IDENTITÉS REMARQUABLES CALCUL LITTÉRAL IDENTITÉS REMARQUABLES exercices mathalecran d'après www.mathsenligne.com

Exercice 1A.1 - Donner le carré de chaque expression : 100t² 16a² x4 25x² Exercice 1A.2 - Réduire chaque produit : 30x² 40x 16x² 6x² 56x 42x 30x² 36x² 240x²

Exercice 1A.3 - Développer en utilisant l’identité remarquable : (a + b)² = a² + 2ab + b²  A = (3 + x)² B = (x + 5)² C = (2x + 1)² A = 3² + 2 3 x + x² × B = x² + 2 x 5 + 5² × C = (2x)² + 2 2x 1 + 1² × A = 9 + 6x + x² B = x² + 10x + 25 C = 4x² + 4x + 1 D = (1 + 3x)² E = (3x + 2)² D = 1² + 2 1 3x + (3x)² × E = (3x)² + 2 3x 2 + 2² × D = 1 + 6x + 9x² E = 9x² + 12x + 4 F = (5x + 3)² G = (x² + 1)² F = (5x)² + 2 5x 3 + 3² × G = (x²)² + 2 x² 1 + 1² × F = 25x² + 30x + 9 G = x4 + 2x² + 1 H = (3 + 4x)² H = 3² + 2 3 4x + (4x)² × H = 9 + 24x + 16x²

Exercice 1A.4 - Développer en utilisant l’identité remarquable : (a - b)² = a² - 2ab + b²  A = (x - 2)² B = (1 - 3x)² C = (3 - x)² A = x² - 2 x 2 + 2² × B = 1² - 2 1 3x + (3x)² × C = 3² - 2 3 x + x² × A = x² - 4x + 4 B = 1 - 6x + 9x² C = 9 - 6x + x² D = (2x - 1)² E = (3 - 5x)² D = (2x)² - 2 2x 1 + 1² × E = 3² - 2 3 5x + (5x)² × D = 4x² - 4x + 1 E = 9 - 30x + 25x² F = (3x - 2)² G = (4x - 3)² F = (3x)² - 2 3x 2 + 2² × G = (4x)² - 2 4x 3 + 3² × F = 9x² - 12x + 4 G = 16x² - 24x + 9 H = (4 - 3x²)² H = 4² - 2 4 3x² + (3x²)² × H = 16 - 24x² + 9x4

Exercice 1A.5 - Développer en utilisant l’identité remarquable : (a + b)(a – b) = a² – b²  A = (x +2)(x – 2) B = (x + 3)(x – 3) C = (3x – 1)(3x + 1) A = x² – 2² B = x² – 3² C = (3x)² – 1² A = x² – 4 B = x² – 9 C = 9x² – 1 D = (2x + 1)(2x – 1) E = (5 + 3x)(5 – 3x) F = (3x – 2)(3x + 2) D = (2x)² – 1² E = 5² – (3x)² F = (3x)² – 2² D = 4x² – 1 E = 25 – 9x² F = 9x² – 4 G = (3 + 4x)(3 – 4x) H = (4x² + 3)(4x² – 3) G = 3² – (4x)² H = (4x²)² – 3² G = 9 – 16x² H = 16x4 – 9

Exercice 1B.1 - Développer en utilisant l’identité remarquable qui convient : A = (x + 4)² B = (2 - x)² C = (x + 1)(x - 1) A = x² + 2 x 4 + 4² × B = 2² - 2 2 x + x² × C = x² - 1² A = x² + 8x + 16 B = 4 - 4x + x² C = x² - 1 D = (2x + 1)² E = (3 - 2x)² F = (7x + 5)² D = (2x)² + 2 2x 1 + 1² × E = 3² - 2 3 2x + (2x)² × F = (7x)² + 2 7x 5 + 5² × D = 4x² + 4x + 1 E = 9 - 12x + 4x² F = 49x² + 70x + 25 G = (5x + 6)(5x - 6) H = (4 - 8x)² I = (3 + 4x)(3 + 4x) G = (5x)² - 6² H = 4² - 2 4 8x + (8x)² × I = (3 + 4x)² G = 25x² - 36 H = 16 - 64x + 64x² I = 3² + 2 3 4x + (4x)² × I = 9 + 24x + 16x² J = (3 + x)(x - 3) K = (2 + 9x)² H = (11x - 12)² J = (x + 3)(x - 3) K = 2² + 2 2 9x + (9x)² × H = (11x)² - 2 11x 12 + 12² × K = 4 + 36x + 81x² J = x² - 3² H = 121x² - 264x + 144 J = x² - 9

