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The Rational Zeros

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IB Higher Level Mathematics Project                                                          Roberto Thais


The Rational Zeros

  1. a)   Graph in turn the following functions (take image05.pngand image06.png):
  1. image45.png


We find that the roots are: image58.png

Therefore, the smallest positive root, expressed as a fraction is



  1. image81.png

We find that the roots are: image85.png

Therefore, the smallest positive root, expressed as a fraction is


  1. image15.png

We find that the roots are: image22.png

Therefore, the smallest positive root, expressed as a fraction is


  1. image40.pngimage39.png

We find that the roots are: image41.png

Therefore, the smallest positive root, expressed as a fraction is


From these functions we can notice that the denominator of the x value at 0 of the smallest positive root is always the same as the first coefficient of the polynomial. While the second and fourth terms of the polynomials are kept constant, so does the numerator and the other two roots of the function keep constant.

b)   Now graph image43.pngand find the smallest positive root of image44.png, and express it as a fraction. Also find the other roots.




We find that the roots are: image46.png

Therefore, the smallest positive root, expressed as a fraction is


        The same principle as that seen before is observed in this curve which has fractional coefficients: the denominator is equal to the first coefficient and the numerator is equal to 1 as the second and fourth terms are kept constant. Simplifying the expression we get the value observed.

c)      Finally graph, with a suitable window image48.png.

                       What are the roots of image44.png?




We find that the roots are: image46.png

...read more.


. Suppose, in addition, that p and q are positive integers with no common factors.
  1. Show that the equation image67.pngcan be rewritten in the form



  • We have:

image70.png and image66.png

  • And:


  • Replacing x:



  • Factorizing the left hand side by p:


  1. Hence show that p must be a factor of d.


  • We begin with:


  • Dividing the expression by p we obtain:


  • Multiplying by –1:


We know from definition that a, b and c are non-zero integers, as well q and p. It therefore follows that the left hand side expression of the equation is also an integer for all allowed values when image76.png. As a result of this, we deduce that the right hand side term is also an integer.

However, we know from definition that the ratio image12.png is in lowest terms, this is, p and q have no common factors. Therefore the ratio image77.png as well as image78.pngare in lowest terms as image79.png is multiple of q.

As d is also an integer, we come to the conclusion that p cannot be diving image79.png: whereas for some values of p and q the division would result in the needed integer (e.g. equation 1(a)(ii) would give us the whole number 8), for some others the value would be a decimal (e.g. for equation 1(c) theresult is image80.png).

If p weren’t a factor of d this would contradict the result of the left hand side expression. Hence, it follows that p must be dividing d

...read more.


  1. List all possible candidates for rational zeros of

Determine if P has any rational zeros, if so, find them and all other remaining zeros.


We know that image23.png and that p must be a factor of 6 and q a factor of 2.

Therefore the possible values for p are: image24.png

And those of q are: image25.png

So the possible candidates for rational zeros are (in ascending order):


Thus we need to test for each individual value in the function and determine if it gives us a zero.

For image27.png


For image28.png


For image29.png

                NOT A ROOT

For image30.png


For image31.png

                NOT A ROOT

For image32.png

                NOT A ROOT

For image34.png


For image35.png


For image36.png

                NOT A ROOT

For image37.png

                NOT A ROOT

Therefore the only real roots are found at image38.png

Summing up, the Rational-zero theorem which we have been able to prove lets us formulate possibilities for rational roots of polynomials of any given degree, whose number would only depend on the amount of factors that the first and last term of the polynomial have. We have found out that the denominator of the root expressed as a fraction must be a factor of the first term coefficient while the numerator a factor of the last term coefficient.

However, this method is limited only to integer coefficients because, as we have seen in section 1(b), the values specifically related to a given term will vary from numerator to denominator or vice-versa, depending on the individual context and therefore cannot be generalized in this manner. This method additionally cannot be used to find irrational roots of polynomials.

...read more.

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