Completion requirements
7. 📚 Exercices
✅
Exercice 1:
a, b et c des nombres réels non nuls tels que : \( ab + bc + ca = 0 \).
Calculer
\[ \frac{b+c}{a} + \frac{c+a}{b} + \frac{a+b}{c} \]
✅ Exercice 2:
Énoncé : \( Z = a^2 + 2ab + b^2 + a + b + 1 \)
Écrire Z sous la forme d'une somme de carrés de trois entiers naturels.
\[ Z = (a^2 + 2ab + b^2) + (a + b) + 1 \] \[ Z = (a + b)^2 + (a + b) + 1 \] \[ Z = (a + b)^2 + (a + b) + \frac{1}{4} + \frac{3}{4} \] \[ Z = \left(a + b + \frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2 \]
Si on pose \( a + b = k \), alors \( Z = k^2 + k + 1 \).
On peut aussi écrire : \( Z = (a+b)^2 + (a+b) + 1 \).
✅ Exercice 3:
Énoncé : Simplifier les expressions.
\( X = \frac{(x^2y)^{-3}xz^2}{xy^{-3}} \)
\[ X = \frac{x^{-6}y^{-3} \cdot x \cdot z^2}{x \cdot y^{-3}} = \frac{x^{-5}y^{-3}z^2}{x y^{-3}} = x^{-6}z^2 = \frac{z^2}{x^6} \]
\( Y = \frac{(-x)^2x^2y}{2y^{-1}} \)
\[ Y = \frac{x^2 \cdot x^2 \cdot y}{2 \cdot y^{-1}} = \frac{x^4 y \cdot y}{2} = \frac{x^4 y^2}{2} \]
\( Z = \frac{(x^2y)^{-3}z^5x^4}{(xy^2)^2y^{-1}} \)
\[ Z = \frac{x^{-6}y^{-3}z^5x^4}{x^2y^4y^{-1}} = \frac{x^{-2}y^{-3}z^5}{x^2y^3} = x^{-4}y^{-6}z^5 = \frac{z^5}{x^4 y^6} \]
\( T = \frac{(-x)^5yz^{-2}}{y^3(-z)^{-3}x^2} \)
\[ T = \frac{-x^5yz^{-2}}{y^3(-z)^{-3}x^2} = \frac{-x^5yz^{-2} \cdot (-z)^3}{y^3x^2} = \frac{-x^5yz^{-2} \cdot (-z^3)}{y^3x^2} \] \[ T = \frac{x^5 y z^{-2} z^3}{y^3 x^2} = \frac{x^3 y z}{y^3} = \frac{x^3 z}{y^2} \]
✅ Exercice 4:
Énoncé : Développer les expressions.
\( A = (x^2 + x + 1)(2x - 5) \)
\[ A = 2x^3 - 5x^2 + 2x^2 - 5x + 2x - 5 = 2x^3 - 3x^2 - 3x - 5 \]
\( B = (7x - 3y)^2 \)
\[ B = 49x^2 - 42xy + 9y^2 \]
\( C = (x - 1)(x + 2)(x - 2) \)
\[ C = (x - 1)(x^2 - 4) = x^3 - 4x - x^2 + 4 = x^3 - x^2 - 4x + 4 \]
\( D = (5a + 3)^3 \)
\[ D = 125a^3 + 3 \cdot 25a^2 \cdot 3 + 3 \cdot 5a \cdot 9 + 27 = 125a^3 + 225a^2 + 135a + 27 \]
\( E = (2x - 5y)^3 \)
\[ E = 8x^3 - 3 \cdot 4x^2 \cdot 5y + 3 \cdot 2x \cdot 25y^2 - 125y^3 = 8x^3 - 60x^2y + 150xy^2 - 125y^3 \]
\( F = (3 + x + a)^2 \)
\[ F = (3 + x + a)^2 = 9 + x^2 + a^2 + 6x + 6a + 2ax \]
\( G = (2\sqrt{3}p + \sqrt{5q})^3 \)
\[ G = 8 \cdot 3\sqrt{3}p^3 + 3 \cdot (2\sqrt{3}p)^2 \cdot \sqrt{5q} + 3 \cdot (2\sqrt{3}p)(\sqrt{5q})^2 + (\sqrt{5q})^3 \] \[ G = 24\sqrt{3}p^3 + 3 \cdot 12p^2 \cdot \sqrt{5q} + 6\sqrt{3}p \cdot 5q + 5q\sqrt{5q} \] \[ G = 24\sqrt{3}p^3 + 36p^2\sqrt{5q} + 30\sqrt{3}pq + 5q\sqrt{5q} \]
\( H = \left(t - \frac{1}{t}\right)^3 \)
\[ H = t^3 - 3t^2 \cdot \frac{1}{t} + 3t \cdot \frac{1}{t^2} - \frac{1}{t^3} = t^3 - 3t + \frac{3}{t} - \frac{1}{t^3} \]
✅ Exercice 5:
Énoncé : Factoriser les expressions.
