What is the quotient in simplest form? State any restrictions on the variable.
z^2−4/z−3 ÷z+2/z^2+z−12
To simplify the given expression:
(z^2-4)/(z-3) ÷ (z+2)/(z^2+z-12)
First, factor the numerator and denominator of the first fraction:
= ((z+2)(z-2))/(z-3)
Factor the denominator of the second fraction:
= (z+2)/((z+4)(z-3))
Now we have:
= ((z+2)(z-2))/(z-3) * (z+4)/((z+4)(z-3))
Cancel out common factors:
= (z-2) / 1
So, the quotient in simplest form is: z-2
Restrictions on the variable:
The variable z cannot be equal to 3 or -4 in order to avoid division by zero in the original expression.