Here is a simple consequence of the quadratic formula: The polynomial f(x) = ax2 + bx + c has two real roots (counting multiplicities) if and only if b2- 4ac ³ 0. In the spring of 1999, Joe and I looked for a corresponding statement for cubic polynomials. We proved the following:
Theorem 1: The polynomial f(x) = x3 + ax2 + bx + c has three real roots (counting multiplicities) if and only if
(2/27)2(3b - a2
)3 + (c - (1/3)ab + (2/27)a3 )2
£ 0.
We were also able to prove a weaker statement for higher degree polynomials:
Theorem 2: If the polynomial
f(x) = xn + dn-1 xn-1 + dn-2 xn-2 + . . .
has n real roots (counting multiplicities), then 2n (dn-2 ) - (n - 1) ( dn-1 )2 £ 0.
Thus if the condition in the theorem is not satisfied, we immediately know that the polynomial has a non-real root.
Joe presented the project at Missouri Western’s Interdisciplinary Research Day in the fall of 1999. In April of 2000, I gave a talk on our project at a meeting of Montana Academy of Sciences in Missoula, Montana. A paper based on our project appeared in the May, 2001 issue of Crux Mathematicorum with Mathematical Mayhem .
In the summer of 2000, Ben Drushell wrote a Java applet that helps illustrate exactly how the coefficients of the cubic affect its roots. See Ben’s applet .
To check a particular cubic, try
the following applet. It refers to “Condition 2,” which is the formula
given in Theorem 1. See the applet
.