Question
Express in simplest form: x2−6x−16x2−64÷2x+4x2+x−56
(1 point)
Responses
(x−7)(x+8)2(x−8)
the fraction with numerator open paren x minus 7 close paren times open paren x plus 8 close paren and denominator 2 times open paren x minus 8 close paren
x−72
the fraction with numerator x minus 7 and denominator 2
−x+72
the fraction with numerator negative x plus 7 and denominator 2
x+72
(1 point)
Responses
(x−7)(x+8)2(x−8)
the fraction with numerator open paren x minus 7 close paren times open paren x plus 8 close paren and denominator 2 times open paren x minus 8 close paren
x−72
the fraction with numerator x minus 7 and denominator 2
−x+72
the fraction with numerator negative x plus 7 and denominator 2
x+72
Answers
GPT-4o mini
To simplify the expression \(\frac{x^2 - 6x - 16}{x^2 - 64} \div (2x + 4x^2 + x - 56)\), we can follow these steps:
1. **Factor the numerator and denominator of the first fraction.**
- **Numerator:** \(x^2 - 6x - 16\) can factor as follows:
\[
x^2 - 6x - 16 = (x - 8)(x + 2)
\]
- **Denominator:** \(x^2 - 64\) is a difference of squares and can be factored as:
\[
x^2 - 64 = (x - 8)(x + 8)
\]
2. **Rewrite the division as multiplication by the reciprocal.**
- The division gives:
\[
\frac{(x - 8)(x + 2)}{(x - 8)(x + 8)} \times \frac{1}{2x + 4x^2 + x - 56}
\]
3. **Factor the polynomial in the denominator of the entire expression:**
- \(2x + 4x^2 + x - 56\) can be rearranged and factored:
\[
4x^2 + 3x - 56
\]
To factor \(4x^2 + 3x - 56\), we can use the trial-and-error method or grouping to find factors:
- This can be written as:
\[
4x^2 + 3x - 56 = (4x - 14)(x + 4) \text{ after finding appropriate factors}
\]
Note: You can verify via expansion.
4. **We can now substitute back:**
\[
\frac{(x - 8)(x + 2)}{(x - 8)(x + 8)} \times \frac{1}{(4x - 14)(x + 4)}
\]
5. **Cancel out the common factors:**
- From \((x - 8)\) in both numerator and denominator:
\[
\frac{(x + 2)}{(x + 8)} \times \frac{1}{(4x - 14)(x + 4)}
\]
6. **Final reassembly:**
\[
\frac{x + 2}{(x + 8)(4x - 14)(x + 4)}
\]
Since we cannot simplify further without specific values for \(x\), let's summarize:
- The expression simplifies to:
\[
\frac{x + 2}{(x + 8)(4x - 14)(x + 4)}
\]
However, after evaluating the responses, none were quite matching the derived expression directly. Please check if you were to factor correctly or recheck the problem context. The true simplest form based directly on provided options can't be thoroughly determined without further evaluations or context.
1. **Factor the numerator and denominator of the first fraction.**
- **Numerator:** \(x^2 - 6x - 16\) can factor as follows:
\[
x^2 - 6x - 16 = (x - 8)(x + 2)
\]
- **Denominator:** \(x^2 - 64\) is a difference of squares and can be factored as:
\[
x^2 - 64 = (x - 8)(x + 8)
\]
2. **Rewrite the division as multiplication by the reciprocal.**
- The division gives:
\[
\frac{(x - 8)(x + 2)}{(x - 8)(x + 8)} \times \frac{1}{2x + 4x^2 + x - 56}
\]
3. **Factor the polynomial in the denominator of the entire expression:**
- \(2x + 4x^2 + x - 56\) can be rearranged and factored:
\[
4x^2 + 3x - 56
\]
To factor \(4x^2 + 3x - 56\), we can use the trial-and-error method or grouping to find factors:
- This can be written as:
\[
4x^2 + 3x - 56 = (4x - 14)(x + 4) \text{ after finding appropriate factors}
\]
Note: You can verify via expansion.
4. **We can now substitute back:**
\[
\frac{(x - 8)(x + 2)}{(x - 8)(x + 8)} \times \frac{1}{(4x - 14)(x + 4)}
\]
5. **Cancel out the common factors:**
- From \((x - 8)\) in both numerator and denominator:
\[
\frac{(x + 2)}{(x + 8)} \times \frac{1}{(4x - 14)(x + 4)}
\]
6. **Final reassembly:**
\[
\frac{x + 2}{(x + 8)(4x - 14)(x + 4)}
\]
Since we cannot simplify further without specific values for \(x\), let's summarize:
- The expression simplifies to:
\[
\frac{x + 2}{(x + 8)(4x - 14)(x + 4)}
\]
However, after evaluating the responses, none were quite matching the derived expression directly. Please check if you were to factor correctly or recheck the problem context. The true simplest form based directly on provided options can't be thoroughly determined without further evaluations or context.