How do you write the partial fraction decomposition of the rational expression # 2 / (x^3 + 1)#?

Question

1104 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-12-14T04:19:11

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

Factor the denominator as a sum of cubes.

#2/((x+1)(x^2-x+1))=A/(x+1)+(Bx+C)/(x^2-x+1)#

#2=A(x^2-x+1)+(Bx+C)(x+1)#

#2=Ax^2-Ax+A+Bx^2+Bx+Cx+C#

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

Determine the following system:
#{(A+B=0),(-A+B+C=0),(A+C=2):}#

Solve to find that #{(A=2/3),(B=-2/3),(C=4/3):}#

Therefore,

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