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
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. 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 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.
To simplify the expression (z^2-4)/(z-3) ÷ (z+2)/(z^2+z-12), we can follow these steps:
Step 1: Factor the numerator and denominator of the first fraction.
(z^2-4) can be factored as (z+2)(z-2).
Step 2: Factor the denominator of the first fraction.
(z-3) is already factored.
Step 3: Invert the second fraction and multiply.
Dividing by a fraction is the same as multiplying by its reciprocal, so we can rewrite the expression as:
((z+2)(z-2)/(z-3)) * ((z^2+z-12)/(z+2))
Step 4: Factor the numerator and denominator of the second fraction.
(z^2+z-12) can be factored as (z+4)(z-3).
Step 5: Cancel out common factors.
In the numerator, the (z-3) factors cancel out with the (z-3) in the denominator, and the (z+2) factors cancel out with the (z+2) in the denominator:
(((z+2)(z-2))/(z-3)) * (((z+4)(z-3))/(z+2))
Step 6: Simplify further if possible.
The (z+2) factors still cancel out in the numerators and denominators, and we are left with:
(z-2) * (z+4)
So the simplified expression is (z-2)(z+4).