Final answer:
The quotient in simplest form is (z−2)(z+4)/(z−3) with the restriction that z cannot equal 3 or −4.
Step-by-step explanation:
To find the quotient in the simplest form of the given expressions z^2−4 divided by z−3, and z+2 divided by z^2+z−12, first, we need to perform the division of the two rational expressions which is equivalent to multiplying the first expression by the reciprocal of the second. Before we do the operations, let's factor in where possible to simplify.
The first expression z^2−4 is a difference of squares and can be factored as (z+2)(z−2). The second expression z^2+z−12 can also be factored as (z+4)(z−3). Therefore, our division problem is now:
((z+2)(z−2)/(z−3)) × ((z+4)(z−3)/(z+2))
After canceling out the common factors, we get:
(z−2)(z+4)/(z−3)
The restrictions on the variable must be stated since we cannot divide by zero. The original denominators give us the restrictions z ≠ 3 from z−3 and z ≠ −4 from z+4.