Exercices corrigés - Tronc commun Sciences BIOF

 

📌  Exercice1:

Diviser \(P(x) = x^3 + 2x^2 - 4x + 3\) par \(Q(x) = x + 1\).
✅ Division euclidienne

\(x^3 + 2x^2 - 4x + 3\) \(x + 1\)
\(- (x^3 + x^2)\) \(x^2 + x - 5\)
\(\overline{\phantom{xxx}} x^2 - 4x + 3\)  
\(- (x^2 + x)\)
\(\overline{\phantom{xxx}} -5x + 3\)
\(- (-5x - 5)\)
\(\overline{\phantom{xxx}} 8\)

\[ P(x) = (x + 1)(x^2 + x - 5) + 8 \]

Quotient : \(Q(x) = x^2 + x - 5\), Reste : \(R = 8\)

📌 Exercice2:

Diviser \(P(x) = x^4 + 2x^3 - 3x^2 - 6\) par \(Q(x) = x + 3\).
✅ Division euclidienne
 
\(x^4 + 2x^3 - 3x^2 + 0x - 6\) \(x + 3\)
\(- (x^4 + 3x^3)\) \(x^3 - x^2 + 0x - 3\)
\(\overline{\phantom{xxx}} -x^3 - 3x^2 + 0x - 6\)  
\(- (-x^3 - 3x^2)\)
\(\overline{\phantom{xxx}} 0x^2 + 0x - 6\)
\(- (0x^2 + 0x)\)
\(\overline{\phantom{xxx}} -6\)
\(- (-6)\)
\(\overline{\phantom{xxx}} 0\)
\[ P(x) = (x + 3)(x^3 - x^2 + 0x - 3) + 0 \]

Quotient : \(Q(x) = x^3 - x^2 - 3\), Reste : \(R = 0\)

📌 Exercice :

Diviser \(P(x) = x^3 - 5x^2 + x + 12\) par \(Q(x) = x - 4\).
✅ Division euclidienne

\(x^3 - 5x^2 + x + 12\) \(x - 4\)
\(- (x^3 - 4x^2)\) \(x^2 - x - 3\)
\(\overline{\phantom{xxx}} -x^2 + x + 12\)  
\(- (-x^2 + 4x)\)
\(\overline{\phantom{xxx}} -3x + 12\)
\(- (-3x + 12)\)
\(\overline{\phantom{xxx}} 0\)
\[ P(x) = (x - 4)(x^2 - x - 3) + 0 \]

Quotient : \(Q(x) = x^2 - x - 3\), Reste : \(R = 0\)

📌 Exercice 4:

Diviser \(P(x) = 2x^3 - 3x^2 + 1\) par \(Q(x) = (x-1)^2 = x^2 - 2x + 1\).
✅ Division euclidienne

\(2x^3 - 3x^2 + 0x + 1\) \(x^2 - 2x + 1\)
\(- (2x^3 - 4x^2 + 2x)\) \(2x + 1\)
\(\overline{\phantom{xxx}} x^2 - 2x + 1\)  
\(- (x^2 - 2x + 1)\)
\(\overline{\phantom{xxx}} 0\)
\[ P(x) = (x^2 - 2x + 1)(2x + 1) + 0 \]

Quotient : \(Q(x) = 2x + 1\), Reste : \(R = 0\)

📌 Exercice 5:

Diviser \(P(x) = x^3 - 3x^2 - 6x + 8\) par \(Q(x) = (x+2)(x-1) = x^2 + x - 2\).
✅ Division euclidienne

\(x^3 - 3x^2 - 6x + 8\) \(x^2 + x - 2\)
\(- (x^3 + x^2 - 2x)\) \(x - 4\)
\(\overline{\phantom{xxx}} -4x^2 - 4x + 8\)  
\(- (-4x^2 - 4x + 8)\)
\(\overline{\phantom{xxx}} 0\)

\[ P(x) = (x^2 + x - 2)(x - 4) + 0 \]

Quotient : \(Q(x) = x - 4\), Reste : \(R = 0\)

📌 Exercice 6:

Diviser \(P(x) = x^3 - x^2 + 2x - 2\) par \(Q(x) = x^2 - 1\).
✅ Division euclidienne


\(x^3 - x^2 + 2x - 2\) \(x^2 - 1\)
\(- (x^3 - x)\) \(x - 1\)
\(\overline{\phantom{xxx}} -x^2 + 3x - 2\)  
\(- (-x^2 + 1)\)
\(\overline{\phantom{xxx}} 3x - 3\)

\[ P(x) = (x^2 - 1)(x - 1) + (3x - 3) \]

Quotient : \(Q(x) = x - 1\), Reste : \(R(x) = 3x - 3\)