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)