Q12. - Algebra 1

Question

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

 $\frac{\mathbf{z}\mathbf{-}\mathbf{5}}{{\mathbf{z}}^{\mathbf{2}}\mathbf{-}\mathbf{7}\mathbf{z}\mathbf{+}\mathbf{10}}$

Step-by-Step Solution

Answer

The simplified expression is $\frac{\mathbf{1}}{\mathbf{z}\mathbf{-}\mathbf{2}}$.

The excluded values are $\mathit{z}\mathbf{=}\mathbf{2}\mathbf{,}\mathbf{5}$.

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 z − 5 z 2 − 7 z + 10 .

Factor the denominator to simplify the expression $\frac{z-5}{z^2-7z+10}$.

 $\begin{array}{l}\frac{z-5}{z^2-7z+10}=\frac{z-5}{z^2-5z-2z+10}\\ =\frac{z-5}{z(z-5)-2(z-5)}\\ =\frac{z-5}{(z-5)(z-2)}\\ =\frac{1}{z-2}\end{array}$

Step 4. State the excluded values of the variable for the expression z − 5 z 2 − 7 z + 10 .

For excluded value,

Denominator $=0$

 $\begin{array}{l}{z}^2-7z+10=0\\ z^2-5z-2z+10=0\\ z(z-5)-2(z-5)=0\\ (z-5)(z-2)=0\\ z=2,5\end{array}$

Therefore,

The simplified expression is $\frac{1}{z-2}$. 

The excluded values are $z=2,5$.