How do you express (x^4+1)/(x^5+6 x^3)x4+1x5+6x3 in partial fractions?
1 Answer
Explanation:
(x^4+1)/(x^5+6x^3) = (x^4+1)/(x^3(x^2+6))x4+1x5+6x3=x4+1x3(x2+6)
color(white)((x^4+1)/(x^5+6x^3)) = A/x+B/x^2+C/x^3+(Dx+E)/(x^2+6)x4+1x5+6x3=Ax+Bx2+Cx3+Dx+Ex2+6
color(white)((x^4+1)/(x^5+6x^3)) = (Ax^2(x^2+6)+Bx(x^2+6)+C(x^2+6)+Dx^4+Ex^3)/(x^2+6)x4+1x5+6x3=Ax2(x2+6)+Bx(x2+6)+C(x2+6)+Dx4+Ex3x2+6
color(white)((x^4+1)/(x^5+6x^3)) = ((A+D)x^4+(B+E)x^3+(6A+C)x^2+6Bx+6C)/(x^2+6)x4+1x5+6x3=(A+D)x4+(B+E)x3+(6A+C)x2+6Bx+6Cx2+6
Equating coefficients gives us this system of linear equations:
{ (A+D = 1), (B+E = 0), (6A+C = 0), (6B = 0), (6C = 1) :}
Hence:
{ (A = -1/36), (B = 0), (C = 1/6), (D = 37/36), (E = 0) :}
So:
(x^4+1)/(x^5+6x^3) = -1/(36x)+1/(6x^3)+(37x)/(36(x^2+6))
