Question

complete the factorization of (3x^2 - 5x + 2).
which two factors can be multiplied together to make this trinomial?

((x - 2))
((x - 1))
((x + 1))
((3x - 2))
((3x + 2))

Explanation:

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<reasoning>

Identify the trinomial to factor

The given trinomial is:
\[
3x^2 - 5x + 2
\]

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

For the quadratic expression \(ax^2 + bx + c\):
\[
a = 3, \quad b = -5, \quad c = 2
\]
The product is:
\[
ac = 3 \times 2 = 6
\]
The sum is:
\[
b = -5
\]
The two numbers that multiply to \(6\) and add to \(-5\) are \(-3\) and \(-2\).

Rewrite and factor by grouping

Rewrite the middle term using \(-3x\) and \(-2x\):
\[
3x^2 - 3x - 2x + 2
\]
Group the terms:
\[
(3x^2 - 3x) - (2x - 2)
\]
Factor out the greatest common factor from each group:
\[
3x(x - 1) - 2(x - 1)
\]
Factor out the common binomial \((x - 1)\):
\[
(3x - 2)(x - 1)
\]
The two factors are \((x - 1)\) and \((3x - 2)\).
</reasoning>

<answer>
<mcq-option>☐ \((x-2)\)</mcq-option>
<mcq-correct>☑ \((x-1)\)</mcq-correct>
<mcq-option>☐ \((x+1)\)</mcq-option>
<mcq-correct>☑ \((3x-2)\)</mcq-correct>
<mcq-option>☐ \((3x+2)\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify the trinomial to factor

The given trinomial is:
\[
3x^2 - 5x + 2
\]

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

For the quadratic expression \(ax^2 + bx + c\):
\[
a = 3, \quad b = -5, \quad c = 2
\]
The product is:
\[
ac = 3 \times 2 = 6
\]
The sum is:
\[
b = -5
\]
The two numbers that multiply to \(6\) and add to \(-5\) are \(-3\) and \(-2\).

Rewrite and factor by grouping

Rewrite the middle term using \(-3x\) and \(-2x\):
\[
3x^2 - 3x - 2x + 2
\]
Group the terms:
\[
(3x^2 - 3x) - (2x - 2)
\]
Factor out the greatest common factor from each group:
\[
3x(x - 1) - 2(x - 1)
\]
Factor out the common binomial \((x - 1)\):
\[
(3x - 2)(x - 1)
\]
The two factors are \((x - 1)\) and \((3x - 2)\).
</reasoning>

<answer>
<mcq-option>☐ \((x-2)\)</mcq-option>
<mcq-correct>☑ \((x-1)\)</mcq-correct>
<mcq-option>☐ \((x+1)\)</mcq-option>
<mcq-correct>☑ \((3x-2)\)</mcq-correct>
<mcq-option>☐ \((3x+2)\)</mcq-option>
</answer>

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