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 simplify the quotient, we need to first factor the numerator and denominator.
z^2 - 4 can be factored as (z+2)(z-2)
z^2 + z - 12 can be factored as (z+4)(z-3)
Now, our expression becomes:
((z+2)(z-2))/(z-3)) ÷ ((z+2)/((z+4)(z-3)))
To divide fractions, we multiply by the reciprocal of the denominator:
((z+2)(z-2))/(z-3)) * ((z+4)(z-3)/(z+2))
(z+2)(z-2)(z+4)(z-3)/(z-3)(z+2)
Cancelling out common factors, we get:
(z+4)(z-2)
Therefore, the simplified quotient is: (z+4)(z-2)
Restrictions on the variable are z ≠ 3, z ≠ -2 and z ≠ -4.