what is the quotient in simplest form? state any restrictions on the variable.
(z^2 - 4)/(z - 3) divided by (z+2)/(z^2 + z - 12)
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)