Factor the denominator.
x^4-2x^2-8=(x^2-4)(x^2+2)=(x+2)(x-2)(x^2+2)x4−2x2−8=(x2−4)(x2+2)=(x+2)(x−2)(x2+2)
(x^2+9)/((x+2)(x-2)(x^2+2))=A/(x+2)+B/(x-2)+(Cx+D)/(x^2+2)x2+9(x+2)(x−2)(x2+2)=Ax+2+Bx−2+Cx+Dx2+2
Find a common denominator of (x+2)(x-2)(x^2+2)(x+2)(x−2)(x2+2).
x^2+9=A(x-2)(x^2+2)+B(x+2)(x^2+2)+(Cx+D)(x^2-4)x2+9=A(x−2)(x2+2)+B(x+2)(x2+2)+(Cx+D)(x2−4)
x^2+9=Ax^3-2Ax^2+2Ax-4A+Bx^3+2Bx^2+2Bx+4B+Cx^3-4Cx+Dx^2-4Dx2+9=Ax3−2Ax2+2Ax−4A+Bx3+2Bx2+2Bx+4B+Cx3−4Cx+Dx2−4D
x^2+9=x^3(A+B+C)+x^2(-2A+2B+D)+x(2A+2B+4C)+1(-4A+4B-4D)x2+9=x3(A+B+C)+x2(−2A+2B+D)+x(2A+2B+4C)+1(−4A+4B−4D)
From this, write the following system:
{(A+B+C=0),(-2A+2B+D=1),(2A+2B+4C=0),(-4A+4B-4D=9):}
Solve the system:
{(A=-13/24),(B=13/24),(C=0),(D=-7/6):}
This gives:
(x^2+9)/(x^4-2x^2-8)=-13/(24(x+2))+13/(24(x-2))-7/(6(x^2+2))
