Final answer:
The partial fraction decomposition of the given rational expression is found by expressing it as a sum of simpler fractions, solving for their coefficients, and then substituting these values back into the expression.
Step-by-step explanation:
The partial fraction decomposition of a given rational expression like -8z^2-14z+73/(3z-4)(z+5)(z+5) involves breaking it down into a sum of simpler fractions called partial fractions. Since we have two linear factors (3z-4 and z+5) and a repeated linear factor (z+5) squared, the partial fraction decomposition will have the following form:
A/(3z-4) + B/(z+5) + C/(z+5)^2
To find the values of A, B, and C, we need to put this expression over a common denominator and equate the numerators of both sides:
-8z^2-14z+73 = A(z+5)^2 + B(3z-4)(z+5) + C(3z-4)
By expanding and simplifying the equation and then equating coefficients of the same powers of z, we can solve for A, B, and C. This process usually involves a system of equations that can be solved using algebraic methods such as substitution or elimination. After finding the coefficients, we substitute them back into the partial fractions for the final expression.