With the standard definition of the discriminant, we can obtain equivalent conditions for the existence of complex roots only up to polynomials of degree three with real coefficients. In this work, we have shown that for any polynomial of degree $n$, there are $n−1$ conditions $\Delta_{1}(f),...,\Delta_{n−1}(f)$ that indicate whether the polynomial has a complex root. Each condition requires only 3 coefficients as input, compared to the classical definition of the discriminant, which requires all coefficients. We also demonstrated that $f(x) = a_{n}(x−r)^n$ holds if and only if all the conditions are equal to $0$. In the case where all the roots are real, $\Delta_{1}(f)$ can be used to set an upper and lower bound for the values of the roots.
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