How do I find the partial-fraction decomposition of #(2x^3+7x^2-2x+6)/(x^4+4)#?
2 Answers
First factor the denominator to find
Then solve
finding
Explanation:
It will have quadratic factors with Real coefficients.
Without loss of generalisation,
Then looking at the coefficient of
Then looking at the coefficient of
If
If
So we have
Now attempt to solve:
If we multiply through by
Equating coefficients we get:
(i)
(ii)
(iii)
(iv)
From (i) and (iv) we get:
(v)
(vi)
Substitute these into (ii) to get:
Hence
Substitute
Divide both sides by
Hence
So
Explanation:
Here is a quick way to find the quadratic factors of
If
Here are the roots in the complex plane:
graph{((x-1)^2+(y-1)^2 - 0.01)((x+1)^2+(y-1)^2 - 0.01)((x-1)^2+(y+1)^2 - 0.01)((x+1)^2+(y+1)^2-0.01) = 0 [-5, 5, -2.5, 2.5]}
So
Now pick pairs of factors to multiply to get Real coefficients.
To do this, pick the ones which are complex conjugates:
and
For the rest of the partial fraction decomposition, see the other answers.