what is the quotient in simplest form? state any restrictions on the variable.

Question

(z^2 - 4)/(z - 3) divided by (z+2)/(z^2 + z - 12)

Answer 1

To divide two fractions, we multiply by the reciprocal of the second fraction.

The expression becomes:

(z^2 - 4)/(z - 3) * (z^2 + z - 12)/(z + 2)

Now, let's factor the numerator and denominator of each fraction:

(z^2 - 4) can be factored as (z + 2)(z - 2)
(z^2 + z - 12) can be factored as (z + 4)(z - 3)

The expression becomes:

[(z + 2)(z - 2)/(z - 3)] * [(z + 4)(z - 3)/(z + 2)]

Now, cancel out the common factors:

= (z - 2)(z + 4)

So, the quotient in simplest form is (z - 2)(z + 4).

Restrictions on the variable:
1. z cannot equal 3 (due to the denominator z - 3)
2. z cannot equal -2 (due to the denominator z + 2)