Cubic Equations in the xb Plane
by
Mike Rosonet
Consider the cubic graph
.
Now graph it in the "xb" plane. What this implies is substituting b = y, a = c = 1 into the equation and graphing it. Doing so will give the following graph.
Let b = 4. Overlay the graph of this function with the graph above. Notice that the function (red line) is parallel to the x-axis and intersects the graph of the polynomial twice. Any points where the line intersects the graph of the polynomial are roots of the polynomial. Thus, when b = 4, the polynomial has 2 negative real roots. Similarly, from this graph, it is plane to see that,
- for b > 2, the polynomial has two negative real roots
- for b = 2, the polynomial has one negative real root
- for -2 < b < 2, the polynomial has no real roots
- for b = -2, the polynomial has one positive real root, and
- for b < -2, the polynomial has two positive real roots.
Now, consider the case when c = -1 instead of +1.
The case is still the same as when c = 1; where the function b = 4 intersects the polynomial (blue hyperbola) is the location of the roots of the polynomial. Hence, when b = 4, there is one negative real root and one positive real root.
Consider changing the values of c to c = -3, -1, 0, 1, and 3.
Notice from the graph the following pattern:
when c > 0, the differing values of b cause there to be two roots, one root, or no roots
when c = 0, the differing values of b cause there to be exactly two roots at x = 0 and x = -b
when c < 0, the differing values of b cause there to be exactly two roots, one positive and one negative.
Secondary Investigation
What is the relationship between the graph of 2x + b = 0 and the graph of the differing values of c as shown above?
First, add the graph of 2x + b = 0 to the graph of the differing c values.
The graph 2x + b = 0 (black line) appears to be going through the middle of each of the functions graphed above. This can be proved through the use of calculus. Find the derivative of the polynomial.
Hence, 2x + b = 0 is the first derivative of the polynomial. By definition, the first derivative of a polynomial will give a critical value when set equal to zero and solved for x. Hence, the line above is the location of the vertex of the polynomial for every value of c.
Also, by definition, the vertex is equidistant from the real roots of a quadratic. Therefore, the line 2x + b = 0 goes through the exact middle point of every graph shown above as well as for all other real values of c.
Extension 1
Consider the pattern of roots in the xc plane.
Extension 2
Consider the pattern of roots in the xd plane.
Extension 3
Consider the pattern of roots in the xa plane.