First, let's simplify each fraction separately:
z^2−4/z−3 can be factored as (z+2)(z-2)/(z-3)
z+2/z^2+z−12 can be factored as (z+2)/(z-3)(z+4)
Now we can rewrite the original expression as:
[(z+2)(z-2)/(z-3)] / [(z+2)/(z-3)(z+4)]
Invert the second fraction and multiply:
[(z+2)(z-2)/(z-3)] * [(z-3)(z+4)/(z+2)]
Simplify:
(z-2) * (z+4) = z^2 + 2z - 8
Answer: (z^2 + 2z - 8)/(z-3)
Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.
z^2−4/z−3 ÷ z+2/z^2+z−12
3 answers
Note: Enter your answer and show all the steps that you use to solve this problem in the space provided. What is the quotient in simplest form? State any restrictions on the variable.
z^2−4/z−3 ÷ z+2/z^2+z−12
z^2−4/z−3 ÷ z+2/z^2+z−12
To divide fractions, we need to multiply by the reciprocal of the second fraction:
(z^2-4)/(z-3) * (z^2+z-12)/(z+2)
Now we can factor each expression:
(z+2)(z-2)/(z-3) * (z+4)(z-3)/(z+2)
Simplify:
(z-2)(z+4) = z^2 + 2z - 8
So the quotient in simplest form is:
z^2 + 2z - 8
The restriction on the variable is that z cannot be equal to -2 or 3, since those values would make the denominator(s) equal to zero.
(z^2-4)/(z-3) * (z^2+z-12)/(z+2)
Now we can factor each expression:
(z+2)(z-2)/(z-3) * (z+4)(z-3)/(z+2)
Simplify:
(z-2)(z+4) = z^2 + 2z - 8
So the quotient in simplest form is:
z^2 + 2z - 8
The restriction on the variable is that z cannot be equal to -2 or 3, since those values would make the denominator(s) equal to zero.