Is there any way to know the number of imaginary and real solutions to a polynomial by just looking at the equation?

Question

For an equation $Ax^4 + Bx^3 + Cx^2 + Dx + E=0$ or any $n$ degree polynomial with $n \geq 4$, can you tell the number of real and imaginary roots without solving or factoring the equation?

Answer 1

For $n=4$ and real coefficients, lookig here, you will see that, for , computing a few numbers, you can conclude about the number and nature of the roots.

Just reproducing results given in the Wikipedia page, given $$ax^4+bx^3+cx^2+dx+e=0$$ compute

$$\Delta=256 a^3 e^3-192 a^2 b d e^2-128 a^2 c^2 e^2+144 a^2 c d^2 e-27 a^2 d^4+144 a b^2 c e^2-6 a b^2 d^2 e-80 a b c^2 d e+18 a b c d^3+16 a c^4 e-4 a c^3 d^2-27 b^4 e^2+18 b^3 c d e-4 b^3 d^3-4 b^2 c^3 e+b^2 c^2 d^2$$ $$P=8 a c-3 b^2 \qquad Q=8 a^2 d-4 a b c+b^3\qquad \Delta_0=12 a e-3 b d+c^2\qquad D=64 a^3 e-16 a^2 b d-16 a^2 c^2+16 a b^2 c-3 b^4$$ Based on these, you will find the different cases described.

Excluding the case of multiple roots (corresponding to $\Delta=0$), the problem is quite simple :

• if $\Delta <0$, then the equation has two distinct real roots and two complex conjugate non-real roots.
• if $\Delta >0$, then either the roots are all real or none is

Then, subcases need to be considered.