Question

What is the quotient in simplest form? State any restrictions on the variable. If anybody can figure this out please help.. medal rewareded.
(z^2-4)/(z-3) divided (z+2)/(z^2+z-12)

Answer 1

@whadduptori

\[\dfrac{\dfrac{z^2-4}{z-3}}{\dfrac{z+2}{z^2+z-12}}\]

First thing to remember is that when dividing by a fraction, we can achieve the same result by taking the reciprocal of the divisor (invert it) and multiplying instead of dividing. For our problem, that would become

\[\frac{z^2-4}{z-3}*\frac{z+2}{z^2+z-12}\]

Next thing to do would be to factor the \(z^2-4\) and \(z^2+z-12\) and see if we end up with some matching items in both numerator and denominator that we can cancel out.

To find the restrictions, if any, take the original denominators (before cancellation) and find all of the values of \(z\) that would result in a division by 0. For example, one is that \(z\ne3\) because that would result in the left hand fraction in the multiplication having a denominator of 0.

Answer 2

Values of \(z\) that make the numerator be 0 are fine, so long as they don't also make the denominator be 0.