Chapter 8.5, Problem 6LC

Solution

To identify: The two rational expressions that simplify to $\frac{x+1}{x-5}$ .

The two rational expressions that simplify to $\frac{x+1}{x-5}$ are $\frac{x^2+2x+2}{x^2-4x-5}$ and $\frac{x^2-4x-5}{x^2-10x+25}$ .s

Given information:

The rational expression $\frac{x+1}{x-5}$ .

Explanation:

To simplify the ration expressions, identify the restrictions of the function.

To multiply the rational expressions, look for the numerator and denominator to see if there area nay common factors that can be cancelled out.

To divide the rational expressions, multiply the dividend by the reciprocal of the divisor.

Consider the rational expression $\frac{x+1}{x-5}$ ,

Multiply and divide the expression by $x+1$ .

  $\begin{array}{c}\frac{x+1}{x-5}=\frac{{\left(x+1\right)}^2}{\left(x-5\right)\left(x+1\right)}\\ =\frac{x^2+2x+2}{x^2-4x-5}\end{array}$

Multiply and divide the expression by $x-5$ .

  $\begin{array}{c}\frac{x+1}{x-5}=\frac{\left(x+1\right)\left(x-5\right)}{{\left(x-5\right)}^2}\\ =\frac{x^2-4x-5}{x^2-10x+25}\end{array}$

Thus, the two rational expressions that simplify to $\frac{x+1}{x-5}$ are $\frac{x^2+2x+2}{x^2-4x-5}$ and $\frac{x^2-4x-5}{x^2-10x+25}$ .