perform the indicated operations and simplify
z-4/z-6 - z+1/z+6 + z-30/z^2-36
answer: 8/(z+6)
To perform the indicated operations and simplify the given expression z-4/z-6 - z+1/z+6 + z-30/z^2-36, we need to find a common denominator and combine the fractions.
First, we need to factor the denominators to understand the common denominator.
For z-4/z-6, the denominator is z-6, which cannot be factored further since it is in the form of z-a, where a is a constant.
For z+1/z+6, the denominator is also z+6, which cannot be factored any further.
For z-30/z^2-36, the denominator is z^2-36, which can be factored as the difference of squares: (z-6)(z+6).
Now that we have the factored denominators, we can determine the common denominator. In this case, the common denominator is (z-6)(z+6).
To combine the fractions, we multiply each fraction by the necessary factors to make the denominators equal to the common denominator.
The first fraction z-4/z-6 already has the common denominator, so we leave it as is.
For the second fraction z+1/z+6, we need to multiply both the numerator and the denominator by (z-6):
(z+1)(z-6)/[(z+6)(z-6)]
For the third fraction z-30/z^2-36, we need to multiply both the numerator and the denominator by (z+6):
(z-30)(z+6)/[(z-6)(z+6)]
Now, the expression becomes:
(z-4)/[z-6] - (z+1)(z-6)/[(z+6)(z-6)] + (z-30)(z+6)/[(z-6)(z+6)]
Next, we must simplify each fraction.
The first fraction (z-4)/[z-6] remains the same.
For the second fraction, we can simplify by canceling out the common factor (z-6) in the numerator and denominator:
(z+1)(z-6)/(z+6)
For the third fraction, we can cancel out the common factor (z+6) in the numerator and denominator:
(z-30)(z+6)/(z-6)
The expression now becomes:
(z-4)/[z-6] - (z+1)/(z+6) + (z-30)/(z-6)
This is the simplified form of the given expression.