find the partrial fraction decomposition 2x31 2Ax1 BxCx21x1

find the partrial fraction decomposition

2/x^3-1

2=A/x-1 + Bx+C/x^2+1x+1????

Solution

2 / (x^3 -1 )

2 / (x^3 -1 ) = A / (x-1) + Bx + C / (x^2+x+1)

multiplying entire equation by the denominator

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

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

on simplifying we get

A = 2/3

2 = 2/3 (x^2+x+1) + (Bx + C)( x-1)

expanding the equation

2 = (2x^2+2x+2) / 3 + Bx^2 - Bx + Cx -C

combining like terms and simplifying

-B + 2/3 = 2

B - C + 2/3 = 0

C + 2/3 = 0

hence on solving we get

B = -4/3 , C = -2/3

therefore, partial fraction decomposition is

( - 2x - 4) / 3 (x^2+x+1) + 2 / 3 (x-1)