\( A = a^2 - 2ab + b^2 - c^2 \)
\[ A = (a - b)^2 - c^2 = (a - b - c)(a - b + c) \]
\( B = (3x^2 - 3) + (x^2 - 2x + 1) \)
\[ B = 3(x^2 - 1) + (x - 1)^2 = 3(x - 1)(x + 1) + (x - 1)^2 = (x - 1)[3(x + 1) + (x - 1)] \] \[ B = (x - 1)(3x + 3 + x - 1) = (x - 1)(4x + 2) = 2(x - 1)(2x + 1) \]
\( C = x^{16} - 16 \)
\[ C = (x^8 - 4)(x^8 + 4) = (x^4 - 2)(x^4 + 2)(x^8 + 4) \]
\( D = 2\sqrt{2}x^3 + 27 \)
\[ D = (\sqrt{2}x)^3 + 3^3 = (\sqrt{2}x + 3)(2x^2 - 3\sqrt{2}x + 9) \]
\( E = x^{12} - 2x^6 + 1 \)
\[ E = (x^6 - 1)^2 = (x^3 - 1)^2(x^3 + 1)^2 = (x - 1)^2(x^2 + x + 1)^2(x + 1)^2(x^2 - x + 1)^2 \]
\( F = x^5 + x^3 - x^2 - 1 \)
\[ F = x^3(x^2 + 1) - (x^2 + 1) = (x^2 + 1)(x^3 - 1) = (x^2 + 1)(x - 1)(x^2 + x + 1) \]
\( I = 125a^3 + 64 \)
\[ I = (5a)^3 + 4^3 = (5a + 4)(25a^2 - 20a + 16) \]
\( J = 2(x + 7)(x + 5) - (x^2 - 25) \)
\[ J = 2(x + 7)(x + 5) - (x - 5)(x + 5) = (x + 5)[2(x + 7) - (x - 5)] \] \[ J = (x + 5)(2x + 14 - x + 5) = (x + 5)(x + 19) \]
\( K = 6x(x - 2) - 3x^2 + 12 \)
\[ K = 6x(x - 2) - 3(x^2 - 4) = 6x(x - 2) - 3(x - 2)(x + 2) \] \[ K = (x - 2)(6x - 3x - 6) = (x - 2)(3x - 6) = 3(x - 2)(x - 2) = 3(x - 2)^2 \]
\( L = x^3 - 8 - 5(x - 2) \)
\[ L = (x - 2)(x^2 + 2x + 4) - 5(x - 2) = (x - 2)(x^2 + 2x + 4 - 5) \] \[ L = (x - 2)(x^2 + 2x - 1) \]
\( M = 64 - (5x - 7)^3 \)
\[ M = 4^3 - (5x - 7)^3 = (4 - (5x - 7))(16 + 4(5x - 7) + (5x - 7)^2) \] \[ M = (-5x + 11)(16 + 20x - 28 + 25x^2 - 70x + 49) \] \[ M = (-5x + 11)(25x^2 - 50x + 37) \]
\( N = (x - 7y)^3 + 27y^3 \)
\[ N = (x - 7y)^3 + (3y)^3 = (x - 7y + 3y)((x - 7y)^2 - 3y(x - 7y) + 9y^2) \] \[ N = (x - 4y)(x^2 - 14xy + 49y^2 - 3xy + 21y^2 + 9y^2) \] \[ N = (x - 4y)(x^2 - 17xy + 79y^2) \]
\( U = x^3 - 8 + 4(x^2 - 4) - 3x + 6 \)
\[ U = (x - 2)(x^2 + 2x + 4) + 4(x - 2)(x + 2) - 3(x - 2) \] \[ U = (x - 2)[x^2 + 2x + 4 + 4x + 8 - 3] = (x - 2)(x^2 + 6x + 9) \] \[ U = (x - 2)(x + 3)^2 \]
\( V = x^3 + 1 + 2(x^2 - 1) - (x + 1) \)
\[ V = (x + 1)(x^2 - x + 1) + 2(x - 1)(x + 1) - (x + 1) \] \[ V = (x + 1)[x^2 - x + 1 + 2x - 2 - 1] = (x + 1)(x^2 + x - 2) \] \[ V = (x + 1)(x + 2)(x - 1) \]
✅ Exercice 6:
Énoncé : \( A = a + \frac{1}{a} \). Calculer en fonction de \( A \).
\( a^2 + \frac{1}{a^2} \)
\[ \left(a + \frac{1}{a}\right)^2 = a^2 + 2 + \frac{1}{a^2} \Rightarrow a^2 + \frac{1}{a^2} = A^2 - 2 \]
\( a^3 + \frac{1}{a^3} \)
\[ a^3 + \frac{1}{a^3} = \left(a + \frac{1}{a}\right)^3 - 3\left(a + \frac{1}{a}\right) = A^3 - 3A \]
\( a^4 + \frac{1}{a^4} \)
\[ a^4 + \frac{1}{a^4} = \left(a^2 + \frac{1}{a^2}\right)^2 - 2 = (A^2 - 2)^2 - 2 = A^4 - 4A^2 + 2 \]
✅ Exercice 7
Énoncé : \( 2^a \times 3^b \times 5^c = 648000 \)
\[ 648000 = 648 \times 1000 = (8 \times 81) \times (10^3) \] \[ 648 = 2^3 \times 3^4 \quad \text{et} \quad 1000 = 2^3 \times 5^3 \] \[ 648000 = 2^3 \times 3^4 \times 2^3 \times 5^3 = 2^6 \times 3^4 \times 5^3 \] \[ \text{Donc } a = 6,\; b = 4,\; c = 3 \]