Investigation of the cubic equation

                                          By Na Young Kwon

 

 

Investigation 1.

 

Consider the equation

                     _____(1)

To consider the equation (1) we construct several graphs of

                   ______(2)

 for different values of a, b, c and d .

From these graphs discussion of the patterns for the roots of the equation (1)

Can be followed. For example, if we set

                         

for b=-3,-2,-1,0,1,2,3, and overlay the graphs, the following picture is obtained.

Graph3-1

 

 

We can discuss the movement of a curve as b is changed. The curve

passes through  the same point on the y-axis (the point (0,1) with this equation).

For b=-3, the curve intersected the x-axis in three points with two positive

 x values and one negative value(i.e. the original equation will have three real

 roots). In my investigation, for –2.7<b<-26, the intersection point is changed

from three to one. For –2<b<3, the curve intersected the x-axis in one point

with negative value.

Consider the extreme points of this curve. For –3<b<-2, the curve had

two extreme points with a positive maximum value and a negative minimum value.

For –1<b<1, it had no extreme point.  For 2<b<3, it had two extreme points

with a positive maximum and a negative minimum value.

 

Investigation 2.

 

Graphs in the xb plane.

Consider again the equation

 

                        x3+bx2+x+1=0

Now we investigate the graph of this relation in the xb plane.

We get the following graph.

 

Graph 3-2

 

If we take any particular value of b, say b=3, and overlay this equation

On the graph we add a line parallel to the x-axis. If it intersects the curve

In the xb plane, the intersection points correspond to the roots of the

original equation for that value of b. We have the following graph.

 

Graph 3-3

 

 As value of b is changed, we can know a change of the roots of the

Original equation. When –2<b, we get one negative real root and

when –3>b, two positive real roots and one negative real root.

For –3<b<-2, right one of the graph  is tangent to a parallel line. In this

case, there is two roots with one negative real roots and one positive root.

 

Let’s investigate when a value of a and c are changing.

Graph 3-3-1                                                           Graph 3-3-2

 

               

    

 

We find, as changing a value of a, the left part of a graph is more closer in y-axis

and the right part of a graph is far away more and more from an original point.

When a>0 and when a<0 , the graph is converse. In the case of moving a value of c,

we can find the similar situation of changing a value of a.

 

Consider the case when d=-1 rather than +1.

 

             Graph 3-4

                

 

We can know this graph is symmetry to a graph of d=+1 by original point.

 

Investigation 3.

Graphs in the xc plane.

Consider the equation

                                 

Now we consider this graph in the xc plane.

We get the following graph.

 

Graph 3-5

                               

 

If we take any particular value of c, say c=-2, and overlay this equation

on the graph we add a line parallel to the x-axis. For c=-2, we get one

negative real root of the equation.

 

Graph 3-6

             

 

Roughly saying, when c>-2.5, there is one real root and when c<-3 one

negative real roots and two positive real roots.

 

Let’s investigate the graph as a value of a is changed.

Graph 3-6-1

  

 

Similar with the graph of, this graph are different in right and left graphs.

As a value of a is increasing, the left graph of y-axis is closer to y-axis and the right graph

is far from the original point.

 

Consider the case when d=-1 rather than +1.

 

Graph 3-7

                

 

In this case, the graph of d=-1 is not symmetry to the graph of d=+1.

However, the asymptotic curve is y-axis the same as the graph of d=+1.

 

Investigation 4.

Graphs in the xd plane.

 

Consider the equation

                                          

Now graph this relation in the xd plane. We get the following graph.

 

Graph 3-8

                         

 

This graph is the graph of general cubic equation.

If we take any particular value of d, we get one intersection point

(i.e. the original equation has one real root). When d>0, there is

one negative real root, when d<0, one positive real root and for d=0,

one real root x=0. This graph has no local extreme value and symmetry to

original point. If a sign of a is changed, a=-1, then the end of the graph is

increasing and this graph has two extreme value with one local maxima

and one local minima. We can find this truth and also see that a value of a is

changed by the following graphs.

 

Graph 3-9

               

 

 

When b=0, the graph is symmetry to the original point for any value of c.

 

Graph 3-10

 

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