Effectuer des divisions euclidiennes

Durée : 15 mn

Note maximale : 20

Question

Trouver le quotient et le reste de la division euclidienne du polynôme \(A\) par le polynôme \(B\) dans les cas suivants :

  1. \(A(X) = X^{5} - 3 X^{4} + 4 X^{2} - 5X + 6 \qquad B(X) = X^{3} - X + 1\)

  2. \(A(X) = 2 X^{4} + 5 X^{3} - 4 X^{2} + 7X - 3 \qquad B(X) = X^{2} - 3X +1\)

  3. \(A(X) = 4 X^{3} + X^{2} - 1 \qquad B(X) = X^{2} + iX - i\)

Solution

L'identité de la division euclidienne s'écrit \(A = B.Q + R\) où \(R = 0\) ou \(\textrm{deg}(R) < \textrm{deg}(B)\).

  1. (7 points)

    \(\begin{array}{l|l}\begin{array}{rrrrrr}\color{blue} X^{5} & \color{blue} -3 X^{4} & & \color{blue} +4 X^{2} & \color{blue} -5X & \color{blue} +6 \\-X^{5} & & +X^{3} & -X^{2} & & \\\hline& \color{blue} -3X^{4} & \color{blue} +X^{3} & \color{blue} +3X^{2} & \color{blue} -5X & \color{blue} +6 \\& +3X^{4} & & -3X^{2} & +3X & \\\hline& & \color{blue} +X^{3} & & \color{blue}-2X & \color{blue}+6 \\& & -X^{3} & & +X & -1\\\hline& & & & \color{red}-X & \color{red} +5\end{array}&\begin{array}{rrr}\color{blue} X^{3} & \color{blue} -X & \color{blue} +1 \\\hline\color{red} X^{2} & \color{red} -3X & \color{red} +1 \\ \\\\\\\\\\\end{array}\end{array}\)

    Donc, \(\color{red} Q(X) = X^{2} - 3X +1, \quad R(X) = -X + 5\).

  2. (6 points)

    \(\begin{array}{l|l}\begin{array}{rrrrr}\color{blue} 2X^{4} & \color{blue} +5 X^{3} & \color{blue} -4X^2 & \color{blue} +7X & \color{blue} -3 \\-2X^{4} & +6X^{3} & -2X^{2} & & \\\hline& \color{blue} 11X^{3} & \color{blue} -6X^{2} & \color{blue} +7X & \color{blue} -3 \\& -11X^{3} & +33X^{2} & -11X & \\\hline& & \color{blue} 27X^{2} & \color{blue}-4X & \color{blue}-3 \\& & -27X^{2} & +81X & -27\\\hline& & & \color{red}77X & \color{red} -30\end{array}&\begin{array}{rrrr}\color{blue} -X & \color{blue} +1 \\\hline\color{red} X^{2} & \color{red} -3X & \color{red} +1 \\ \\\\\\\\\\\end{array}\end{array}\)

    Donc \(\color{red} Q(X)=2X^2+11X+27, R(X)=77X-30\)

  3. (7 points)

    \(\begin{array}{l|l}\begin{array}{rrrr}\color{blue} 4X^{3} & \color{blue} + X^{2} & & \color{blue} -1 \\-4X^{3} & -4iX^{2} & +4iX & \\\hline& \color{blue} (1-4i)X^{2} & \color{blue} +4iX & \color{blue} -1\\& -(1-4i)X^{2} & -i(1-4i)X & +i(1-4i)\\\hline& & \color{red} (3i-4)X & \color{red}+(i+3)\end{array}&\begin{array}{ll}\color{blue} X^2 & \color{blue} +iX-i \\\hline\color{red} 4X+(1-4i \\ \\\\\\\end{array}\end{array}\)

    Donc \(\color{red} Q(X)=4X+(-4i+1),~R(X)=(3i-4)X+(i+3)\)