Question

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

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

x+88
Start Fraction x plus 8 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 will first factor both the numerator and the denominator.

**Step 1: Factor the numerator.**

The numerator is \(x^3 + 4x^2 - 32x\). We can factor out an \(x\):

\[
x(x^2 + 4x - 32)
\]

Next, we will factor \(x^2 + 4x - 32\). We need two numbers that multiply to \(-32\) and add to \(4\), which are \(8\) and \(-4\):

\[
x^2 + 4x - 32 = (x + 8)(x - 4)
\]

So, the numerator becomes:

\[
x(x + 8)(x - 4)
\]

**Step 2: Factor the denominator.**

The denominator is \(32x - 8x^2\). We can factor out \(-8x\):

\[
-8x(4 - x)
\]

**Step 3: Rewrite the entire expression.**

Now we have:

\[
\frac{x(x + 8)(x - 4)}{-8x(4 - x)}
\]

**Step 4: Simplify.**

Notice that \(4 - x = -(x - 4)\), so we can rewrite the denominator:

\[
-8x(4 - x) = -8x(-1)(x - 4) = 8x(x - 4)
\]

Now we can simplify the expression:

\[
\frac{x(x + 8)(x - 4)}{8x(x - 4)}
\]

We can cancel \(x\) (assuming \(x \neq 0\)) and \(x - 4\) (assuming \(x \neq 4\)):

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

Thus, the expression simplifies to:

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

**Conclusion:**

The simplest form of the original expression is \(\frac{x + 8}{8}\). This corresponds to the response:

\[
\text{Start Fraction x plus 8 over 8 End Fraction}
\]