Question

write the partial fraction decomposition step by step

(-5x^2-8x-3)/(x^3+x^2+x)

Answer 1

First factor the denominator:
x^3+x^2+x=x(x^2+x+1)

Then assume constants A, B and C as follows:

(-5x^2-8x-3)/(x^3+x^2+x) = A/x + (Bx+C)/(x^2+x+1)
The steps are to find constants A, B and C (the solution) by adding together the right hand side and comparing coefficients of each power of x.

Using common denominator,
RHS=A/x + (Bx+C)/(x^2+x+1)
=[(A+B)x^2+(A+C)x + A]/(x^3+x^2+x)
Compare with LHS, which is
(-5x^2-8x-3)/(x^3+x^2+x)
If LHS is the same as RHS,
then the coefficient of each power of x should be equal, thus
A=-3
A+C=-8
A+B=-5
=> A=-3, C=-5, B=-2
or
the partial fractions are:
-3/x + (-2x-5)/(x^2+x+1)