Chapter 11.1, Problem 2LC

To determine

To simplify the expression $\frac{5-x}{x^2-2x-15}$ and state any excluded values.

Answer to Problem 2LC

  $\frac{-1}{x+3},x\ne -3,5$

Explanation of Solution

Given:

The expression, $\frac{5-x}{x^2-2x-15}$

Concept Used:

• Least Common Denominator (LCD) is the smallest number that is divisible by the denominator of all the fractions that are being added or subtracted.
• Solving an equation containing rational expressions
1. Multiply both sides of equations by LCD(least common denominator) of all denominators.
2. Remove any grouping symbols and solve the resulting equation for the variable asked in equation

Calculation:

In order to simplify the given expression, first write the denominator in factored form and then cancel out any common factor from numerator and denominator to simplify further as shown below,

  $\begin{array}{c}\frac{5-x}{x^2-2x-15}=\frac{5-x}{\left(x-5\right)\left(x+3\right)}\\ =\frac{-1\left(x-5\right)}{\left(x-5\right)\left(x+3\right)}\\ =\frac{-1}{\left(x+3\right)}\end{array}$

Thus, the given expression simplifies to $\frac{-1}{x+3}$ .

The excluded values of the expression can be found by setting the denominator zero. That is,

  $\begin{array}{c}{x}^2-2x-15=0\\ \left(x-5\right)\left(x+3\right)=0\\ x+3=0\text{or}x-5=0\\ x=-3\text{or}x=5\end{array}$

Thus, the excluded values are $x\ne -3,5$ .