Question

multiply the rational expression and write the product in simplest form x2-9/x^2-4 • 4x-8/12x+36

Answer 1

To multiply the given rational expressions \(\frac{x^2 - 9}{x^2 - 4}\) and \(\frac{4x - 8}{12x + 36}\), we first factor all the expressions involved.

1. Factor \(x^2 - 9\) and \(x^2 - 4\):
- \(x^2 - 9\) is a difference of squares:
\[
x^2 - 9 = (x - 3)(x + 3)
\]
- \(x^2 - 4\) is also a difference of squares:
\[
x^2 - 4 = (x - 2)(x + 2)
\]

2. Factor the numerator and the denominator of the second expression:
- \(4x - 8\) can be factored as:
\[
4x - 8 = 4(x - 2)
\]
- \(12x + 36\) can be factored as:
\[
12x + 36 = 12(x + 3) = 12(x + 3)
\]

Now we can rewrite the expression as:
\[
\frac{(x - 3)(x + 3)}{(x - 2)(x + 2)} \cdot \frac{4(x - 2)}{12(x + 3)}
\]

Now, combine the fractions by multiplying the numerators and the denominators:
\[
= \frac{(x - 3)(x + 3) \cdot 4(x - 2)}{(x - 2)(x + 2) \cdot 12(x + 3)}
\]

Simplifying gives us:
\[
= \frac{4(x - 3)(x + 3)(x - 2)}{12(x - 2)(x + 2)(x + 3)}
\]

Next, we can cancel the common factors in the numerator and the denominator:
- The factor \(x - 2\) cancels out.
- The factor \(x + 3\) also cancels out.

This leaves us with:
\[
= \frac{4(x - 3)}{12(x + 2)}
\]

Now we can simplify further:
\[
= \frac{4}{12} \cdot \frac{(x - 3)}{(x + 2)} = \frac{1}{3} \cdot \frac{(x - 3)}{(x + 2)}
\]

Thus, the simplest form of the product is:
\[
\frac{x - 3}{3(x + 2)}
\]