Rational Zeros Portfolio Assignment

Not only does the y-intercept stay the same, but so do the negative root and the larger positive root. If you notice from the Table 1.1 and Figure 1.1, the negative root is consistently and the larger positive root is consistently . Since the only other term, besides the constant term 4, that stays the same is the b coefficient , we can deduce that it is this b coefficient that affects the negative root and the larger positive roots of the function of the form . We show and test this by graphing and another function that is different only in the b coefficient (Figure 1.2) on the same grid. Notice the change in the negative root and the larger positive root.

Also notice how an identical half of each graph seems to be flipped across the y-axis and then flipped across the line .

It seems too that the denominator of the smallest positive root is the a coefficient. For example, the smallest positive root of is . The denominator of is 3. Coincidentally, the leading coefficient a of the function is 3. From this we can conclude, that the q value of the rational root must be somehow directly related to the value of a the leading coefficient.

Figure 1.2

Legend:

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Next we will graph (Figure 1.3) and find the zeros of two functions (Table 1.3), one which is a vertical stretch of the other.  The function  is the functionvertically stretched by a factor of 3. The function can also be written as . Thus we can more clearly see the vertical stretch factor.

Figure 1.3

Legend:

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Table 1.3

From Figure 1.3 and Table 1.3 we can see that vertically stretching a polynomial function does not alter its x-intercept and therefore does not affect the roots of the function. We can also say that for a function of the form , the variable k, denoting a vertical stretch or compression, does not affect the zeros of a polynomial function.  Thus for several functions of the form where a, b, c and d remain the same, k can have different values and still those functions will have the same zeros.

PART  2

Now we are going to look at two sets of functions, their graphs, their smallest positive roots, and the relationships between these things, if we can find any. In each set, the functions are simple transformations of each other.

In Figure 2.1, we have graphs of functions which vary in their leading term a and third (or second to the last term) term c, if we describe their form as.  Table 2.1 shows how these changes affect their smallest positive roots.

Figure 2.1

Legend:

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Table 2.1

It seems that as the value of the a coefficient increases, the arch of the left side of the graph heightens. In other words, when the value of the a coefficient increases, the y-value of the maximum point of the left side of the graph increases as well. Similarly, it seems that as the value of the c coefficient decreases, the arch of the right side of the graph deepens. In other words, it seems that as the value of the c coefficient decreases, the y-value of the minimum point of the right side of the graph decreases as well.

Also notice how an identical half of each graph seems to be flipped across the y-axis and then flipped across the line .

In Figure 2.2, we have graphs of functions which vary in almost all their terms, except for the last two functions which have the same a coefficient and c coefficient. Table 2.2, shows how this affects their smallest positive roots.

Figure 2.2

Legend:

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Table 2.2

The pink graph () in Figure 2.2 which looks like three random lines is actually shaped like the two other functions except that it crosses the y-axis at  positive 180 and the minimum value on the right side reaches a very low number, compared to the others.

If you notice too that the graphs in Figure 2.2 and Figure 1.3 are unlike the graphs of Figure 1.1, 1.2, and 2.1, which seem to have “double flips”.  

More importantly, I would like to bring to attention the interesting fact that all of the functions (of the form ) we have so far displayed shows that the smallest positive root equals the negative ratio of its constant term to the c coefficient () . Also, the smallest positive root equals the negative ratio of the b coefficient to the a coefficient ().

Let us use some functions from Table 2.2 as proof of this conjecture:

: ,,

        

        To test: the smallest positive root  equals  and .

        

: ,,

To test: the smallest positive root  equals  and.

        

PART 3

Through out Part 1 and Part 2, we have been noticing that there is a pattern relating the rational root of the equation to the coefficients of a and d. In this part of the portfolio we shall set out to discover this relationship and prove it.

Suppose a, b, c and d are integers with a ≠ 0, and that f(x) is the polynomial defined by . Suppose, in addition, that p and q are positive integers with no common factors.

First we rewrite the equation  into the form  so that we can see the relationship of numerator p to the constant d, by placing the two variables on opposite sides of the equal sign.

We start by substituting  into the equation .

We subtract d from both sides…

…and multiply the whole equation by .

Then we factor out p from the right side of the equation

Now the left side is divisible by p; consequently,  is divisible by p. Since p does not divide q, it does not divide , so p divides d.

Next, we rewrite the equation  into the form  so that we can see the relationship of denominator q to the leading coefficient a, by placing the two variables on opposite sides of the equal sign.

We start over and substitute into the equation .

Then we isolate on one side of the equation by subtracting the other terms from the entire equation.  

