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Zeros of cubic functions

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Introduction

Andrei B. Alexandra

Mathematics                                                                                                                                

Zeros of Cubic Functions

image00.png

I am going to investigate the zeros of a cubic function. Zeros of functions are in other words roots of functions. A cubic function might have a root, two roots or three roots. An easy way to find the roots of a function is by using the Remainder Theorem, which states that a is aroot of image01.png if and only ifimage96.png. I will make a very god use of this Theorem troughout the essay.

My mission is to find the equation of the tangent lines to the average of two of the three roots, by  taking the roots two at a time. Then to find where the tangent line intersect the curve again in order to be able to state a conjecture concerning the roots of the cubic function and the tangent line at the average value of these roots.

Let us consider the cubic function image97.png and take a look at the graph of it:

image127.png

* Window values:

image137.png

First I will state theroots of the function as follow

  • -3.0
  • -1.5
  •  1.5

And then prove this using the Remainder Theorem:

image146.png is a root of image154.png

Substituting x with the values of the roots:

image97.png

  • image169.png

image02.png

image11.png

  • image21.png

image29.png

image35.png

  • image44.png

image55.png

image66.png

...read more.

Middle

  • image155.png

image156.pngimage06.pngThe intersection will be at (-3.0, 0)

image157.png

As I observe in the graph and in my calculations the tangent line at the average of two of the roots will intersect the graph at the other root. An observation which I can base it on my conjecture: The equation of a tangent at the average of any two roots will intersect the y-axis at the third root.

The next step will be to test my conjecture in another similar cubic function using the same steps as previous.

Let us consider the function: image158.png

The graph of the function:                                                        

image160.pngimage159.png

*Window values:

* The average: image113.png

* The equation of tangent: image118.pngimage06.pngimage61.png

* The slope (image161.png) of the function

*  image162.png

The roots of the function are:

  • -1
  • image163.png
  •  2

Proof using the Remainder Theorem:

image158.png

  • image164.png

image165.png

image166.png

  • image167.png

image168.png

image170.png

  • image171.png

image172.png

image173.png

Verification on the calculator :

image174.png

  • Roots -1 and image23.png

Average image175.png

image176.png

image177.png

Slope at (-0.25, 2.53) will be: image178.png

Equation of the tangent whereimage03.png, image04.png and image05.pngimage06.pngimage07.png

image08.png

Intersection occurs whenimage09.pngimage06.pngimage10.pngimage06.png the intersection will be at (2, 0)

  • Roots -1 and 2

Averageimage12.png

image13.png

image14.png

Slope at (0.5, 0) will be: image15.png

Equation of the tangent whereimage16.png, image17.png and image18.pngimage06.pngimage19.png

image20.png

Intersection occurs whenimage09.pngimage06.pngimage22.pngimage06.png the intersection will be at (0.5, 0)

  • Roots  2 and image23.png

Averageimage24.png

image25.png

image26.png

Slope at (1.25, -2.53) will be: image27.png

Equation of the tangent whereimage28.png, image30.png and image05.pngimage06.pngimage31.png

image32.png

Intersection occurs whenimage09.pngimage06.pngimage33.pngimage06.png the intersection will be at (-1, 0)

...read more.

Conclusion

the equation of tangent at the average of any two roots will intersect the y–axis at the third root.

Forward I will investigate the above properties with cubic functions which have one root or two roots.

I will first start with the cubic function which has only one root:

Let us consider the function:image74.png

image75.png

*Window values:

image77.png

The root of the equation is:

  • image78.png

Proof using the remainder theorem:

  • image79.png

image80.png

image81.png

Equation of the tangent line

  • Roots -1.1456 and -1.1456
    Average
    image82.png
    image83.png

image84.png
     
image85.png
Slope at (-1.1456, 0) will be:image86.png

Equation of the tangent where image87.pngand image88.png  at image09.png

image89.png

image90.png

This shows that the conjecture is applicable to a cubic equation with one root, although not required.

Now I will investigate a cubic function with two roots.

Let us consider the function: image91.png

image92.png

                        *Window values:

image93.png

The roots of the function are:

  • image94.png
  • image95.png

Proof using the remainder theorem:

image98.png

  • image99.png

image100.png

  • image101.png

image102.png

Equation of the tangent line (average of two of roots):

  • Roots -2 and -1
    Average
    image103.png
    image104.png

image105.png

image106.png
Slope at (-1.5, 0.125) will be: image107.png
Equation of the tangent where image109.png and image110.pngimage06.pngat y = 0,image111.png

image112.png

This again shows that the conjecture is applicable to a cubic equation with two roots.

With this I will conclude that the conjecture stated is valid.

...read more.

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