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Polynomials

Learning tasks
  
Lesson
Text book
Exercises Correction
Convention
Throughout this exercise sheet, unless otherwise stated, \(\mathbb{K}\) denotes either \(\mathbb{R}\) or \(\mathbb{C}\).
A) Ring \(\mathbb{K}[X]\)\(\virgule\) degree\(\virgule\) composition
    1) Computing degrees and operating on polynomialsEx 1 Ex 2 Ex 3 Ex 4 Ex 5 Ex 6
B) Divisibility and Euclidean division
    2) Performing Euclidean division and using itEx 7 Ex 8 Ex 9 Ex 10 Ex 11 Ex 12
C) Polynomial functions\(\virgule\) roots\(\virgule\) Viète relations
    3) Finding roots\(\virgule\) factoring\(\virgule\) computing multiplicityEx 13 Ex 14 Ex 15 Ex 16 Ex 17
    4) Using Viète's relationsEx 18 Ex 19 Ex 20 Ex 21
D) Formal derivative and Taylor formula
    5) Computing derivatives and Leibniz formulaEx 22 Ex 23 Ex 24
    6) Using Taylor formula and multiplicity testEx 25 Ex 26 Ex 27 Ex 28 Ex 29 Ex 30
E) Factorization in \(\mathbb{C}\lbrack X\rbrack\) and \(\mathbb{R}\lbrack X\rbrack\)
    7) Factoring \(X^n - 1\) and \(X^n + 1\)Ex 31 Ex 32 Ex 33 Ex 34
    8) Using conjugate roots and identifying irreduciblesEx 35 Ex 36 Ex 37 Ex 38
F) Lagrange interpolation
    9) Computing the Lagrange interpolation polynomialEx 39 Ex 40 Ex 41
    10) Using Lagrange in proofsEx 42 Ex 43 Ex 44 Ex 45 Ex 46
G) Going further
    11) Synthesis problems combining several sectionsEx 47 Ex 48 Ex 49