# Finding the discriminant of a quartic polynomial, and determining whether it has complex conjugate roots.

I have learnt the if the discriminant of a polynomial is less than 0, then the polynomial has complex conjugate roots. I am trying to find a function for a quartic polynomial with a quadratic factor and complex conjugate roots. How do you find the discriminant of a quartic polynomial? I have only been familiarised with the discriminant of a quadratic.

The discriminant of the polynomial $(x-r_1)(x-r_2)\ldots(x-r_n)$ is $\prod_{i<j}(r_i-r_j)^2$. Therefore, if you take $P(x)=x(x-1)(x-i)(x+i)$, its discriminant is
$(0-1)^2(0+i)^2(0-i)^2(1-i)^2(1+i)^2(i+i)^2=-16.$
On the other hand, $P(x)=(x^2-x)(x^2+1)=x^4-x^3+x^2-x$.