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
(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 given expression
\[
\left( \frac{2x^2 - 5x - 3}{x^2 + x - 20} \right) \left( \frac{x^2 + 8x + 15}{2x^2 + 9x + 4} \right),
\]
we need to factor all the polynomials involved.
1. **Factoring \(2x^2 - 5x - 3\)**:
To factor \(2x^2 - 5x - 3\), we look for two numbers that multiply to \(2 \times -3 = -6\) and add to \(-5\). The numbers are \(-6\) and \(1\).
\[
2x^2 - 6x + x - 3 = 2x(x - 3) + 1(x - 3) = (2x + 1)(x - 3).
\]
2. **Factoring \(x^2 + x - 20\)**:
For \(x^2 + x - 20\), we want two numbers that multiply to \(-20\) and add up to \(1\). The numbers are \(5\) and \(-4\).
\[
x^2 + 5x - 4x - 20 = (x + 5)(x - 4).
\]
3. **Factoring \(x^2 + 8x + 15\)**:
Here we need two numbers that multiply to \(15\) and add to \(8\). The numbers are \(3\) and \(5\).
\[
x^2 + 8x + 15 = (x + 3)(x + 5).
\]
4. **Factoring \(2x^2 + 9x + 4\)**:
Lastly, for \(2x^2 + 9x + 4\), we find two numbers that multiply to \(2 \times 4 = 8\) and add to \(9\). The numbers are \(8\) and \(1\).
\[
2x^2 + 8x + x + 4 = 2x(x + 4) + 1(x + 4) = (2x + 1)(x + 4).
\]
Now, substituting these factors back into the expression, we have:
\[
\frac{(2x + 1)(x - 3)}{(x + 5)(x - 4)} \cdot \frac{(x + 3)(x + 5)}{(2x + 1)(x + 4)}.
\]
Next, we can cancel out the common factors:
- \( (2x + 1) \)
- \( (x + 5) \)
Thus, what remains is:
\[
\frac{(x - 3)(x + 3)}{(x - 4)(x + 4)}.
\]
Now, we notice that \( (x - 3)(x + 3) = x^2 - 9 \) and \( (x - 4)(x + 4) = x^2 - 16 \).
So the simplified expression is:
\[
\frac{x^2 - 9}{x^2 - 16}.
\]
Thus, the answer is
\[
\boxed{\frac{x^2 - 9}{x^2 - 16}}.
\]
\[
\left( \frac{2x^2 - 5x - 3}{x^2 + x - 20} \right) \left( \frac{x^2 + 8x + 15}{2x^2 + 9x + 4} \right),
\]
we need to factor all the polynomials involved.
1. **Factoring \(2x^2 - 5x - 3\)**:
To factor \(2x^2 - 5x - 3\), we look for two numbers that multiply to \(2 \times -3 = -6\) and add to \(-5\). The numbers are \(-6\) and \(1\).
\[
2x^2 - 6x + x - 3 = 2x(x - 3) + 1(x - 3) = (2x + 1)(x - 3).
\]
2. **Factoring \(x^2 + x - 20\)**:
For \(x^2 + x - 20\), we want two numbers that multiply to \(-20\) and add up to \(1\). The numbers are \(5\) and \(-4\).
\[
x^2 + 5x - 4x - 20 = (x + 5)(x - 4).
\]
3. **Factoring \(x^2 + 8x + 15\)**:
Here we need two numbers that multiply to \(15\) and add to \(8\). The numbers are \(3\) and \(5\).
\[
x^2 + 8x + 15 = (x + 3)(x + 5).
\]
4. **Factoring \(2x^2 + 9x + 4\)**:
Lastly, for \(2x^2 + 9x + 4\), we find two numbers that multiply to \(2 \times 4 = 8\) and add to \(9\). The numbers are \(8\) and \(1\).
\[
2x^2 + 8x + x + 4 = 2x(x + 4) + 1(x + 4) = (2x + 1)(x + 4).
\]
Now, substituting these factors back into the expression, we have:
\[
\frac{(2x + 1)(x - 3)}{(x + 5)(x - 4)} \cdot \frac{(x + 3)(x + 5)}{(2x + 1)(x + 4)}.
\]
Next, we can cancel out the common factors:
- \( (2x + 1) \)
- \( (x + 5) \)
Thus, what remains is:
\[
\frac{(x - 3)(x + 3)}{(x - 4)(x + 4)}.
\]
Now, we notice that \( (x - 3)(x + 3) = x^2 - 9 \) and \( (x - 4)(x + 4) = x^2 - 16 \).
So the simplified expression is:
\[
\frac{x^2 - 9}{x^2 - 16}.
\]
Thus, the answer is
\[
\boxed{\frac{x^2 - 9}{x^2 - 16}}.
\]