Question #85073

1 Answer
May 9, 2016

Answer:

#(x^2+1)/(x(x-1)^3) = -1/x + 1/(x-1)+2/(x-1)^3#

Explanation:

Performing the decomposition, we first set up the equation with the unknown constants.

#(x^2+1)/(x(x-1)^3) = A/x + B/(x-1) + C/(x-1)^2 + D/(x-1)^3#

Multiplying both sides by #x(x-1)^3#, we get

#x^2+1 = A(x-1)^3 + Bx(x-1)^2 + Cx(x-1) + Dx#

#=(A+B)x^3+(-3A-2B+C)x^2+(3A+B-C+D)x+(-A)#

Equating the coefficients of corresponding powers of #x#, we end up with the following system of equations:

#{(A+B=0),(-3A-2B+C=1),(3A+B-C+D=0),(-A=1):}#

From the fourth equation, we have #A=-1#

Substituting that into the first equation and solving for #B# gives #B=1#

Substituting both of those into the second equation and solving for #C# gives #C = 0#

Substituting all of those into the third equation and solving for #D# gives #D = 2#

Thus, from our original equation, we have the decomposition

#(x^2+1)/(x(x-1)^3) = -1/x + 1/(x-1)+2/(x-1)^3#