Question

current objective
multiply and divide more than two rational expressions

question

simplify the rational expression: \\(\frac{3q^2 - 12q}{3q - 3} \cdot \frac{q^2 + q - 2}{q^2 + q - 6} \div \frac{3q^2 + 3q}{3q - 6}\\)

provide your answer below:

Explanation:

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<reasoning>

Convert division to multiplication

Using the Dividing Rational Expressions knowledge point
\[
\frac{3q^2 - 12q}{3q - 3} \cdot \frac{q^2 + q - 2}{q^2 + q - 6} \div \frac{3q^2 + 3q}{3q - 6} = \frac{3q^2 - 12q}{3q - 3} \cdot \frac{q^2 + q - 2}{q^2 + q - 6} \cdot \frac{3q - 6}{3q^2 + 3q}
\]

Factor all numerators and denominators

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

$$\begin{aligned} 3q^2 - 12q &= 3q(q - 4) \\ 3q - 3 &= 3(q - 1) \\ q^2 + q - 2 &= (q + 2)(q - 1) \\ q^2 + q - 6 &= (q + 3)(q - 2) \\ 3q - 6 &= 3(q - 2) \\ 3q^2 + 3q &= 3q(q + 1) \end{aligned}$$

\]

Substitute factored forms into the expression

Using the Multiplying Rational Expressions knowledge point
\[
\frac{3q(q - 4)}{3(q - 1)} \cdot \frac{(q + 2)(q - 1)}{(q + 3)(q - 2)} \cdot \frac{3(q - 2)}{3q(q + 1)}
\]

Combine and simplify the expression

Using the Multiplying Rational Expressions knowledge point
\[

$$\begin{aligned} &\frac{3q(q - 4) \cdot (q + 2)(q - 1) \cdot 3(q - 2)}{3(q - 1) \cdot (q + 3)(q - 2) \cdot 3q(q + 1)} \\ &= \frac{3 \cdot 3q \cdot (q - 4)(q + 2)(q - 1)(q - 2)}{3 \cdot 3q \cdot (q - 1)(q + 3)(q - 2)(q + 1)} \\ &= \frac{(q - 4)(q + 2)}{(q + 3)(q + 1)} \end{aligned}$$

\]
</reasoning>

<answer>
Simplify the rational expression: \(\frac{3q^2 - 12q}{3q - 3} \cdot \frac{q^2 + q - 2}{q^2 + q - 6} \div \frac{3q^2 + 3q}{3q - 6}\) is <blank>\(\frac{(q - 4)(q + 2)}{(q + 3)(q + 1)}\)</blank>.
</answer>

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Answer:

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<reasoning>

Convert division to multiplication

Using the Dividing Rational Expressions knowledge point
\[
\frac{3q^2 - 12q}{3q - 3} \cdot \frac{q^2 + q - 2}{q^2 + q - 6} \div \frac{3q^2 + 3q}{3q - 6} = \frac{3q^2 - 12q}{3q - 3} \cdot \frac{q^2 + q - 2}{q^2 + q - 6} \cdot \frac{3q - 6}{3q^2 + 3q}
\]

Factor all numerators and denominators

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

$$\begin{aligned} 3q^2 - 12q &= 3q(q - 4) \\ 3q - 3 &= 3(q - 1) \\ q^2 + q - 2 &= (q + 2)(q - 1) \\ q^2 + q - 6 &= (q + 3)(q - 2) \\ 3q - 6 &= 3(q - 2) \\ 3q^2 + 3q &= 3q(q + 1) \end{aligned}$$

\]

Substitute factored forms into the expression

Using the Multiplying Rational Expressions knowledge point
\[
\frac{3q(q - 4)}{3(q - 1)} \cdot \frac{(q + 2)(q - 1)}{(q + 3)(q - 2)} \cdot \frac{3(q - 2)}{3q(q + 1)}
\]

Combine and simplify the expression

Using the Multiplying Rational Expressions knowledge point
\[

$$\begin{aligned} &\frac{3q(q - 4) \cdot (q + 2)(q - 1) \cdot 3(q - 2)}{3(q - 1) \cdot (q + 3)(q - 2) \cdot 3q(q + 1)} \\ &= \frac{3 \cdot 3q \cdot (q - 4)(q + 2)(q - 1)(q - 2)}{3 \cdot 3q \cdot (q - 1)(q + 3)(q - 2)(q + 1)} \\ &= \frac{(q - 4)(q + 2)}{(q + 3)(q + 1)} \end{aligned}$$

\]
</reasoning>

<answer>
Simplify the rational expression: \(\frac{3q^2 - 12q}{3q - 3} \cdot \frac{q^2 + q - 2}{q^2 + q - 6} \div \frac{3q^2 + 3q}{3q - 6}\) is <blank>\(\frac{(q - 4)(q + 2)}{(q + 3)(q + 1)}\)</blank>.
</answer>

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