Question
Question
Simplify the expression: left parenthesis Start Fraction 2 x squared minus 5 x minus 3 over x squared plus x minus 20 End Fraction right parenthesis left parenthesis Start Fraction x squared plus 8 x plus 15 over 2 x squared plus 9 x plus 4 End Fraction right parenthesis
(1 point)
Responses
Start Fraction x squared minus 9 over x squared minus 16 End Fraction
Image with alt text: Start Fraction x squared minus 9 over x squared minus 16 End Fraction
Start Fraction 9 over 16 End Fraction
Image with alt text: Start Fraction 9 over 16 End Fraction
Start Fraction x squared minus 6 x plus 9 over x squared minus 8 x plus 16 End Fraction
Image with alt text: Start Fraction x squared minus 6 x plus 9 over x squared minus 8 x plus 16 End Fraction
Start Fraction x squared plus 6 x plus 9 over x squared plus 8 x plus 16 End Fraction
Simplify the expression: left parenthesis Start Fraction 2 x squared minus 5 x minus 3 over x squared plus x minus 20 End Fraction right parenthesis left parenthesis Start Fraction x squared plus 8 x plus 15 over 2 x squared plus 9 x plus 4 End Fraction right parenthesis
(1 point)
Responses
Start Fraction x squared minus 9 over x squared minus 16 End Fraction
Image with alt text: Start Fraction x squared minus 9 over x squared minus 16 End Fraction
Start Fraction 9 over 16 End Fraction
Image with alt text: Start Fraction 9 over 16 End Fraction
Start Fraction x squared minus 6 x plus 9 over x squared minus 8 x plus 16 End Fraction
Image with alt text: Start Fraction x squared minus 6 x plus 9 over x squared minus 8 x plus 16 End Fraction
Start Fraction x squared plus 6 x plus 9 over x squared plus 8 x plus 16 End Fraction
Answers
GPT-4o mini
To simplify the expression
\[
\left( \frac{2x^2 - 5x - 3}{x^2 + x - 20} \right) \left( \frac{x^2 + 8x + 15}{2x^2 + 9x + 4} \right),
\]
we will factor each polynomial in the numerators and denominators where possible.
**Step 1: Factor the polynomials**
1. For \(2x^2 - 5x - 3\):
\[
2x^2 - 5x - 3 = (2x + 1)(x - 3)
\]
2. For \(x^2 + x - 20\):
\[
x^2 + x - 20 = (x + 5)(x - 4)
\]
3. For \(x^2 + 8x + 15\):
\[
x^2 + 8x + 15 = (x + 3)(x + 5)
\]
4. For \(2x^2 + 9x + 4\):
\[
2x^2 + 9x + 4 = (2x + 1)(x + 4)
\]
**Step 2: Rewrite the expression with the factored forms**
Substituting in the factored forms, we have:
\[
\left( \frac{(2x + 1)(x - 3)}{(x + 5)(x - 4)} \right) \left( \frac{(x + 3)(x + 5)}{(2x + 1)(x + 4)} \right)
\]
**Step 3: Multiply the fractions**
Now, combine the numerators and the denominators:
\[
\frac{(2x + 1)(x - 3)(x + 3)(x + 5)}{(x + 5)(x - 4)(2x + 1)(x + 4)}
\]
**Step 4: Cancel common factors**
We can cancel the common factors:
- \( (2x + 1) \) cancels with \( (2x + 1) \)
- \( (x + 5) \) cancels with \( (x + 5) \)
After cancellation, we have:
\[
\frac{(x - 3)(x + 3)}{(x - 4)(x + 4)}
\]
**Step 5: Recognize the resultant expression**
The numerator can be simplified further:
\[
(x - 3)(x + 3) = x^2 - 9
\]
The denominator can be addressed as:
\[
(x - 4)(x + 4) = x^2 - 16
\]
Thus, our simplified expression is:
\[
\frac{x^2 - 9}{x^2 - 16}
\]
**Final Answer:**
So the resulting simplified expression is
\[
\frac{x^2 - 9}{x^2 - 16}.
\]
This corresponds to the first response option.
\[
\left( \frac{2x^2 - 5x - 3}{x^2 + x - 20} \right) \left( \frac{x^2 + 8x + 15}{2x^2 + 9x + 4} \right),
\]
we will factor each polynomial in the numerators and denominators where possible.
**Step 1: Factor the polynomials**
1. For \(2x^2 - 5x - 3\):
\[
2x^2 - 5x - 3 = (2x + 1)(x - 3)
\]
2. For \(x^2 + x - 20\):
\[
x^2 + x - 20 = (x + 5)(x - 4)
\]
3. For \(x^2 + 8x + 15\):
\[
x^2 + 8x + 15 = (x + 3)(x + 5)
\]
4. For \(2x^2 + 9x + 4\):
\[
2x^2 + 9x + 4 = (2x + 1)(x + 4)
\]
**Step 2: Rewrite the expression with the factored forms**
Substituting in the factored forms, we have:
\[
\left( \frac{(2x + 1)(x - 3)}{(x + 5)(x - 4)} \right) \left( \frac{(x + 3)(x + 5)}{(2x + 1)(x + 4)} \right)
\]
**Step 3: Multiply the fractions**
Now, combine the numerators and the denominators:
\[
\frac{(2x + 1)(x - 3)(x + 3)(x + 5)}{(x + 5)(x - 4)(2x + 1)(x + 4)}
\]
**Step 4: Cancel common factors**
We can cancel the common factors:
- \( (2x + 1) \) cancels with \( (2x + 1) \)
- \( (x + 5) \) cancels with \( (x + 5) \)
After cancellation, we have:
\[
\frac{(x - 3)(x + 3)}{(x - 4)(x + 4)}
\]
**Step 5: Recognize the resultant expression**
The numerator can be simplified further:
\[
(x - 3)(x + 3) = x^2 - 9
\]
The denominator can be addressed as:
\[
(x - 4)(x + 4) = x^2 - 16
\]
Thus, our simplified expression is:
\[
\frac{x^2 - 9}{x^2 - 16}
\]
**Final Answer:**
So the resulting simplified expression is
\[
\frac{x^2 - 9}{x^2 - 16}.
\]
This corresponds to the first response option.