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

Question

what is the quotient in simplest form. state any restrictions on the variable. show all steps

Answer 1

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

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

Factor the expressions:
=((z - 2)(z + 2)/(z - 3)) * ((z + 4)(z - 3)/(z + 2))

Next, cancel out common factors in the numerator and denominator:
=((z - 2)(z + 4))/(z + 2)

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

The restrictions on the variable are z cannot be equal to 3 (due to the denominator of (z - 3)) and z cannot be equal to -2 (due to the denominator of (z + 2)) as these values would result in division by zero.