Question

analyzing a method of factorization

ginny factored \\(6x^2 - 31x - 30\\) as shown:

1. \\(ac = -180\\) and \\(b = 31\\)
2. \\(36(-5) = -180\\) and \\(36 + (-5) = 31\\)
3. \\(6x^2 + 36x - 5x - 30\\)
4. \\(6x(x + 6) - 5(x + 6)\\)
5. \\((x + 6)(6x - 5)\\)

determine if ginny factored correctly. if not, explain where she made an error.

• ginny made a mistake in step 1 when she identified \\(b = 31\\). it should be \\(b = -31\\).
• ginny was correct until step 3 when she used 36 and -5 as the coefficients of \\(x\\).
• ginny was correct until step 4 when she incorrectly factored the gcf from \\(5x - 30\\).
• ginny factored the trinomial correctly.

Explanation:

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<reasoning>

Identify the trinomial coefficients

The given quadratic trinomial is \(6x^2 - 31x - 30\).
A quadratic trinomial in standard form is written as \(ax^2 + bx + c\).
Comparing the coefficients, we find:
\(a = 6\)
\(b = -31\)
\(c = -30\)

Using the Trinomial Coefficients knowledge point, we calculate the product \(ac\):
\[ac = 6 \times (-30) = -180\]
The linear coefficient is \(b = -31\).

Analyze Ginny's Step 1

In Step 1, Ginny wrote:
"1. \(ac = -180\) and \(b = 31\)"

Comparing this to our identified coefficients, Ginny correctly identified \(ac = -180\), but she incorrectly identified \(b = 31\). The correct value is \(b = -31\) because of the negative sign in front of the linear term \(-31x\).

Using the Factoring Polynomials knowledge point, identifying the correct sign of the coefficients is essential for finding the correct factors that add up to \(b\).

Evaluate the options

Let's check the given multiple-choice options:

1. "Ginny made a mistake in step 1 when she identified \(b = 31\). It should be \(b = -31\)."
2. "Ginny was correct until step 3 when she used 36 and -5 as the coefficients of \(x\)."
3. "Ginny was correct until step 4 when she incorrectly factored the GCF from \(5x - 30\)."
4. "Ginny factored the trinomial correctly."

Since Ginny's error occurs immediately in Step 1 where she wrote \(b = 31\) instead of \(b = -31\), the first option is the correct choice.
</reasoning>

<answer>
<mcq-correct>(A) Ginny made a mistake in step 1 when she identified \(b = 31\). It should be \(b = -31\).</mcq-correct>
<mcq-option>(B) Ginny was correct until step 3 when she used 36 and -5 as the coefficients of \(x\).</mcq-option>
<mcq-option>(C) Ginny was correct until step 4 when she incorrectly factored the GCF from \(5x - 30\).</mcq-option>
<mcq-option>(D) Ginny factored the trinomial correctly.</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify the trinomial coefficients

The given quadratic trinomial is \(6x^2 - 31x - 30\).
A quadratic trinomial in standard form is written as \(ax^2 + bx + c\).
Comparing the coefficients, we find:
\(a = 6\)
\(b = -31\)
\(c = -30\)

Using the Trinomial Coefficients knowledge point, we calculate the product \(ac\):
\[ac = 6 \times (-30) = -180\]
The linear coefficient is \(b = -31\).

Analyze Ginny's Step 1

In Step 1, Ginny wrote:
"1. \(ac = -180\) and \(b = 31\)"

Comparing this to our identified coefficients, Ginny correctly identified \(ac = -180\), but she incorrectly identified \(b = 31\). The correct value is \(b = -31\) because of the negative sign in front of the linear term \(-31x\).

Using the Factoring Polynomials knowledge point, identifying the correct sign of the coefficients is essential for finding the correct factors that add up to \(b\).

Evaluate the options

Let's check the given multiple-choice options:

1. "Ginny made a mistake in step 1 when she identified \(b = 31\). It should be \(b = -31\)."
2. "Ginny was correct until step 3 when she used 36 and -5 as the coefficients of \(x\)."
3. "Ginny was correct until step 4 when she incorrectly factored the GCF from \(5x - 30\)."
4. "Ginny factored the trinomial correctly."

Since Ginny's error occurs immediately in Step 1 where she wrote \(b = 31\) instead of \(b = -31\), the first option is the correct choice.
</reasoning>

<answer>
<mcq-correct>(A) Ginny made a mistake in step 1 when she identified \(b = 31\). It should be \(b = -31\).</mcq-correct>
<mcq-option>(B) Ginny was correct until step 3 when she used 36 and -5 as the coefficients of \(x\).</mcq-option>
<mcq-option>(C) Ginny was correct until step 4 when she incorrectly factored the GCF from \(5x - 30\).</mcq-option>
<mcq-option>(D) Ginny factored the trinomial correctly.</mcq-option>
</answer>

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