Question

Express in simplest form: x3+4x2−32x32x−8x2
.(1 point)
Responses

x+88
Start Fraction x plus 8 over 8 End Fraction

x3−18
Start Fraction x cubed minus 1 over 8 End Fraction

−(x+8)8
Start Fraction negative left parenthesis x plus 8 right parenthesis over 8 End Fraction

−x

Answer 1

To simplify the expression \( \frac{x^3 + 4x^2 - 32x}{32x - 8x^2} \), we can start by factoring both the numerator and the denominator.

1. **Factor the numerator**: \( x^3 + 4x^2 - 32x \)
\[
= x(x^2 + 4x - 32)
\]
Now, we can factor the quadratic \( x^2 + 4x - 32 \). We look for two numbers that multiply to \(-32\) and add to \(4\). The numbers \(8\) and \(-4\) work.
\[
= x(x + 8)(x - 4)
\]

2. **Factor the denominator**: \( 32x - 8x^2 \)
\[
= -8x^2 + 32x
\]
Factoring out \(-8x\):
\[
= -8x(x - 4)
\]

Now we rewrite our original expression:
\[
\frac{x(x + 8)(x - 4)}{-8x(x - 4)}
\]

3. **Cancel common terms**:
- The \(x - 4\) cancels out.
- The \(x\) in the numerator and the \(x\) in the denominator also cancels out, assuming \(x \neq 0\).

After cancelling, we have:
\[
= \frac{x + 8}{-8} = -\frac{x + 8}{8}
\]

So, the simplified expression is:
\[
-\frac{(x + 8)}{8}
\]

Therefore, the correct response is:
**−(x + 8) / 8**.