# How do you express #1/ (x^4 +1)# in partial fractions?

##### 2 Answers

The polynomial

#### Explanation:

The polynomial

#### Explanation:

However, it is possible to factor it into quadratic factors:

#x^4+1 = (x^2-sqrt(2)+1)(x^2+sqrt(2)x+1)#

Hence there is a partial fraction decomposition in the form:

#1/(x^4+1) = (Ax+B)/(x^2-sqrt(2)+1)+(Cx+D)/(x^2+sqrt(2)x+1)#

#=((Ax+B)(x^2+sqrt(2)x+1)+(Cx+D)(x^2-sqrt(2)x+1))/(x^4+1)#

#=((A+C)x^3+(sqrt(2)A+B-sqrt(2)C+D)x^2+(A+sqrt(2)B+C-sqrt(2)D)x+(B+D))/(x^4+1)#

Equating coefficients, we find:

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

Subtracting the first of these from the third, then dividing by

#B-D = 0#

Combining this with the fourth equation, we find

Substituting

#sqrt(2)A-sqrt(2)C+1 = 0#

Combining this with the first equation, we find:

#2sqrt(2)A = -1#

Hence

Hence we find:

#1/(x^4+1) = (-sqrt(2)x+2)/(x^2-sqrt(2)+1) + (sqrt(2)x+2)/(x^2+sqrt(2)+1)#