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What is the quotient of the rational expressions shown below? Make sure your answer is in reduced form.

[tex]\[ \frac{x^2-16}{x+5} \div \frac{x^2-8x+16}{2x+10} \][/tex]

A. [tex]\[ \frac{2(x+4)}{x-4} \][/tex]

B. [tex]\[ \frac{2(x+4)^2}{x-4} \][/tex]

C. [tex]\[ \frac{2(x-4)}{x+4} \][/tex]

D. [tex]\[ \frac{2(x-4)^2}{x+4} \][/tex]

Sagot :

To find the quotient of the given rational expressions, let's follow these steps:

1. Understand the given problem:
We need to find the quotient of two rational expressions:
[tex]\[ \frac{x^2-16}{x+5} \div \frac{x^2-8x+16}{2x+10} \][/tex]

2. Rewrite the division as multiplication by the reciprocal:
[tex]\[ \frac{x^2-16}{x+5} \div \frac{x^2-8x+16}{2x+10} = \frac{x^2-16}{x+5} \times \frac{2x+10}{x^2-8x+16} \][/tex]

3. Factorize the numerators and denominators:
- [tex]\(x^2 - 16\)[/tex] is a difference of squares:
[tex]\[ x^2 - 16 = (x-4)(x+4) \][/tex]
- [tex]\(x^2 - 8x + 16\)[/tex] is a perfect square trinomial:
[tex]\[ x^2 - 8x + 16 = (x-4)^2 \][/tex]
- [tex]\(2x + 10\)[/tex] can be factored out:
[tex]\[ 2x + 10 = 2(x + 5) \][/tex]

4. Substitute the factored forms into the expressions:
[tex]\[ \frac{(x-4)(x+4)}{x+5} \times \frac{2(x+5)}{(x-4)^2} \][/tex]

5. Simplify by canceling common factors:
Notice that [tex]\((x+5)\)[/tex] and [tex]\((x-4)\)[/tex] appear in both the numerator and the denominator, so they can be canceled out:
[tex]\[ \frac{(x-4)(x+4)}{\cancel{x+5}} \times \frac{2\cancel{(x+5)}}{(x-4)^2} = \frac{(x-4)(x+4)}{1} \times \frac{2}{(x-4)^2} \][/tex]
Simplify further:
[tex]\[ \frac{(x+4)}{1} \times \frac{2}{x-4} = \frac{2(x+4)}{x-4} \][/tex]

Therefore, the quotient of the given rational expressions in reduced form is:
[tex]\[ \boxed{\frac{2(x+4)}{x-4}} \][/tex]

The correct answer is [tex]\(\boxed{A}\)[/tex].
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