We multiply the whole equation by to eliminate the fractions.

        

Then we factor out   from the left side of the equation.

Now the right side is divisible by q; consequently,  is divisible by q. Since q does not divide p, it does not divide , so q divides a.

We can form a generalization about the rational zeros for a general polynomial function  now that we have shown that p must be a factor of d and that q must be a factor of d.

For the rational zero,  (supposing that p and q are integers), of a general polynomial function  (supposing integral coefficients and), p must be a factor of the constant term () and q must be a factor of the leading coefficient ().

Let’s take a random polynomial function for proof: .

Since this polynomial function is random, we don’t’ know yet if it possesses any rational zeros. Therefore, by using the conjecture above we can find out whether this function has any rational zeros and identify them.

We take the leading coefficient and the constant term and find its factors.

 

We could test each fraction by substituting it into the equation  but that would take rather long. Instead we will graph the function using a graphics display calculator and calculate the roots. Then we will select that fractions that closest resemble the decimal roots.

The calculator gave  as the only zero. It seems then that the function  has no rational zeros as none of the possible come close to . Since the conjecture has shown that a polynomial function has failed to have any rational zeros, we have proven our conjecture by counter-example.

PART 4

We shall now apply the rational zeros theorem to an equation that has given values for the coefficients, using the theorem to help us find out the rational zeros of the equation.  

The given equation:

The basic equation of the rational zero is   when .

First, we identify the leading coefficient and the constant term, which are all that is needed from the function when looking for the rational zeros.

:

• leading coefficient is 6
• constant term is 2.

Using the rational zeros theorem, we will find all the factors of the leading coefficient, 6, which will become the possible q values. And then we will find all the factors of the constant term, 2, which will become the possible p values.

The factors of 6 are ±1, ±2, ±3, and ±6.

The factors of 2 are ±1 and ±2.

Therefore

        

We could test each possible rational number formable from this equation, namely , 1, , 2, , , , , , ,  and, by inputting each one into the function to see if it will make the polynomial equal zero. However, this will prove to be a long and tedious task, therefore it is better to graph the function (Figure 4.1) and then find the x-values of its x-intercepts.

Figure 4.1

Legend:

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Using a graphing calculator we calculate the x-value of the x-intercepts of this function. In decimal form, these are  and . If we convert these decimals to fraction form, we get  and . So of all the possible candidates for the rational zeros of , we are down to  and . Next is that we confirm that they are indeed correct them by substitution.

For :

First we input the x-value into the equation.

Then we multiply the entire equation by .

We simplify.

Then we perform the operations.

Hence  as a rational zero of  is correct.

For :

First we input the x-value into the equation.

Then we multiply the entire equation by .

We simplify.

Then we perform the operations.

Hence  as a rational zero of  is correct.

It is important to perform the algebraic check through substitution to determine whether the given x-value is indeed a rational zero of a given function because it may not be clear simply by looking a graph that a value is rational. It could be that the x-value is irrational but that it comes very close to a rational number and/or that the calculator automatically rounds to a specific number of decimals, thereby creating the impression that it is a terminating decimal. So it is best to perform an algebraic check to eliminate the inaccuracies of graphic representations.

Since  has only two x-intercepts and both of them have been found and are confirmed to be rational, the function has no other remaining zeros.

CONCLUSION

The conjecture we had made earlier in Part 3 is commonly known as the Rational Roots Theorem. The Rational Roots Theorem can be used to identify possible roots of the polynomial equations with integer coefficients, as we had done in Part 4. This theorem is very useful; in a variety of real-world fields, such as the field of finance where polynomial functions are used often to find an effective yield, or interest rate, for an investment. The Rational Roots Theorem is more formally stated below.

RATIONAL ROOTS THEOREM

Suppose    

is a polynomial with integer coefficients, and is a rational zero of P(x).

Then.

This portfolio has only touched on the topic of rational zeros since it really does go a long way further. Let me enumerate again what we have accomplished in this portfolio. We have managed to discover the logic and process of reaching the Rational Roots Theorem through investigation. We first studied the graphs and zeros of several functions. Then we studied how certain transformations affected the graphs and smallest positive roots of a new set of functions. Next, we tried to discover the relationship between a general polynomial function and its rational roots through manipulating value-less formulas and rearranging them in such a way that we could see how p (of ) related to d (of ) and how q related to a. After that we formed a conjecture based on our findings and set out to prove it. Lastly we used our conjecture, which we now know is called the Rational Roots Theorem, to find the rational zeros of a given polynomial function.

And that is all we did.

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