

\[

$$\begin{aligned} 3x^2 - 8x + 5 &= 3x^2 - 3x - 5x + 5\\ &= 3x(x - 1) - 5(x - 1)\\ &= (3x - 5)(x - 1) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \((3x - 1)(x + 5)\)</mcq-option>
<mcq-option>(B) \((3x + 5)(x - 1)\)</mcq-option>
<mcq-option>(C) \((3x - 1)(x - 5)\)</mcq-option>
<mcq-correct>(D) \((3x - 5)(x - 1)\)</mcq-correct>
</answer>

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Answer:

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<reasoning>

Identify the coefficients of the quadratic trinomial

\[
a = 3, \quad b = -8, \quad c = 5
\]

Find two numbers that multiply to \(ac\) and add to \(b\)

\[

$$\begin{aligned} &ac = 3 \times 5 = 15\\ &b = -8\\ &\text{The two numbers are } -3 \text{ and } -5 \text{ since:}\\ &(-3) \times (-5) = 15\\ &(-3) + (-5) = -8 \end{aligned}$$

\]

Rewrite and factor the trinomial by grouping

\[

$$\begin{aligned} 3x^2 - 8x + 5 &= 3x^2 - 3x - 5x + 5\\ &= 3x(x - 1) - 5(x - 1)\\ &= (3x - 5)(x - 1) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \((3x - 1)(x + 5)\)</mcq-option>
<mcq-option>(B) \((3x + 5)(x - 1)\)</mcq-option>
<mcq-option>(C) \((3x - 1)(x - 5)\)</mcq-option>
<mcq-correct>(D) \((3x - 5)(x - 1)\)</mcq-correct>
</answer>

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