Question

What is the quotient in simplest form? State any restrictions on the variable.

Answer 1

The given expression:

$\frac{z^2-4}{z-3}\div \frac{z+2}{z^2+z-12}$

Apply the fraction rule:

$\begin{gathered} \frac{\frac{a}{b}}{\frac{c}{d}}=\frac{a\cdot \:d}{b\cdot \:c} \\ =\frac{\left(z^2-4\right)\left(z^2+z-12\right)}{\left(z-3\right)\left(z+2\right)} \end{gathered}$

Cancel out (z - 3)(z + 2):

$=\left(z-2\right)\left(z+4\right)$

Therefore,

$(z-2)(z+4)=z^2+2z-8$

Therefore, the quotient in simplest form is

$z^2+2z-8$

The restrictions are that none of the cancelled factors can be 0, so z ≠ 3 and z ≠- 4.