Question

Divide. Write the rational expression in simplest form. x2−6x−16x+2÷x2−643x (1 point) Responses 3xx−2 the fraction with numerator 3 x and denominator x minus 2 3xx+8 the fraction with numerator 3 x and denominator x plus 8 x−23x the fraction with numerator x minus 2 and denominator 3 x x+83x

Answer 1

To simplify the expression \( \frac{x^2 - 6x - 16}{x + 2} \div \frac{x^2 - 64}{3x} \), we'll start by rewriting the division as multiplication by the reciprocal.

So, we have:

\[
\frac{x^2 - 6x - 16}{x + 2} \times \frac{3x}{x^2 - 64}
\]

Now, we need to factor each part of the expression where possible.

1. **Factor \( x^2 - 6x - 16 \)**:
\[
x^2 - 6x - 16 = (x - 8)(x + 2)
\]

2. **Recognize \( x + 2 \)** (already factored).

3. **Factor \( x^2 - 64 \)** (difference of squares):
\[
x^2 - 64 = (x - 8)(x + 8)
\]

Now, substitute the factored forms back into the expression:

\[
\frac{(x - 8)(x + 2)}{x + 2} \times \frac{3x}{(x - 8)(x + 8)}
\]

Now, we can cancel out the common factors:

- The \( x + 2 \) in the numerator and denominator cancels.
- The \( x - 8 \) in the numerator and denominator also cancels.

After canceling, we have:

\[
\frac{3x}{x + 8}
\]

So the simplified form of the original expression is:

\[
\frac{3x}{x + 8}
\]

Among your given options, this corresponds to:

**3xx+8** the fraction with numerator 3x and denominator x plus 8.