Question

Determine the values for which the rational expression is undefined.

$\frac{\mathrm{b}-7}{{\mathrm{b}}^2-25}.$

Short answer

Rational function can be undefined at$\mathrm{b}=5,-5.$

Step-by-step solution

Step 1. Given data

Given that the rational function is$\mathrm{R}\left(\mathrm{x}\right)=\frac{\mathrm{b}-7}{{\mathrm{b}}^2-25}.$

Step 2. Rational function

Take $\mathrm{R}\left(\mathrm{x}\right)=\frac{\mathrm{b}-7}{{\mathrm{b}}^2-25}$

On applying the property of difference of two squares,

$$\begin{gathered} {\mathrm{x}}^2-{\mathrm{y}}^2=(\mathrm{x}-\mathrm{y})(\mathrm{x}+\mathrm{y}) \\ {\mathrm{b}}^2-25 \\ {\mathrm{b}}^2-5^2 \\ (\mathrm{b}-5)(\mathrm{b}+5) \\ \mathrm{R}\left(\mathrm{x}\right)=\frac{(\mathrm{b}-7)}{(\mathrm{b}-5)(\mathrm{b}+5)} \end{gathered}$$

When the denominator is zero, the rational function can be undefined.

$$\begin{gathered} (\mathrm{b}-5)(\mathrm{b}+5)=0 \\ \mathrm{b}=5,-5 \\ \mathrm{At}\mathrm{b}=5,\mathrm{R}\left(\mathrm{x}\right)=\frac{5-7}{25-25}=\infty \\ \mathrm{At}\mathrm{b}=-5,\mathrm{R}\left(\mathrm{x}\right)=\frac{-5-7}{25-25}=\infty \end{gathered}$$

when$\mathbf{b}\mathbf{=}\mathbf{5}\mathbf{,}\mathbf{-}\mathbf{5}$rational function is undefined.