# How do you express #(x^3+x^2+2x+1)/((x^2+1)(x^2+2))# in partial fractions?

##### 1 Answer

Jun 12, 2016

#### 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))#

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

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

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