Factoring a polynomial of fourth degree with false roots: $x^4+4$

Question

I want to write this polynomial in factored form:

$$x^4+4$$

The reason I want to do this is to be able to make partial-fraction decomposition on it to make an integrand easier to integrate. What's the general method?

In addition to this, I searched on how to figure out the numerators $A, B...etc$ for this decomposed fraction, is Inverse Laplace the way to go? Or is there another way? (which doesnt make it any more difficult).

Answer 1

Hint

$$x^4+4=(x^4\color{red}{+4x^2}+4) \color{red}{-4x^2}$$ and notice that $$a^2-b^2=(a-b)(a+b)$$

Answer 2

$x^4 + 4 = (x^2 + 2i)(x^2 - 2i)$

or without complex numbers :

$x^4 + 4 = (x^2 - 2x + 2)(x^2 + 2x + 2)$

Maybe that helps you ?

Answer 3

You equate the polynomial to zero and solve the equation. In this case, you find that the roots are

$$x_n = \sqrt{2}\exp(i \pi n/2)$$

for $n = 0,1,2,3$

The partial fraction expansion will then be of the form:

$$\sum_{n=0}^3 \frac{A_n}{x-x_n}$$

where $A_n$ is the residue of the integrand at $x = x_n$.