Let's consider the third problem:
Let's first multiply both sides by the least common denominator :
Then we bring the right hand side into "standard form'':
We need the left side polynomial and the right side polynomial to coincide; but two polynomials coincide if and only if all their coefficients are the same! Thus
implies the following system of linear equations:
Let's solve this system and we're done! The third equation implies A=-7, using this information in the first equation yields B=7; last not least, solving the second equation for C we obtain:
This is it. We have computed that
(2) Solving systems of linear equations with "pencil and paper'' is quite error-prone! You can check your answer by combining the partial fractions you obtain into the complicated fraction you started out with.
(2) The computations can be quite cumbersome. For instance, rewriting the fourth problem
using Mathematica yields the partial fraction decomposition
You definitely don't want to do this "by hand''!