# Find the condition that fourth degree equation $x^4+rx+s=0$ will have no real roots.

Find the condition that fourth degree equation $x^4+rx+s=0$ will have no real roots.

I know that the fourth degree equation may not have real roots whereas odd degree equation have at least one real roots by IVT. But I don't know how to find the condition that fourth degree equation have no real roots.

• do you know calculus? – Thoth Dec 27 '15 at 11:22
• What are $r$ and $s$? Real, rational, complex, integer- What? – SchrodingersCat Dec 27 '15 at 11:24
• You can probably do this using the discriminant. See here: en.wikipedia.org/wiki/Discriminant – Gregory Grant Dec 27 '15 at 11:30
• @user26977 there are discriminants for quartic equations – Varun Iyer Dec 27 '15 at 11:32
• – lab bhattacharjee Dec 27 '15 at 11:49

As $|x|$ tends to infinity/grows large the value of $x^4+ rx + s$ will tend to (positive) infinity.
The minimum will be attained at a root of the derivative $4x^3 + r$. The only real root is $\sqrt[3]{-r/4}$.
Thus the condition for no real roots is $(\sqrt[3]{-r/4})^4 + r \sqrt[3]{-r/4} + s > 0$, and this can still be somewhat simplified.