Question

Under what conditions is the expression$\frac{2d\left(d+1\right)}{\left(d+1\right)\left(d^2-4\right)}$undefined.

Short answer

The expression is undefined for $d=-2,-1,2$.

Step-by-step solution

Step 1. Write down the given information.

The given expression is$\frac{2d\left(d+1\right)}{\left(d+1\right)\left(d^2-4\right)}$

Step 2. Concept used.

The expression will be undefined for which the denominator of the given expression will be equal to zero.

Step 3. Calculation.

The given expression $\frac{2d\left(d+1\right)}{\left(d+1\right)\left(d^2-4\right)}$has denominator $\left(d+1\right)\left(d^2-4\right)$

Therefore, equating $\left(d+1\right)\left(d^2-4\right)$gives,

$\begin{array}{c}\left(d+1\right)\left(d^2-4\right)=0\\ \left(d+1\right)\left(d-2\right)\left(d+2\right)=0\\ d+1=0\text{\hspace{0.33em}or}d-2=0\text{or}d+2=0\\ d=-1\text{\hspace{0.33em}or}d=2\text{or}d=-2\\ d=\left\{-2,-1,2\right\}\\ d=-2,-1,2\end{array}$

Since the expression is undefined for $d=-2,-1,2$.

Step 4. Conclusion.

The expression is undefined for $d=-2,-1,2$.