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.


As $|x|$ tends to infinity/grows large the value of $x^4+ rx + s$ will tend to (positive) infinity.

Thus, the question boils down to whether the minimum of the function is positive or not.

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.


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