Question

Write the polynomial expression in simplest form: x2−9x2−4⋅4x−812x+36.(1 point) Responses 2(x−3)6(x+2) Start Fraction 2 left parenthesis x minus 3 right parenthesis over 6 left parenthesis x plus 2 right parenthesis End Fraction 13 Start Fraction 1 over 3 End Fraction (x−3)(x−2)3(x+2)(x+2) Start Fraction left parenthesis x minus 3 right parenthesis left parenthesis x minus 2 right parenthesis over 3 left parenthesis x plus 2 right parenthesis left parenthesis x plus 2 right parenthesis End Fraction x−33(x+2)

Answer 1

To simplify the expression \(\frac{x^2 - 9}{x^2 - 4} \cdot \frac{4x - 8}{12x + 36}\), we will first factor the polynomials in the numerator and the denominator.

1. **Factoring \(x^2 - 9\)**:
- This is a difference of squares: \(x^2 - 9 = (x - 3)(x + 3)\).

2. **Factoring \(x^2 - 4\)**:
- This is also a difference of squares: \(x^2 - 4 = (x - 2)(x + 2)\).

3. **Factoring \(4x - 8\)**:
- Factor out a 4: \(4x - 8 = 4(x - 2)\).

4. **Factoring \(12x + 36\)**:
- Factor out a 12: \(12x + 36 = 12(x + 3)\).

Now we can rewrite the expression using these factorizations:

\[
\frac{(x - 3)(x + 3)}{(x - 2)(x + 2)} \cdot \frac{4(x - 2)}{12(x + 3)}
\]

Next, we can combine this into one fraction:

\[
\frac{(x - 3)(x + 3) \cdot 4(x - 2)}{(x - 2)(x + 2) \cdot 12(x + 3)}
\]

Now we'll cancel out common factors in the numerator and the denominator:

1. \(x - 2\) in the numerator and denominator.
2. \(x + 3\) in the numerator and denominator.

After canceling these common factors, we are left with:

\[
\frac{4(x - 3)}{12(x + 2)}
\]

We can simplify this further by reducing the coefficients:

\[
\frac{4}{12} = \frac{1}{3}
\]

Thus, we end up with:

\[
\frac{(x - 3)}{3(x + 2)}
\]

Therefore, the final expression in simplest form is:

\[
\frac{x - 3}{3(x + 2)}
\]

From the given options, this corresponds to:

\[
\text{Start Fraction left parenthesis x minus 3 right parenthesis over 3 left parenthesis x plus 2 right parenthesis End Fraction}
\]