Let's say that I have an incomplete quartic equation with real coefficients, which is $$x^4 - 3x^3 + ... - 10 = 0$$
And also given 2 complex roots, $a + 2i$ and $1 + bi$ where $a$ and $b$ are real numbers.
The problem asks the sum of the real roots, but firstly I don't know how to determine if the equation even has a real root.
I can't do Rule of Signs because obviously the polynomial is incomplete. Although I can do a (heavy) assumption that $1 + bi$ is the conjugate of $a + 2i$ or not, that still would give me $0$ or $2$ real roots.
How can I know if it has a real root or not?
EDIT : I got the second term wrong, it should be $-3x^3$ and not $-3x^2$ ! I have edited the original equation. Sorry!