Question

How do you write the partial fraction decomposition of the rational expression `(2x^3-x^2+x+5)/(x^2+3x+2)`?

Answer 1

`(2x^3-x^2+x+5)/(x^2+3x+2) = 2x-7+1/(x+1)+17/(x+2)`

Explanation:

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

`=(color(blue)(2x^3+6x^2+4x)color(green)(-7x^2-21x-14)+18x+19)/(x^2+3x+2)`

`=(color(blue)(2x(x^2+3x+2))color(green)(-7(x^2+3x+2))+18x+19)/(x^2+3x+2)`

`=2x-7+(18x+19)/(x^2+3x+2)`

Focusing on the remaining rational expression:

`(18x+19)/(x^2+3x+2)`

`=(18x+19)/((x+1)(x+2))`

`=A/(x+1)+B/(x+2)`

Using Heaviside's cover-up method we find:

`A=(18(-1)+19)/((-1)+2) = 1/1 = 1`

`B=(18(-2)+19)/((-2)+1) = (-17)/(-1) = 17`

So:

`(18x+19)/(x^2+3x+2)=1/(x+1)+17/(x+2)`

Putting it all together:

`(2x^3-x^2+x+5)/(x^2+3x+2) = 2x-7+1/(x+1)+17/(x+2)`