Question

Simplify the given expression. State the excluded values of the variables.

$\frac{\mathbf{3}\mathbf{x}\mathbf{-}\mathbf{15}}{{\mathbf{x}}^{\mathbf{2}}\mathbf{-}\mathbf{25}}$

Short answer

The simplified expression is $\frac{\mathbf{3}}{\mathbf{x}\mathbf{+}\mathbf{5}}$.

The excluded values are $\mathit{x}\mathbf{=}\mathbf{-}\mathbf{5}\mathbf{,}\mathbf{5}$.

Step-by-step solution

Step 1. Define a rational function.

A function of the form $y=\frac{p}{q}$ is called a rational function, where $p$and $q$ are the polynomials and $q\ne 0$.

Step 2. Define the excluded values for the rational function.

Any value of a variable that results in a denominator of zero in a rational function is excluded from the domain of the function and these values are called the excluded values of the rational function.

Step 3. Simplify the expression 3x−15x2−25. 

Factor the numerator and denominator to simplify the expression $\frac{3x-15}{x^2-25}$.

$\begin{array}{l}\frac{3x-15}{x^2-25}=\frac{3(x-5)}{x^2-5^2}\\ =\frac{3(x-5)}{(x+5)(x-5)}\\ =\frac{3}{x+5}\end{array}$

Step 4. State the excluded values of the variable for the expression  3x−15x2−25.

For excluded value,

Denominator $=0$

$\begin{array}{l}{x}^2-25=0\\ x^2-5^2=0\\ (x+5)(x-5)=0\\ x=-5,5\end{array}$

Therefore,

The simplified expression is $\frac{3}{x+5}$.

The excluded values are $x=-5,5$.