Why the Sum of the Roots of the Reduced Polynomial is Zero

Zeroing the coefficient of the squared term (setting A=0) resulted in the sum of the roots being zero. There is a simple explanation for this. If we let p_k=x-x_k with k=0,1,2 then multiplying three monomials containing the roots together involves selecting either the unknown x or the root which is designated by 0 or 1 respectively in the table below. There are eight combinations.

There is one combination with three 0s for the x³ term, three combinations with two 0s and a 1 for the x² term, three combinations with one 0 and two 1s for the x term and one combination with three 1s for a constant term. So when we multiply the three monomials together we get coefficients that involve the roots.

Setting the coeffient of the x² term, A, equal to zero in the reduced polynomial also requires that the sum of the roots is also zero. If we did the same for a higher degree polynomial we would find that the sum of all the roots is also zero. Smith says that the cubic was first solved about 1515 nearly 500 years ago.