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Zeros of cubic functions

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Andrei B. Alexandra


Zeros of Cubic Functions


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:


* Window values:


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:


  • image169.png



  • image21.png



  • image44.png



...read more.


  • image155.png

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


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:                                                        


*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:


  • image164.png



  • image167.png



  • image171.png



Verification on the calculator :


  • Roots -1 and image23.png

Average image175.png



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

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


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

  • Roots -1 and 2




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

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


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

  • Roots  2 and image23.png




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

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


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

...read more.


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


*Window values:


The root of the equation is:

  • image78.png

Proof using the remainder theorem:

  • image79.png



Equation of the tangent line

  • Roots -1.1456 and -1.1456

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

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


                        *Window values:


The roots of the function are:

  • image94.png
  • image95.png

Proof using the remainder theorem:


  • image99.png


  • image101.png


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

  • Roots -2 and -1


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


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.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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