Question

Multiplication & Division of Rational Expressions Quick Check
2 of 52 of 5 Items
Question
Write the polynomial expression in simplest form: x2−9x2−4⋅4x−812x+36 .(1 point)
Responses

(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)
Start Fraction x minus 3 over 3 left parenthesis x plus 2 right parenthesis End Fraction

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

Answer 1

To simplify the expression \(\frac{x^2 - 9}{x^2 - 4} \cdot \frac{4x - 8}{12x + 36}\), we'll start by factoring each polynomial where possible.

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

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

3. **Factor \(4x - 8\)**: This can be factored by taking out a common factor of 4.
\[
4x - 8 = 4(x - 2)
\]

4. **Factor \(12x + 36\)**: We can also factor this by taking out a common factor of 12.
\[
12x + 36 = 12(x + 3)
\]

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

Next, combine the fractions:
\[
\frac{(x - 3)(x + 3) \cdot 4(x - 2)}{(x - 2)(x + 2) \cdot 12(x + 3)}
\]

Now let's cancel out common factors in the numerator and the denominator:
- The \(x - 2\) in the numerator and denominator cancels out.
- The \(4\) and \(12\) can be simplified to \(\frac{1}{3}\), so we can express this as:
\[
\frac{(x - 3)(x + 3)}{3(x + 2)(x + 3)}
\]

Now, we can cancel the \(x + 3\) from the numerator and the denominator:
\[
\frac{x - 3}{3(x + 2)}
\]

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

This matches better with the option you provided:

**Answer**: \( \frac{x - 3}{3(x + 2)} \)