Položaj realnih nultočaka polinoma

master's thesis

• Author: Lucija Vinceljak
• Mentor: Rajna Rajić (mentor)
• Committee member: Rajna Rajić (predsjednik povjerenstva)
• Committee member: Mladen Vuković (član povjerenstva)
• Committee member: Dijana Ilišević (član povjerenstva)
• Committee member: Vjekoslav Kovač (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-09-24, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Tema ovog rada je određivanje položaja realnih nultočaka polinoma, tj. određivanje granica u kojima se pojavljuju nultočke danih polinoma. U prvom poglavlju uvodimo pojam polinoma te dajemo neka svojstva djeljivosti polinoma. Definiramo najveću zajedničku mjeru dvaju polinoma i primjenom Euklidova algoritma dokazujemo kako je ona jednoznačno određena. Definiramo kratnost nultočaka te pokazujemo da polinom s realnim koeficijentima može imati realne nultočke i/ili parove konjugirano kompleksnih nultočaka. Promatramo kako vodeći i konstantan član utječu na vrijednost polinoma te kako se koriste pri određivanju granica nultočaka. Dajemo i dokaz Bolzanovog teorema za polinome koji je potreban za proučavanje raspodjele nultočaka polinoma. U drugom poglavlju proučavamo tri korisne metode razdvajanja realnih nultočaka polinoma. Opisujemo metodu, koju dugujemo Sturmu, za nalaženje Sturmovog niza funkcija potrebnog za određivanje broja različitih realnih nultočaka polinoma koje se nalaze u nekom intervalu. Primjenom Fourierovog teorema razdvajamo realne nultočke polinoma korištenjem samo njegovih derivacija. Primjenom Descartesovog pravila predznaka određujemo pozitivne i negativne nultočke. U posljednjoj točki drugog poglavlja dani su dokazi Sturmovog i Fourierovog teorema. U radu je dano nekoliko primjera kojima se ilustrira primjena navedenih teorema i metoda.

Abstract (english)

The topic of this thesis is to determine the position of the real roots of polynomials, i.e. to determine the bounds in which the real roots of polynomials might occur. In the first chapter we introduce the term polynomial and give some properties of polynomial divisibility. We define the highest common factor of the two polynomials and using Euclidean algorithm we prove that it is uniquely determined. We define the multiplicity of roots and show that the polynomial with real coefficients can have real roots and / or pairs of conjugated complex roots. We observe how the leading and constant term influence the value of the polynomial and can be used to determine the boundaries of the roots. We also give the proof of Bolzano’s theorem for polynomials needed to study the distribution of polynomial roots. In the second chapter we study three useful methods of separation the real roots of a polynomial. We describe a method, due to Sturm, for finding Sturm’s sequence of functions needed to determine the number of distinct real roots of a polynomial located in an interval. Using Fourier’s theorem we separate the real roots of a polynomial using only its derivatives. By applying the Descartes rule of signs, we determine positive and negative roots. In the last section of the second chapter, the proofs of Sturm’s and Fourier’s theorems are given. The paper presents a few examples illustrating the application of the aforementioned theorems and methods.

• Keywords: nultočke polinoma; Euklidov algoritam; Bolzanov teorem za polinome; Sturmov niz funkcija; Fourierov teorem; Descartesovo pravilo predznaka
• Keywords (english): roots of polynomials; Euclidean algorithm; Bolzano’s theorem for polynomials; Sturm’s sequence of functions; Fourier’s theorem; Descartes rule of signs
• Language: croatian
• URN:NBN: urn:nbn:hr:217:289790
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-07-03 11:56:38