Question

Write the polynomial expression in simplest form:

x^2−9/x^2−4⋅4x−812x+36

(1 point)
Responses

2(x−3)/6(x+2)
the fraction with numerator 2 times open paren x minus 3 close paren and denominator 6 times open paren x plus 2 close paren

x−3/3(x+2)
the fraction with numerator x minus 3 and denominator 3 times open paren x plus 2 close paren

1/3
1 third

(x−3)(x−2)/3(x+2)(x+2)
the fraction with numerator open paren x minus 3 close paren times open paren x minus 2 close paren and denominator 3 open paren x plus 2 close paren open paren x plus 2 close paren
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Answer 1

To simplify the polynomial expression

\[
\frac{x^2 - 9}{x^2 - 4} \cdot \frac{4x - 8}{12x + 36},
\]

let's break it down step-by-step.

1. **Factor the polynomials where possible.**

The first fraction:
\[
x^2 - 9 = (x - 3)(x + 3) \quad \text{(difference of squares)}
\]
\[
x^2 - 4 = (x - 2)(x + 2) \quad \text{(difference of squares)}
\]

The second fraction:
\[
4x - 8 = 4(x - 2)
\]
\[
12x + 36 = 12(x + 3) = 12 \cdot 3 \cdot (x + 3) = 3(x + 3)
\]

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

2. **Combine the fractions:**
\[
= \frac{(x - 3)(x + 3) \cdot 4(x - 2)}{(x - 2)(x + 2) \cdot 12(x + 3)}.
\]

3. **Cancel out common factors:**
- The \( (x - 2) \) cancels with one in the numerator and denominator.
- The \( (x + 3) \) cancels with one in the numerator and denominator.

After canceling, the expression becomes:
\[
\frac{4(x - 3)}{12(x + 2)}.
\]

4. **Simplify the coefficients:**
\[
\frac{4}{12} = \frac{1}{3},
\]
thus we have:
\[
\frac{x - 3}{3(x + 2)}.
\]

The final simplified expression is:
\[
\frac{x - 3}{3(x + 2)}.
\]

Thus, the answer is:

**x−3/3(x+2)**