Certainly! Let's walk through the solution step-by-step to divide and simplify the given rational expressions:
0. Initialization:
We are given:
[tex]\[
\frac{x+1}{x^2 - 16} \div \frac{x^2 - 6x - 7}{x^2 + 2x - 8}
\][/tex]
By the division of fractions rule, we convert the division into multiplication by taking the reciprocal of the second fraction:
[tex]\[
\frac{x+1}{x^2 - 16} \div \frac{x^2 - 6x - 7}{x^2 + 2x - 8} = \frac{x+1}{x^2 - 16} \cdot \frac{x^2 + 2x - 8}{x^2 - 6x - 7}
\][/tex]
1. Factoring all numerators and denominators:
Factor [tex]\( x^2 - 16 \)[/tex]:
[tex]\[
x^2 - 16 = (x + 4)(x - 4)
\][/tex]
Factor [tex]\( x^2 - 6x - 7 \)[/tex]:
[tex]\[
x^2 - 6x - 7 = (x - 7)(x + 1)
\][/tex]
Factor [tex]\( x^2 + 2x - 8 \)[/tex]:
[tex]\[
x^2 + 2x - 8 = (x + 4)(x - 2)
\][/tex]
Thus, the expressions can be written as:
[tex]\[
\frac{x+1}{(x+4)(x-4)} \cdot \frac{(x+4)(x-2)}{(x-7)(x+1)}
\][/tex]
2. Combine the expressions into a single fraction:
[tex]\[
\frac{(x + 1)(x + 4)(x - 2)}{(x + 4)(x - 4)(x - 7)(x + 1)}
\][/tex]
3. Simplify by removing common factors:
We notice that both the numerator and the denominator have the factors [tex]\(x + 1\)[/tex] and [tex]\(x + 4\)[/tex].
Cancelling these common factors:
[tex]\[
= \frac{(x - 2)}{(x - 4)(x - 7)}
\][/tex]
4. Final simplified form:
After canceling the common factors, the simplified expression becomes:
[tex]\[
= \frac{(x - 2)}{(x - 4)(x - 7)}
\][/tex]
Therefore, the simplified form of the given division of rational expressions is:
[tex]\[
\boxed{\frac{x - 2}{(x - 4)(x - 7)}}
\][/tex]
