How do I find the partial fraction decomposition of x4x41 ?

1 Answer
Sep 18, 2014

By Partial Fraction Decomposition, we can write

x4x41=114x+1+14x112x2+1.

Let us look at some details.

By rewriting a bit,

x4x41=1+1x41

Let us find the partial fractions of

1x41

by factoring out the denominator,

=1(x+1)(x1)(x2+1)

by splitting into the partial fraction form,

=Ax+1+Bx1+Cx+Dx2+1

by taking the common denominator,

=A(x1)(x2+1)+B(x+1)(x2+1)+(Cx+D)(x21)(x+1)(x1)(x2+1)

by simplifying the numerator,

=(A+B+C)x3+(A+B+D)x2+(A+BC)x+(A+BD)x41

Since the numerator is originally 1, by matching the coefficients,

(1) A+B+C=0
(2) A+B+D=0
(3) A+BC=0
(4) A+BD=1

By adding (1) and (3),

(5) 2A+2B=0

By adding (2) and (4),

(6) 2A+2B=1

By adding (5) and (6),

(7) B=14

By plugging (7) into (5),

(8) A=14

By plugging (7) and (8) into (1),

(9) C=0

By plugging (7) and (8) into (2),

(10) D=12

By (5), (6), (9), and (10),

x4x41=114x+1+14x112x2+1