Question

Express the following difference in the simplest form $?$

$\frac{{\mathrm{d}}^2+3}{{\mathrm{d}}^2-2\mathrm{d}}-\frac{\mathrm{d}-4}{\mathrm{d}}$

Answer 1

Find the difference in the simplest form:

$\frac{{\mathrm{d}}^2+3}{{\mathrm{d}}^2-2\mathrm{d}}-\frac{\mathrm{d}-4}{\mathrm{d}}$ (Take the common term)

$=\frac{{\mathrm{d}}^2+3}{\mathrm{d}(\mathrm{d}-2)}-\frac{\mathrm{d}-4}{\mathrm{d}}$ (simplify by taking the least common denominator)

$=\frac{{\mathrm{d}}^2+3-(\mathrm{d}-2)(\mathrm{d}-4)}{\mathrm{d}(\mathrm{d}-2)}$ (simplify by distribution)

$$\begin{gathered} =\frac{{\mathrm{d}}^2+3-\mathrm{d}(\mathrm{d}-4)-2(\mathrm{d}-4)}{\mathrm{d}(\mathrm{d}-2)} \\ =\frac{{\mathrm{d}}^2+3-{\mathrm{d}}^2+4\mathrm{d}-2\mathrm{d}+8}{\mathrm{d}(\mathrm{d}-2)} \end{gathered}$$ (group the like term together)

$=\frac{({\mathrm{d}}^2-{\mathrm{d}}^2)+(4\mathrm{d}-2\mathrm{d})+(3+8)}{\mathrm{d}(\mathrm{d}-2)}$ (combine the like terms together)

$=\frac{2\mathrm{d}+11}{\mathrm{d}(\mathrm{d}-2)}$

Hence, the difference in the simplest form is $\frac{\mathbf{2}\mathbf{d}\mathbf{+}\mathbf{11}}{\mathbf{d}\mathbf{(}\mathbf{d}\mathbf{-}\mathbf{2}\mathbf{)}}$.