The Midpoints Between Roots Reveal the Quartic Equation

The midpoints between roots provide the key to understanding the geometry, in the complex plane, behind the roots of a quartic polynomial. In reduced form (i.e., with no cubic term), midpoints come in three pairs, with opposite signs, as solutions to a resolvent cubic. At any midpoint, a startlingly simple expression of the polynomial derivative indicates the vectors from the midpoint to the corresponding pair of roots. This approach simplifies Euler's method for solving the quartic, since there is no need to make a suitable choice of the plus or minus signs in the pairs of midpoints.