When do Cubics have Real Roots?
Every cubic function: may be transformed by a linear change of variable
to the form . In the latter form, the abscissa of the inflection point is 0, so the average of the roots of the new equation is 0. Such an equation has three real solutions if and only if the original equation did. One solves g(u) = 0 for u and then sets the roots to
This procedure might be called completing the cube but it amounts to shifting the function so that the graph has inflection point at u = 0. The actual shape of the cubic graph and its position above or below the x-axis is determined by the two parameters: d and e.
When we develop techniques for solving cubic equations later in this Microworld we will show that the equation with d and e real has all real solutions if and only if the discriminant: . The points in the plane that satisfy this condition are the points inside the cusp:
In this exercise, you will get a little experience with how the choice of equation (hence the choice of d and e) determine the shape of the graph and whether it has all real roots.
To do this experiment, just click at any point on the right-hand screen to determine the values of d and e. The corresponding graph will be drawn on the left. You may clear the screens separately using the pushbuttons below.
You may notice that all graphs drawn have abscissa of inflection point equal to 0. Now when the roots are not all real, the average of the real roots (the single real root) is in general not equal to 0. In order to understand this, we will have to take a close look at complex numbers. And we shall do that in the next section.
But for those of you who know, you may recall that we mentioned earlier that the abscissa of the inflection point is the average of the real parts of the roots. If one root is real, then the real parts of each of the other two are equal (since they are conjugate). Thus the real part of each complex (not real) root is 1/2 the negative of the real root. What is going on here? Everything will be clear when we introduce the "roots of unity." They are special complex numbers with some remarkable properties. When you finish this exercise, you will be ready to move to Chapter 2: Euler's Formula.