How do you write the partial fraction decomposition of the rational expression #(7x2)/((x3)^2(x+1))#?
2 Answers
The answer is
Explanation:
Let's perform the decomposition into partial fractions
The denominators are the same, we compare the numerators
Let
Let
Coefficients of
Coefficients of
Therefore,
Explanation:
We recognize that there will be three different fractions with denominators of

#x3# 
#(x3)^2# 
#x+1#
So we set it up as
#(7x2)/((x3)^2(x+1)) = A/(x3) + B/((x3)^2) + C/(x+1)#
Multiply both sides by the quantity
#7x2 = A(x3)(x+1) + B(x+1) + C(x3)^2#
Expand the terms:
#7x2 = A(x^22x+3) + B(x+1) + C(x^26x+9)#
Collect the terms by power of
#7x2 = x^2(A+C) + x(2A + B  6C) + 3A + B + 9C#
Equate the coefficients on both sides (there is no
Now we set up an augmented matrix (sorry for the poor quality):
In order to avoid mayhem, I won't work out the solving of the matrix (maybe you already know how, if not there's plenty of other sources, maybe some of Socratic!), but the solutions are
Plugging these into the original equation, we have