Question

question

divide: \\(\frac{y^2 - y - 6}{6y + 66} \div \frac{-y^2 - y + 12}{12y + 132}\\).

provide your answer below:

Explanation:

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<reasoning>

Convert division to multiplication

Using the Dividing Rational Expressions knowledge point
\[
\frac{y^2 - y - 6}{6y + 66} \div \frac{-y^2 - y + 12}{12y + 132} = \frac{y^2 - y - 6}{6y + 66} \cdot \frac{12y + 132}{-y^2 - y + 12}
\]

Factor the numerators

Using the Factoring Trinomials and Greatest Common Factor Factoring knowledge points
\[

$$\begin{aligned} y^2 - y - 6 &= (y - 3)(y + 2) \\ -y^2 - y + 12 &= -(y^2 + y - 12) = -(y + 4)(y - 3) \end{aligned}$$

\]

Factor the denominators

Using the Greatest Common Factor Factoring knowledge point
\[

$$\begin{aligned} 6y + 66 &= 6(y + 11) \\ 12y + 132 &= 12(y + 11) \end{aligned}$$

\]

Substitute and simplify the expression

Using the Opposite Factors knowledge point
\[

$$\begin{aligned} \frac{(y - 3)(y + 2)}{6(y + 11)} \cdot \frac{12(y + 11)}{-(y + 4)(y - 3)} &= \frac{(y - 3)(y + 2) \cdot 12(y + 11)}{6(y + 11) \cdot [-(y + 4)(y - 3)]} \\ &= \frac{12}{6} \cdot \frac{y + 11}{y + 11} \cdot \frac{y - 3}{y - 3} \cdot \frac{y + 2}{-(y + 4)} \\ &= 2 \cdot 1 \cdot 1 \cdot \frac{y + 2}{-(y + 4)} \\ &= -\frac{2(y + 2)}{y + 4} \end{aligned}$$

\]
</reasoning>

<answer>
Divide: \(\frac{y^2 - y - 6}{6y + 66} \div \frac{-y^2 - y + 12}{12y + 132}\).

Provide your answer below:
<blank>\(-\frac{2(y + 2)}{y + 4}\)</blank>
</answer>

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</post_analysis>

Answer:

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<reasoning>

Convert division to multiplication

Using the Dividing Rational Expressions knowledge point
\[
\frac{y^2 - y - 6}{6y + 66} \div \frac{-y^2 - y + 12}{12y + 132} = \frac{y^2 - y - 6}{6y + 66} \cdot \frac{12y + 132}{-y^2 - y + 12}
\]

Factor the numerators

Using the Factoring Trinomials and Greatest Common Factor Factoring knowledge points
\[

$$\begin{aligned} y^2 - y - 6 &= (y - 3)(y + 2) \\ -y^2 - y + 12 &= -(y^2 + y - 12) = -(y + 4)(y - 3) \end{aligned}$$

\]

Factor the denominators

Using the Greatest Common Factor Factoring knowledge point
\[

$$\begin{aligned} 6y + 66 &= 6(y + 11) \\ 12y + 132 &= 12(y + 11) \end{aligned}$$

\]

Substitute and simplify the expression

Using the Opposite Factors knowledge point
\[

$$\begin{aligned} \frac{(y - 3)(y + 2)}{6(y + 11)} \cdot \frac{12(y + 11)}{-(y + 4)(y - 3)} &= \frac{(y - 3)(y + 2) \cdot 12(y + 11)}{6(y + 11) \cdot [-(y + 4)(y - 3)]} \\ &= \frac{12}{6} \cdot \frac{y + 11}{y + 11} \cdot \frac{y - 3}{y - 3} \cdot \frac{y + 2}{-(y + 4)} \\ &= 2 \cdot 1 \cdot 1 \cdot \frac{y + 2}{-(y + 4)} \\ &= -\frac{2(y + 2)}{y + 4} \end{aligned}$$

\]
</reasoning>

<answer>
Divide: \(\frac{y^2 - y - 6}{6y + 66} \div \frac{-y^2 - y + 12}{12y + 132}\).

Provide your answer below:
<blank>\(-\frac{2(y + 2)}{y + 4}\)</blank>
</answer>

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"Mathematics",
"Algebra",
"Dividing Rational Expressions"
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