Question 447148
Find the partial fraction decomposition of the following rational expression:
{{{(5x^2+x+5)/(x^3+x^2+2x+2)}}} 
Factor the denominator, I used synthetic division 
 ________________
-1|1 + 1 + 2 + 2
and got
{{{(5x^2+x+5)/(x^2+2)(x+1))}}} = {{{A/((x^2+2))}}} + {{{B/((x+1))}}} = {{{(A(x+1)+B(x^2+2))/((x^2+2)(x+1))}}}
If the denominators are equal the numerators equal, so we have:
5x^2 + x + 5 = A(x+1) + B(x^2+2)
:
let x=-1, then one factor will drop out, we can solve for B
5(-1)^2 - 1 + 5 = A(-1+1) + B(-1^2+2)
5 - 1 + 5 = 0 + 3B
9 = 3B
B = 3
:
Replace B with 3, Find A
5x^2 + x + 5 = A(x+1) + 3(x^2+2)
5x^2 + x + 5 = A(x+1) + 3x^2 + 6
5x^2 - 3x^2 + x + 5 - 6 = A(x+1)
2x^2 + x - 1 = A(x+1)
Factor
(2x-1)(x+1) = A(x+1)
Divide both sides by (x+1)
(2x - 1) = A
:
so we have
{{{(5x^2+x+5)/(x^3+x^2+2x+2)}}} = {{{(2x-1)/((x^2+2))}}} + {{{3/((x+1))}}}