Question

use factoring to find the simplified product of these rational expressions.

\\\frac{x^2 - 81}{x^2 + 11x + 28} \cdot \frac{x^2 - 16}{x^2 + 6x - 27}\\

\\\frac{(x - 9)(x - ?)}{(x + \quad)(x - 3)}\\

Explanation:

Factor all numerators and denominators

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Write the product and simplify common factors

$$ \frac{(x - 9)(x + 9)}{(x + 7)(x + 4)} \cdot \frac{(x - 4)(x + 4)}{(x + 9)(x - 3)} = \frac{(x - 9)(x + 9)(x - 4)(x + 4)}{(x + 7)(x + 4)(x + 9)(x - 3)} $$

Cancel the common factors \( (x + 9) \) and \( (x + 4) \):

$$ \frac{(x - 9)(x - 4)}{(x + 7)(x - 3)} $$

Match with the given template

The template is:

$$ \frac{(x - 9)(x - [ ? ])}{(x + [ \quad ])(x - 3)} $$

Comparing this to our simplified expression:

$$ \frac{(x - 9)(x - 4)}{(x + 7)(x - 3)} $$

We find:

• The green box with [ ? ] corresponds to \(4\).
• The grey box with [ \quad ] corresponds to \(7\).

Answer:

Use factoring to find the simplified product of these rational expressions.

$$ \frac{x^2 - 81}{x^2 + 11x + 28} \cdot \frac{x^2 - 16}{x^2 + 6x - 27} $$

$$ \frac{(x - 9)(x - 4)}{(x + 7)(x - 3)} $$