Exercices de Mathématiques
Niveau 3ème
Exercice 1 : Développement simple
1. Développer et réduire :
a) \( 3(x + 4) \)
b) \( -2(5 – x) \)
c) \( x(2x + 3) \)
Correction :
a) \( 3(x + 4) = 3 \times x + 3 \times 4 = \boxed{3x + 12} \)
b) \( -2(5 – x) = -2 \times 5 + (-2) \times (-x) = \boxed{-10 + 2x} \)
c) \( x(2x + 3) = x \times 2x + x \times 3 = \boxed{2x^2 + 3x} \)
2. Développer et réduire :
a) \( (x + 3)(x + 2) \)
b) \( (2x – 1)(3x + 4) \)
Indice : Utilise la méthode “FOIL” (First, Outer, Inner, Last)
Correction :
a) \( (x + 3)(x + 2) = x \times x + x \times 2 + 3 \times x + 3 \times 2 = x^2 + 2x + 3x + 6 = \boxed{x^2 + 5x + 6} \)
b) \( (2x – 1)(3x + 4) = 6x^2 + 8x – 3x – 4 = \boxed{6x^2 + 5x – 4} \)
Exercice 2 : Identités remarquables
1. Développer en utilisant les identités remarquables :
a) \( (x + 5)^2 \) (a + b)² = a² + 2ab + b²
b) \( (3x – 2)^2 \)
c) \( (x + 4)(x – 4) \) (a + b)(a – b) = a² – b²
Correction :
a) \( (x + 5)^2 = x^2 + 2 \times x \times 5 + 5^2 = \boxed{x^2 + 10x + 25} \)
b) \( (3x – 2)^2 = (3x)^2 – 2 \times 3x \times 2 + 2^2 = \boxed{9x^2 – 12x + 4} \)
c) \( (x + 4)(x – 4) = x^2 – 4^2 = \boxed{x^2 – 16} \)
2. Calculer mentalement :
a) \( 102^2 \)
b) \( 99 \times 101 \)
Astuce : 102 = 100 + 2 et 99 = 100 – 1, 101 = 100 + 1
Correction :
a) \( 102^2 = (100 + 2)^2 = 100^2 + 2 \times 100 \times 2 + 2^2 = 10000 + 400 + 4 = \boxed{10404} \)
b) \( 99 \times 101 = (100 – 1)(100 + 1) = 100^2 – 1^2 = 10000 – 1 = \boxed{9999} \)
Exercice 3 : Factorisation
1. Factoriser :
a) \( 6x + 9 \)
b) \( 4x^2 – 7x \)
c) \( 15x^3 – 5x^2 + 10x \)
Correction :
a) \( 6x + 9 = 3(2x + 3) \)
b) \( 4x^2 – 7x = x(4x – 7) \)
c) \( 15x^3 – 5x^2 + 10x = 5x(3x^2 – x + 2) \)
2. Factoriser en utilisant les identités remarquables :
a) \( x^2 + 6x + 9 \)
b) \( 16x^2 – 9 \)
c) \( 4x^2 – 12x + 9 \)
Correction :
a) \( x^2 + 6x + 9 = (x)^2 + 2 \times x \times 3 + 3^2 = \boxed{(x + 3)^2} \)
b) \( 16x^2 – 9 = (4x)^2 – 3^2 = \boxed{(4x – 3)(4x + 3)} \)
c) \( 4x^2 – 12x + 9 = (2x)^2 – 2 \times 2x \times 3 + 3^2 = \boxed{(2x – 3)^2} \)
Exercice 4 : Problèmes concrets
1. Un rectangle a une longueur de (x + 5) cm et une largeur de (x – 2) cm.
a) Exprimer l’aire du rectangle sous forme développée.
b) Calculer cette aire si x = 4 cm.
Correction :
a) Aire = longueur × largeur = \( (x + 5)(x – 2) = x^2 – 2x + 5x – 10 = \boxed{x^2 + 3x – 10} \) cm²
b) Si x = 4 : \( 4^2 + 3 \times 4 – 10 = 16 + 12 – 10 = \boxed{18} \) cm²
2. On considère l’expression \( A = (3x + 1)^2 – (2x – 5)^2 \)
a) Développer A
b) Factoriser A
c) Calculer A pour x = 0
Correction :
a) Développement :
\( A = (9x^2 + 6x + 1) – (4x^2 – 20x + 25) = 9x^2 + 6x + 1 – 4x^2 + 20x – 25 = \boxed{5x^2 + 26x – 24} \)
b) Factorisation (identité a² – b²) :
\( A = [(3x + 1) + (2x – 5)][(3x + 1) – (2x – 5)] = (5x – 4)(x + 6) \)
c) Pour x = 0 :
\( A = (0 + 1)^2 – (0 – 5)^2 = 1 – 25 = \boxed{-24} \)
