Factoring a quartic polynomial over the reals

[Mr Davis 97]

I have the simple quartic polynomial $x^4+1$. How in general do I determine whether this is factorable over the reals or not? Since it has no real roots, it could only factor into two quadratic polynomials, but I am not sure what I can do to determine whether this is possible or not.

 

[StoneTemplePython]

All coefficients of the polynomial are real. Thus any roots with non-zero imaginary components come in conjugate pairs... Take advantage of this structure to split into 2 quadratics that you can multiply.

i.e.

for complex $\lambda$ (with non-zero imaginary component) we have

$(x - \lambda)(x - \bar{\lambda}) = x^2 - 2\Big(\text{real}(\lambda)\Big)x + \vert \lambda \vert^2$

 

[jedishrfu]

There's an old discussion here on this very problem:

https://www.physicsforums.com/threads/factoring-x-4-1.371741/

 

[mfb]

You can factor every (non-trivial) polynomial with only real coefficients into linear and quadratic terms.
This is a direct consequence of the full factorization in the complex numbers and the conjugate pairs of complex roots.