Polynômes à coefficients dans $	ext{K}$

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Questions and Answers

Si $P(X) = X^3 - 2X^2 + X$ et $Q(X) = 2X + 1$, quelle est la valeur de $\deg(P \circ Q)$ ?

  • 6
  • 3
  • 5 (correct)
  • 4

Si $P$ et $Q$ sont deux polynmes avec $\deg(P) = m$ et $\deg(Q) = n$, quelle est la valeur de $\deg(P + Q)$ si $m > n$ ?

  • n
  • m + n
  • m - n
  • m (correct)

Quel est le degr du polynme driv de $P(X) = 3X^4 - 2X^3 + 5X - 1$ ?

  • 5
  • 4
  • 6
  • 3 (correct)

Si $A = X^4 + 2X^3 - X^2$ et $B = X^2 + 1$, quel est le reste de la division euclidienne de $A$ par $B$ ?

<p>$X - 1$ (D)</p> Signup and view all the answers

Soit $P(X) = X^3 - 3X^2 + 3X - 1$. Quel est l'ordre de multiplicit de la racine $x = 1$ ?

<p>3 (B)</p> Signup and view all the answers

Quel est le degr du polynme $P(X)=3+5X^2-7X^5$ ?

<p>5 (B)</p> Signup and view all the answers

Quel est le coefficient dominant du polynme $P(X)=2X^3+4X-1$ ?

<p>2 (C)</p> Signup and view all the answers

Lequel des polynmes suivants est unitaire ?

<p>X^2-5X+3 (B)</p> Signup and view all the answers

Soit $P(X)=2X^2+1$ et $Q(X)=X+3$. Quel est le rsultat de $P(X)\circ Q(X)$ ?

<p>$2X^2+12X+19$ (C)</p> Signup and view all the answers

Si $P(X)$ et $Q(X)$ sont deux polynmes, quel est le degr de $P(X) + Q(X)$ ?

<p>Le maximum des degrs de $P(X)$ et $Q(X)$ (A)</p> Signup and view all the answers

Si $P(X) = 3X^2 + 2X - 1$ et $B(X) = X + 1$, quel est le rsultat de $B(X) \circ P(X)$ ?

<p>$3X^2 + 8X + 2$ (A)</p> Signup and view all the answers

Quel est le compos du polynme $P(X) = X^2 + 1$ par le polynme $Q(X) = X + 2$ ?

<p>$X^2 + 4X + 5$ (B)</p> Signup and view all the answers

Flashcards

Degré d'une somme de polynômes

Pour deux polynômes P et Q non nuls, le degré de leur somme est au plus le maximum des degrés des deux polynômes.

Degré d'un produit de polynômes

Pour deux polynômes P et Q dans K[X], le degré de leur produit est la somme des degrés des deux polynômes.

Polynôme dérivé

Le polynôme dérivé P' d'un polynôme P est obtenu en dérivant P, et son degré est un de moins que celui de P si P a un degré d'au moins 1.

Divisibilité de polynômes

Un polynôme B divise un polynôme A si il existe un polynôme Q tel que A = BQ avec B non nul.

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Racine d'ordre de multiplicité

a est une racine d'ordre de multiplicité m pour un polynôme P si P(a) et ses premières m-1 dérivées s'annulent, mais pas la m-ème.

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Polynôme

Suite finie d'éléments de $ extbf{K}$, notée $ orall n ext{, } P(X) = extstyle euilleZ(a_0, ext{...},a_N)$.

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Indéterminée

Variable utilisée dans un polynôme, représentée par $X$.

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Anneau de Polynômes

L'ensemble $ extbf{K}[X]$ où l'addition et la multiplication sont définies.

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Addition de Polynômes

Pour $P(X)$ et $Q(X)$, $(P+Q)(X)= extstyle euilleZ(a_n + b_n)X^n$.

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Multiplication de Polynômes

Pour $P(X)$ et $Q(X)$, $c_n= extstyle euilleZ(a_k b_{n-k})$.

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Degré du Polynôme

Le plus grand indice $n$ tel que $a_n eq 0$, noté $ ext{deg}(P)$.

