Question

How do you write the partial fraction decomposition of the rational expression `3/(x ^(2) + 16x +5)`?

Answer 1

`3/(x^2+16x+5)`

Explanation:

Notice how the denominator cannot be factored (within the real numbers).

This fraction is already as simple as it can get.

Answer 2

`3/(x^2+16x+5) = 3/(2 sqrt(59)(x+8-sqrt(59)))-3/(2 sqrt(59)(x+8+sqrt(59)))`

Explanation:

`x^2+16x+5 = x^2+16x+64-59`

`=(x+8)^2-(sqrt(59))^2`

`=(x+8-sqrt(59))(x+8+sqrt(59))`

So to find the partial fraction decomposition, solve:

`3/((x+8-sqrt(59))(x+8+sqrt(59)))`

`= A/(x+8-sqrt(59)) + B/(x+8+sqrt(59))`

`= (A(x+8+sqrt(59))+B(x+8-sqrt(59)))/((x+8-sqrt(59))(x+8+sqrt(59)))`

`=((A+B)x + 8(A+B) + (A-B)sqrt(59))/((x+8-sqrt(59))(x+8+sqrt(59)))`

Equating the coefficient of `x`, we find `A+B = 0`. So `B = -A`.

Equating the remaining constant term we find `2 A sqrt(59) = 3`

Hence `A=3/(2 sqrt(59))` and:

`3/(x^2+16x+5) = 3/(2 sqrt(59)(x+8-sqrt(59)))-3/(2 sqrt(59)(x+8+sqrt(59)))`