Question

Write the polynomial expression in simplest form:

(6x / (x - 4)) * ((x^2 - 16) / (4x + 16))

A. (3x / 2)

B. (6x / 4)

C. (6x (x + 4) / (4 (x - 4)))

D. (3x (x + 4) / (2 (x - 4)))

Answer 1

To simplify the given expression \((6x / (x - 4)) * ((x^2 - 16) / (4x + 16))\), we start by factoring where possible.

1. **Factor the second fraction**:
The expression \(x^2 - 16\) can be factored as a difference of squares:
\[
x^2 - 16 = (x - 4)(x + 4)
\]

The expression \(4x + 16\) can be factored as:
\[
4x + 16 = 4(x + 4)
\]

2. **Substituting the factored forms into the expression**:
The entire expression now looks like this:
\[
\left(\frac{6x}{x - 4}\right) * \left(\frac{(x - 4)(x + 4)}{4(x + 4)}\right)
\]

3. **Combine the fractions**:
\[
\frac{6x \cdot (x - 4)(x + 4)}{(x - 4) \cdot 4(x + 4)}
\]

Here, we can cancel \(x - 4\) in the numerator and denominator, assuming \(x \neq 4\), and \(x + 4\) as well:
\[
= \frac{6x}{4}
\]

4. **Simplifying further**:
\[
= \frac{3x}{2}
\]

So the expression simplifies to \(\frac{3x}{2}\).

Hence, the answer is:

**A. \((3x / 2)\)**.