Question

Multiply the rational expression and write the product in simplest form:

x2+2x−3x2+3x−4⋅x2+6x+8x2−9


(1 point)
Responses

(x+4)(x+2)(x+3)(x−3)
the fraction with numerator open paren x plus 4 close paren times open paren x plus 2 close paren and denominator open paren x plus 3 close paren times open paren x minus 3 close paren

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

x+2x−3
the fraction with numerator x plus 2 and denominator x minus 3

−23
negative 2 thirds
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page 16 of 16

Answer 1

To multiply the rational expression \(\frac{x^2 + 2x - 3}{x^2 + 3x - 4} \cdot \frac{x^2 + 6x + 8}{x^2 - 9}\), we first need to factor each polynomial.

1. **Factor each expression:**

- **Numerator of the first fraction \(x^2 + 2x - 3\)**:
\[
x^2 + 2x - 3 = (x + 3)(x - 1)
\]
- **Denominator of the first fraction \(x^2 + 3x - 4\)**:
\[
x^2 + 3x - 4 = (x + 4)(x - 1)
\]
- **Numerator of the second fraction \(x^2 + 6x + 8\)**:
\[
x^2 + 6x + 8 = (x + 4)(x + 2)
\]
- **Denominator of the second fraction \(x^2 - 9\)** (difference of squares):
\[
x^2 - 9 = (x + 3)(x - 3)
\]

2. **Rewrite the original expression with the factored forms**:
\[
\frac{(x + 3)(x - 1)}{(x + 4)(x - 1)} \cdot \frac{(x + 4)(x + 2)}{(x + 3)(x - 3)}
\]

3. **Multiply the numerators and the denominators**:
\[
\frac{(x + 3)(x - 1)(x + 4)(x + 2)}{(x + 4)(x - 1)(x + 3)(x - 3)}
\]

4. **Cancel out the common factors**:
- \( (x + 3) \) appears in both numerator and denominator.
- \( (x + 4) \) appears in both numerator and denominator.
- \( (x - 1) \) appears in both numerator and denominator.

Thus, we can simplify:
\[
= \frac{(x + 2)}{(x - 3)}
\]

5. **Final result**:
The product in simplest form is:
\[
\frac{x + 2}{x - 3}
\]

Thus, the final answer is:

**x + 2 / x - 3**