Polinomi

master's thesis

• Title (english): Polynomials
• Author: Zrinka Peko-Lončar
• Mentor: Ana Prlić (mentor)
• Committee member: Ana Prlić (predsjednik povjerenstva)
• Committee member: Goran Muić (član povjerenstva)
• Committee member: Maja Resman (član povjerenstva)
• Committee member: Marko Tadić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2023-09-28, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom diplomskom radu opisujemo povijesni razvoj polinoma i osnovnog teorema algebre, od Al-Khwarizmijeve Algebre do Galoisove teorije. Zanimljivosti iz povijesti nastavljamo u sljedećem poglavlju s tri klasična problema starogrčke matematike. Uz uvođenje konstruktibilnih brojeva dokazujemo nerješivost tri klasična problema pomoću polinoma. U nastavku rada detaljno proučavamo polinome i njihova svojstva. Posebno se osvrćemo na nultočke i faktorizaciju polinoma, a zatim proučavamo kako pomoću Hornerovog algoritma efikasno dijelimo polinom polinomom prvog stupnja i pronalazimo Taylorov razvoj polinoma. Proučavamo simetrične polinome i iskazujemo osnovni teorem o simetričnim polinomima kao ključni za dokaz osnovnog teorema algebre. Uvodimo pojam polja razlaganja polinoma te prezentiramo jedan algebarski dokaz osnovnog teorema algebre. Na kraju rada obrađujemo razne zadatke vezane za polinome koji su se pojavili na domaćim, stranim i međunarodnim natjecanjima iz matematike.

Abstract (english)

In this master’s thesis, we describe the historical development of polynomials and the Fundamental Theorem of Algebra, from Al-Khwarizmi’s Algebra to Galois theory. We continue the next chapter with the three classical Greek problems of mathematics, another popular historical problem. By introducing constructible numbers, we prove the unsolvability of the three classical problems using polynomials. In the next chapter, we study polynomials and their properties in detail. In particular, we look at zeros and factorization of polynomials following with Horner’s algorithm and examples how to efficiently divide a polynomial by the first degree polynomial. We study symmetric polynomials and state the Fundamental Theorem of Symmetric Polynomials as a key to proving the Fundamental Theorem of Algebra. Upon introducing a splitting field of a polynomial, we present an algebraic proof of the Fundamental Theorem of Algebra. At the end of the thesis, we show various problems related to polynomials that appeared at national, foreign and international competitions in mathematics.

• Keywords: Al-Khwarizmijeva Algebra; Galoisova teorija; nultočke i faktorizacija polinoma; Hornerov algoritam; Taylorov razvoj polinoma
• Keywords (english): Al-Khwarizmi’s Algebra; Galois theory; Horner’s algorithm
• Language: croatian
• URN:NBN: urn:nbn:hr:217:517152
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: sveučilišni magistar edukacije matematike (sveučilišni magistar edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2024-02-05 12:18:27