Question

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

\\\frac{4x^2 - 81}{3x^2 - 7x - 20} \div \frac{2x^2 + 3x - 54}{x^2 - 8x + 16}\\

\\\frac{(2x + ?)(x - 4)}{(3x + 5)(x + \box)}\\

Explanation:

Factor each polynomial expression

Using the Factoring Trinomials knowledge point

$$
LATEXBLOCK0
$$

Rewrite division as multiplication

Using the Dividing Rational Expressions knowledge point

$$ \frac{4x^2 - 81}{3x^2 - 7x - 20} \cdot \frac{x^2 - 8x + 16}{2x^2 + 3x - 54} $$

Substitute factored forms and simplify

Using the Simplifying Rational Expressions knowledge point

$$
LATEXBLOCK1
$$

Identify the missing value

Comparing the simplified quotient to the template:

$$ \frac{(2x + [?])(x - 4)}{(3x + 5)(x + \text{___})} $$

The value in the green box with the question mark \([?]\) is \(9\).

Answer:

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

$$\frac{4x^2 - 81}{3x^2 - 7x - 20} \div \frac{2x^2 + 3x - 54}{x^2 - 8x + 16}$$

$$\frac{(2x + 9)(x - 4)}{(3x + 5)(x + \text{___})}$$