# Factoring a quartic polynomial over the reals

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
Gold Member
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
Mentor
Last edited:
• Janosh89
mfb
Mentor
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.

• Mr Davis 97