How do you express (x^3+x^2+2x+1)/((x^2+1)(x^2+2))x3+x2+2x+1(x2+1)(x2+2) in partial fractions?
1 Answer
Explanation:
Neither of the quadratic factors in the denominator have linear factors with Real coefficients.
So we are looking for a partial fraction decomposition of the form:
(x^3+x^2+2x+1)/((x^2+1)(x^2+2))x3+x2+2x+1(x2+1)(x2+2)
=(Ax+B)/(x^2+1) + (Cx+D)/(x^2+2)=Ax+Bx2+1+Cx+Dx2+2
=((Ax+B)(x^2+2) + (Cx+D)(x^2+1))/((x^2+1)(x^2+2))=(Ax+B)(x2+2)+(Cx+D)(x2+1)(x2+1)(x2+2)
=((A+C)x^3+(B+D)x^2+(2A+C)x+(2B+D))/((x^2+1)(x^2+2))=(A+C)x3+(B+D)x2+(2A+C)x+(2B+D)(x2+1)(x2+2)
Equating coefficients we get the following system of linear equations:
{ (A+C=1), (B+D=1), (2A+C=2), (2B+D=1) :}
From the first and third equations we find
From the second and fourth equations we find
So:
(x^3+x^2+2x+1)/((x^2+1)(x^2+2))=x/(x^2+1) + 1/(x^2+2)
