Question

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

\\\frac{x^2 - 25}{x^2 - 8x + 15} \cdot \frac{2x^2 - 11x - 6}{x^2 - 36}\\

\\\frac{(x + ?)(2x + 1)}{(x - 3)(x + \quad)}\\

Explanation:

Factor each polynomial expression

Factor the numerators and denominators of both rational expressions:

• \(x^2 - 25 = (x - 5)(x + 5)\)
• \(x^2 - 8x + 15 = (x - 5)(x - 3)\)
• \(2x^2 - 11x - 6 = (2x + 1)(x - 6)\)
• \(x^2 - 36 = (x - 6)(x + 6)\)

Substitute and simplify the product

Write the product of the rational expressions with their factored forms:

$$ \frac{(x - 5)(x + 5)}{(x - 5)(x - 3)} \cdot \frac{(2x + 1)(x - 6)}{(x - 6)(x + 6)} $$

Cancel common factors in the numerator and denominator:

• Cancel \((x - 5)\)
• Cancel \((x - 6)\)

This leaves:

$$ \frac{(x + 5)(2x + 1)}{(x - 3)(x + 6)} $$

Identify the missing values

Compare the simplified product with the given template:

$$ \frac{(x + [ ? ])(2x + 1)}{(x - 3)(x + [ \quad ])} $$

Matching the terms:

• The numerator term \((x + [ ? ])\) corresponds to \((x + 5)\), so the green box with the question mark \([ ? ]\) is \(5\).
• The denominator term \((x + [ \quad ])\) corresponds to \((x + 6)\), so the grey box is \(6\).

Answer:

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

$$\frac{x^2 - 25}{x^2 - 8x + 15} \cdot \frac{2x^2 - 11x - 6}{x^2 - 36}$$

$$\frac{(x + 5)(2x + 1)}{(x - 3)(x + 6)}$$