What is the partial fraction expansion of (x+2)/((x+1)^2)?
First we write the fraction in terms of partial fractions with two unknown numerators, A and B, as follows: (x+2)/(x+1)2 = A/(x+1) + B/(x+1)2
Note that since the denominator of the original fraction is of index two, we need to have two different fractions in our partial fraction expanison. Now we multiply through by (x+1)2 to get rid of all of the fractions and turn the problem into a more well known problem, solving a quadratic equation. We get: x+2 = A(x+1) + B. This is now simple to solve. We compare 'x' terms on the left and right hand side: x=Ax. This tells us A=1. Substituting this in, we have the equation: x+2=x+1+B. We can subtract x+1 from both sides and we get: 1=B. Therefore, our partial fraction expansion is:
(x+2)/(x+1)2 = 1/(x+1) + 1/(x+1)2
