Question

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

\\\frac{4x^2 - 17x - 15}{x^2 + 10x + 25} \div \frac{16x^2 - 9}{2x^2 + 3x - 35}\\

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

Explanation:

Factor each polynomial expression

We begin by factoring each numerator and denominator in the division problem.

Using the Factoring Trinomials knowledge point:

1. First numerator: \(4x^2 - 17x - 15\)

We look for two numbers that multiply to \(4 \times (-15) = -60\) and add to \(-17\). These numbers are \(-20\) and \(3\).

$$ 4x^2 - 20x + 3x - 15 = 4x(x - 5) + 3(x - 5) = (4x + 3)(x - 5) $$

1. First denominator: \(x^2 + 10x + 25\)

This is a perfect square trinomial:

$$ x^2 + 10x + 25 = (x + 5)^2 = (x + 5)(x + 5) $$

1. Second numerator: \(16x^2 - 9\)

This is a difference of squares:

$$ 16x^2 - 9 = (4x - 3)(4x + 3) $$

1. Second denominator: \(2x^2 + 3x - 35\)

We look for two numbers that multiply to \(2 \times (-35) = -70\) and add to \(3\). These numbers are \(10\) and \(-7\).

$$ 2x^2 + 10x - 7x - 35 = 2x(x + 5) - 7(x + 5) = (2x - 7)(x + 5) $$

Rewrite the division as multiplication

Using the Dividing Rational Expressions knowledge point, we multiply by the reciprocal of the divisor:

$$ \frac{4x^2 - 17x - 15}{x^2 + 10x + 25} \div \frac{16x^2 - 9}{2x^2 + 3x - 35} = \frac{4x^2 - 17x - 15}{x^2 + 10x + 25} \times \frac{2x^2 + 3x - 35}{16x^2 - 9} $$

Substitute the factored forms into the expression:

$$ \frac{(4x + 3)(x - 5)}{(x + 5)(x + 5)} \times \frac{(2x - 7)(x + 5)}{(4x - 3)(4x + 3)} $$

Simplify the rational expression

Using the Simplifying Rational Expressions knowledge point, we cancel the common factors in the numerator and denominator:

• Cancel \((4x + 3)\) from the numerator and denominator.
• Cancel \((x + 5)\) from the numerator and denominator.

This leaves:

$$ \frac{(x - 5)(2x - 7)}{(x + 5)(4x - 3)} $$

Identify the missing values

We compare our simplified expression to the given template:

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

By matching the terms:

• The term \((2x - [ ? ])\) corresponds to \((2x - 7)\), so \([ ? ] = 7\).
• The term \((x + [ \quad ])\) corresponds to \((x + 5)\), so the blank is \(5\).

Answer:

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

$$\frac{4x^2 - 17x - 15}{x^2 + 10x + 25} \div \frac{16x^2 - 9}{2x^2 + 3x - 35}$$

The simplified quotient is:

$$\frac{(x - 5)(2x - \text{7})}{(x + \text{5})(4x - 3)}$$