Rational Zeros Portfolio Assignment
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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 , where b = 0.i 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.ii 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.
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.
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). Thenv. 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. i "Rational and irrational numbers - topics in precalculus", Copyright (c) 2001-2008 Lawrence Spector, http://www.themathpage.com/aPreCalc/rational-irrational-numbers.htm. ii Stapel, Elizabeth. "The Rational Roots Test: Introduction." Purplemath. Available from http://www.purplemath.com/modules/rtnlroot.htm. Accessed 14 December 2008 iii "Rational Zeros of Polynomials", S.O.S. Math, Available from: http://www.sosmath.com/algebra/factor/fac10/fac10.html. Accessed on: 14 December 2008 iv "Rational Zeros of Polynomials", S.O.S. Math, Available from: http://www.sosmath.com/algebra/factor/fac10/fac10.html. Accessed on: 14 December 2008 v "Rational Zeros of Polynomials", S.O.S. Math, Available from: http://www.sosmath.com/algebra/factor/fac10/fac10.html. Accessed on: 14 December 2008 ?? ?? ?? ??
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