Question

Write the polynomial expression in simplest form: 6x/x−4⋅x^2 − 16/4x+16
(1 point)
Responses

3x2
3 x over 2

6x4
6 x over 4

6x(x+4)4(x−4)
the fraction with numerator 6 x open paren x plus 4 close paren and denominator 4 times open paren x minus 4 close paren

3x(x+4)2(x−4)

Answer 1

To simplify the expression \( \frac{6x}{x - 4} \cdot \frac{x^2 - 16}{4x + 16} \), we will follow these steps:

1. **Factor the expressions where possible**.
2. **Simplify the product of the fractions**.

Start with the expression:

\[
\frac{6x}{x - 4} \cdot \frac{x^2 - 16}{4x + 16}
\]

### Step 1: Factor \(x^2 - 16\) and \(4x + 16\)

- \(x^2 - 16\) is a difference of squares and can be factored as:
\[
x^2 - 16 = (x - 4)(x + 4)
\]

- \(4x + 16\) can be factored by taking out a common factor:
\[
4x + 16 = 4(x + 4)
\]

### Step 2: Substitute the factored forms

Now, substitute these factored forms into the expression:

\[
\frac{6x}{x - 4} \cdot \frac{(x - 4)(x + 4)}{4(x + 4)}
\]

### Step 3: Cancel common terms

In the fraction, we can cancel \(x - 4\) and \(x + 4\):

- The \((x - 4)\) in the numerator of the second fraction cancels with the denominator of the first fraction.
- The \((x + 4)\) in the numerator of the second fraction cancels with the denominator of the second fraction.

The remaining expression is:

\[
\frac{6x}{4} = \frac{3x}{2}
\]

### Conclusion

Thus, the simplified form of the polynomial expression is:

\[
\frac{3x}{2}
\]

Correspondingly, the correct response from the options provided is:

**3 x over 2**.