Question

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

\\\frac{x^2 - 12x + 36}{x^2 - 4} \div \frac{4x^2 - 21x - 18}{3x^2 + 2x - 16}\\

\\\frac{(x - 6)(3x + ? )}{(x + \quad)(4x + 3)}\\

Explanation:

Factor the numerators and denominators

Using the Factoring Trinomials knowledge point

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Rewrite division as multiplication

Using the Dividing Rational Expressions knowledge point

$$ \frac{(x - 6)^2}{(x - 2)(x + 2)} \div \frac{(x - 6)(4x + 3)}{(x - 2)(3x + 8)} = \frac{(x - 6)^2}{(x - 2)(x + 2)} \cdot \frac{(x - 2)(3x + 8)}{(x - 6)(4x + 3)} $$

Simplify the expression

Using the Simplifying Rational Expressions knowledge point

$$ \frac{(x - 6)^2 \cdot (x - 2)(3x + 8)}{(x - 2)(x + 2) \cdot (x - 6)(4x + 3)} = \frac{(x - 6)(3x + 8)}{(x + 2)(4x + 3)} $$

Match with the given template

Comparing the simplified quotient to the template:

$$ \frac{(x - 6)(3x + [ ? ])}{(x + [ \quad ])(4x + 3)} = \frac{(x - 6)(3x + 8)}{(x + 2)(4x + 3)} $$

Thus, the green box with the question mark \([ ? ]\) is \(8\), and the grey box is \(2\).

Answer:

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

$$\frac{x^2 - 12x + 36}{x^2 - 4} \div \frac{4x^2 - 21x - 18}{3x^2 + 2x - 16}$$

$$\frac{(x - 6)(3x + 8)}{(x + 2)(4x + 3)}$$