Assignment 3: Exploring Coefficients and
Roots

by

Tom Cooper

Let's explore how viewing an equation in a plane other than the regular xy plane reveals information about its roots.

We will begin by examining y = x^{2} + bx
+ 1. What happens as we vary the values
of b?

The following animation shows the graph as b runs from -5
to 5 with 100 steps. The moving parabola is y = x^{2} + bx + 1 and the other parabola is y = -x^{2} + 1.

It appears that as b varies, the graph of y = x^{2} + bx + 1 is translated along y = -x^{2} + 1.

In fact this will generalize to varying b in y = ax^{2} + bx + c
causes translations along y = -ax^{2} + c. Why?

How do the types of roots change with b? Instead of animation, let's look at several plots together.

One thing that we can observe from the above graph is that y = x^{2}
+ bx + 1 = 0 has 2 (real) positive roots when b<-2,
1 (real) positive roots when b = -2, no real roots when -2<b<2, 1 (real)
negative root when b = 2, and 2 (real) negative roots when b>2.

Another way to see this would be to graph the equation in the xb plane. In
order to do this, we will plot x^{2} + yx + 1 = 0, since our software
likes to use the xy plane.

The purple graph is x^{2} + bx + 1 = 0. Since all points on the
purple graph satisfy x^{2} + bx + 1 = 0 (i.e., they are roots), the
intersections with b = k give the roots when b is a given value k. So this
graph also shows that y = x^{2} + bx + 1
= 0 has 2 (real) positive roots when b<-2, 1 (real) positive roots when b =
-2, no real roots when -2<b<2, 1 (real) negative root when b = 2, and 2
(real) negative roots when b>2.

Here is a plot that animates the curve and shows the correspondence in the
xb plane. The parabola is y = x^{2} + bx + 1.

Of course we could fix any two of the coefficients and observe the behavior of the roots upon varying the third.

Instead, let's look at a similar procedure with a
cubic equation, y = x^{3} + bx^{2} + x + 1 = 0.

Here is an animation as b varies from -5 to 5.

Now here is the animation combined with x^{3} + bx^{2} + x +
1 = 0 in the xb plane.

It appears that the function will have 3 real roots (2 positive, 1 negative) when b < -2.6 (approximately), 2 real roots (1 positive, 1 negative) when b = -2.6, and 1 real negative root when b > -2.6