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Coefficient Dominant

Le coefficient correspondant au degré maximal $a_n$ dans un polynôme.

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Polynôme Unitaire

Polynôme dont le coefficient dominant est égal à 1.

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Study Notes

Polynômes à coefficients dans $\mathbb{K}$

  • Polynômes: Finite sequences $(a_0, \dots, a_N)$ of elements in $\mathbb{K}$. Notably represented as $\sum_{n\ge0} a_nX^n$, where $X$ is the indeterminate.
  • Set of Polynomials: $\mathbb{K}[X]$ is the set of all such polynomials with coefficients from $\mathbb{K}$.
  • Operations on Polynomials:
    • Addition: $(P+Q)(X)=\sum_{n\ge0}(a_n+b_n)X^n$.
    • Multiplication: $(PQ)(X)=\sum_{n\ge0}c_nX^n$ where $c_n=\sum_{k=0}^n a_kb_{n-k}$.
  • $\mathbb{K}[X]$ forms a ring under these operations.
  • Composition of Polynomials: If $A, B \in \mathbb{K}[X]$, with $B = \sum_{n=0}^N b_nX^n$, then $B\circ A = \sum_{n=0}^N b_n A^n$.

Degree and Dominant Coefficient

  • Degree: For a non-zero polynomial $P=\sum_{n\ge0} a_nX^n$, the largest index $n$ for which $a_n \neq 0$ is the degree, denoted $\deg(P)$. If $P$ is the zero polynomial, its degree is $-\infty$.
  • Dominant Coefficient: The coefficient $a_n$ corresponding to the degree is the dominant coefficient.
  • Univariate Polynomials: A polynomial with dominant coefficient 1 is called a monic polynomial (or sometimes unitaire in some texts).

Properties of Degrees

  • Degree of Sum: $\deg(P+Q) \le \max(\deg(P), \deg(Q))$.
  • Degree of Product: $\deg(PQ) = \deg(P) + \deg(Q)$.
  • Degree of Composition: $\deg(P\circ Q) = \deg(P) \times \deg(Q)$.

Derivative of a Polynomial

  • Derivative: The derivative of $P = \sum_{n\ge0}a_nX^n$ is $P' = \sum_{n\ge1}na_nX^{n-1}$.
  • Degree of Derivative: If $\deg(P) \ge 1$, then $\deg(P') = \deg(P) - 1$.

Divisibility and Associated Polynomials

  • Divisibility: $B$ divides $A$ if there exists $Q\in\mathbb{K}[X]$ such that $A = BQ$.
  • Associated Polynomials: Two non-zero polynomials $A$ and $B$ are associated if $A$ divides $B$ and $B$ divides $A$. This is equivalent to the existence of $\lambda \in \mathbb{K}^*$ such that $A = \lambda B$.

Polynomial Long Division

  • Remainder Theorem: For $A, B \in \mathbb{K}[X]$ with $B$ non-zero, there exists a unique pair $(Q, R) \in \mathbb{K}[X]$ such that $A = BQ + R$ and $\deg(R) < \deg(B)$.

Polynomial Functions

  • Polynomial Function: A polynomial $P = \sum_{n=0}^N a_n X^n$ defines a polynomial function $\tilde{P}: \mathbb{K} \to \mathbb{K}$ by $\tilde{P}(z) = \sum_{n=0}^N a_n z^n$. Typically, the polynomial and the polynomial function are identified.

Roots of Polynomials and Multiplicity

  • Root with Multiplicity: A value $a \in \mathbb{K}$ is a root of a polynomial $P$ with multiplicity $m$ if $P(a) = P'(a) = \dots = P^{(m-1)}(a) = 0$ and $P^{(m)}(a) \neq 0$.
  • Equivalence Statements (about Roots): Several Equivalent statements are mentioned in the text relate to properties of roots and their multiplicity (but not fully detailed or presented here). The equivalence involves statements about derivates of the polynomial evaluated at the root and its existence.

Relation between Coefficients and Roots (if applicable)

  • If a polynomial is factored (has all its roots), the coefficients can be expressed in terms of these roots. The text mentions a formula based on the roots. (The full details of the formula are not summarized here as the provided notes are incomplete.)

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