IB Higher Level Mathematics Project                                                          Roberto Thais

The Rational Zeros

1. a)   Graph in turn the following functions (take and ):

We find that the roots are:

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

We find that the roots are:

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

We find that the roots are:

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

We find that the roots are:

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 and find the smallest positive root of , and express it as a fraction. Also find the other roots.

We find that the roots are:

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 .

                       What are the roots of ?

We find that the roots are:

1. Show that the equation can be rewritten in the form

• We have:

 and

• And:

• Replacing x:

• Factorizing the left hand side by p:

2. 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 . 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  is in lowest terms, this is, p and q have no common factors. Therefore the ratio  as well as are in lowest terms as  is multiple of q.

As d is also an integer, we come to the conclusion that p cannot be diving : 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 ).

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

4. 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  and that p must be a factor of 6 and q a factor of 2.

Therefore the possible values for p are:

And those of q are:

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

NOT A ROOT

For

NOT A ROOT

For

                NOT A ROOT

For

NOT A ROOT

For

                NOT A ROOT

For

                NOT A ROOT

For

ROOT

For

        ROOT

For

                NOT A ROOT

For

                NOT A ROOT

Therefore the only real roots are found at

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.