Question

question

multiply: \\(\frac{2x - 6}{x^2 - 8x + 15} \cdot \frac{x^2 - 25}{2x + 10}\\).

provide your answer below:

Explanation:

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<reasoning>

Factor the numerators

Using the Greatest Common Factor Factoring and Difference of Squares knowledge points
\[

$$\begin{aligned} 2x - 6 &= 2(x - 3) \\ x^2 - 25 &= (x - 5)(x + 5) \end{aligned}$$

\]

Factor the denominators

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

$$\begin{aligned} x^2 - 8x + 15 &= (x - 3)(x - 5) \\ 2x + 10 &= 2(x + 5) \end{aligned}$$

\]

Multiply and simplify the expression

Using the Multiplying Rational Expressions knowledge point
\[

$$\begin{aligned} \frac{2x - 6}{x^2 - 8x + 15} \cdot \frac{x^2 - 25}{2x + 10} &= \frac{2(x - 3)}{(x - 3)(x - 5)} \cdot \frac{(x - 5)(x + 5)}{2(x + 5)} \\ &= \frac{2(x - 3)(x - 5)(x + 5)}{2(x - 3)(x - 5)(x + 5)} \\ &= 1 \quad (x eq 3, 5, -5) \end{aligned}$$

\]
</reasoning>

<answer>
Multiply: \(\frac{2x - 6}{x^2 - 8x + 15} \cdot \frac{x^2 - 25}{2x + 10} =\) <blank>\(1\)</blank>
</answer>

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Answer:

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<reasoning>

Factor the numerators

Using the Greatest Common Factor Factoring and Difference of Squares knowledge points
\[

$$\begin{aligned} 2x - 6 &= 2(x - 3) \\ x^2 - 25 &= (x - 5)(x + 5) \end{aligned}$$

\]

Factor the denominators

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

$$\begin{aligned} x^2 - 8x + 15 &= (x - 3)(x - 5) \\ 2x + 10 &= 2(x + 5) \end{aligned}$$

\]

Multiply and simplify the expression

Using the Multiplying Rational Expressions knowledge point
\[

$$\begin{aligned} \frac{2x - 6}{x^2 - 8x + 15} \cdot \frac{x^2 - 25}{2x + 10} &= \frac{2(x - 3)}{(x - 3)(x - 5)} \cdot \frac{(x - 5)(x + 5)}{2(x + 5)} \\ &= \frac{2(x - 3)(x - 5)(x + 5)}{2(x - 3)(x - 5)(x + 5)} \\ &= 1 \quad (x eq 3, 5, -5) \end{aligned}$$

\]
</reasoning>

<answer>
Multiply: \(\frac{2x - 6}{x^2 - 8x + 15} \cdot \frac{x^2 - 25}{2x + 10} =\) <blank>\(1\)</blank>
</answer>

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