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  • Level: GCSE
  • Subject: Maths
  • Word count: 2769

Rational Zeros Portfolio Assignment

Extracts from this document...

Introduction

 Math Honors 2 Portfolio Assignment

Rational Zeros

Type I – Mathematical Investigation

Nicole K. Calo

Math Honors 2

Block D

Ms. Kopp

12/ 16/ 2008

INTRODUCTION

This portfolio will be dealing briefly with the topic of rational zeros.

In mathematics, a rational number is a number which can be expressed as a ratio of two integers. Non-integer rational numbers (commonly called fractions) are usually written as the basic fraction image00.png, where b ≠ 0.[1]

The zero of a polynomial is an input value (usually an x-value) that returns a value of zero for the whole polynomial when you plug it into the polynomial. When a zero is a real (that is, not complex) number, it is also an x-intercept of the graph of the polynomial function.[2]

Hence, a rational zero is a rational number that makes the entire polynomial equal to zero when it is substituted into the polynomial. Graphically, it is the x-value of the intersection point of the function f (x) and the line y = 0.

OUTLINE

In Part 1 of the portfolio, we will be graphing closely related function to discover the polynomials’ relations to their graphs and their zeros. We will also try to spot certain patterns whilst not trying to form generalizations through them, but only for the purpose of observation. We cannot be sure yet at this stage that these apparent consistencies are of huge importance or simply nuisances, therefore we shall merely bring them to attention and not attempt to prove them.

In Part 2, we will be more closely examining the smallest positive roots of several functions.

...read more.

Middle

1

image85.png

image40.png

image86.png

image44.png

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 coefficientincreases, 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 image88.png.

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:

----image89.png

----image90.png

----image91.png

image92.png

Table 2.2

Function

Smallest positive root (x-coordinate)

image89.png

image44.png

image90.png

image82.png

image91.png

image93.png

The pink graph (image89.png) 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 image74.png)

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Conclusion

image01.pngis a rational zero of P(x).

Thenimage67.png[5].

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 image01.png) related to d (of image05.png) 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.


[1] “Rational and irrational numbers – topics in precalculus”, Copyright © 2001-2008 Lawrence Spector, http://www.themathpage.com/aPreCalc/rational-irrational-numbers.htm.

[2] Stapel, Elizabeth. "The Rational Roots Test: Introduction." Purplemath. Available from    http://www.purplemath.com/modules/rtnlroot.htm. Accessed 14 December 2008

[3] “Rational Zeros of Polynomials”, S.O.S. Math, Available from: http://www.sosmath.com/algebra/factor/fac10/fac10.html. Accessed on: 14 December 2008

[4] “Rational Zeros of Polynomials”, S.O.S. Math, Available from: http://www.sosmath.com/algebra/factor/fac10/fac10.html. Accessed on: 14 December 2008

[5] “Rational Zeros of Polynomials”, S.O.S. Math, Available from: http://www.sosmath.com/algebra/factor/fac10/fac10.html. Accessed on: 14 December 2008

...read more.

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