Exercice 1B.2 - Développer puis réduire : A = (x + 1)² + (x – 3)² B = (3 – x)² + (x + 5)² A = x² + 2x + 1 + ( x² – 6x + 9) B = 9 – 6x + x² + ( x² + 10x + 25) A = x² + 2x + 1 + x² – 6x + 9 B = 9 – 6x + x² + x² + 10x + 25 A = 2x² – 4x + 10 B = 2x² + 4x + 34 C = (x – 2)² + (x + 4)(x – 4) D = (x + 1)(x – 1) + (x + 4)² C = x² – 4x + 4 + ( x² – 16) D = x² – 1 + ( x² + 8x + 16) C = x² – 4x + 4 + x² – 16 D = x² – 1 + x² + 8x + 16 C = 2x² – 4x – 12 D = 2x² + 8x + 15 E = (x – 5)² + (2x + 7)(2x – 7) E = x² – 10x + 25 + ( 4x² – 49) E = x² – 10x + 25 + 4x² – 49 E = 5x² – 10x – 24

Exercice 1B.3 - Développer puis réduire : A = (2x + 1)² – (x + 3)² B = (2x + 3)² – (x – 7)(x + 7) A = 4x² + 2x + 1 – ( x² + 6x + 9) B = 4x² + 12x + 9 – ( x² – 49) A = 4x² + 2x + 1 – x² – 6x – 9 B = 4x² + 12x + 9 – x² + 49 A = 3x² – 4x – 8 B = 3x² + 12x + 58 C = (x + 2)(x – 2) – (x + 3)² C = x² – 4 – ( x² + 6x + 9) C = x² – 4 – x² – 6x – 9 C = 0x² – 6x – 13 D = (x – 5)² – (2x – 7)(x – 5) D = x² – 10x + 25 – ( 2x² – 10x – 7x + 35) D = x² – 10x + 25 – 2x² + 10x + 7x – 35 D = – x² + 7x – 10 E = (3x + 1)(x – 2) + (2x – 3)² E = 3x² – 6x + x – 2 – ( 4x² – 12x + 9) E = 3x² – 6x + x – 2 – 4x² + 12x – 9 E = – x² + 7x – 11

EXERCICE 2A.1 A = 3x + 3y A = 3x + 3y B = -3a + 3 b B = -3a + 3 b C = 7 x + 12 x C = 7 x + 12 x D = -6(3x – 2) – (3x – 2)(x – 4) D = -6(3x – 2) – (3x – 2)(x – 4) E = (x + 2)(x + 1) + (x + 2)(7x – 5) E = (x + 2)(x + 1) + (x + 2)(7x – 5) F = (2x + 1)² + (2x + 1)(x + 3) F = (2x + 1)² + (2x + 1)(x + 3) G = (x + 1)(2x – 3) + (x + 1)(5x + 1) G = (x + 1)(2x – 3) + (x + 1)(5x + 1) H = (3x – 4)(2 – x) – (3x – 4)² H = (3x – 4)(2 – x) – (3x – 4)² I = (6x + 4)(2 + 3x) + (2 + 3x)(7 – x) I = (6x + 4)(2 + 3x) + (2 + 3x)(7 – x) J = (3 + x)(5x + 2) + (x + 3)² J = (3 + x)(5x + 2) + (x + 3)²

4x 1 4x x EXERCICE 2A.2 A = 4x + 4y = A = 4x + 4y = 4 (x + y ) B = 6 9 + 6 3 = × B = 6 9 + 6 3 = × 6 (9 + 3 ) C = 8a + 8b = C = 8a + 8b = 8 (a + b ) D = 5 3 + 3 14 = × D = 5 3 + 3 14 = × 3 ( 5 + 14) E = 2 + 2x = × 1 2 ( 1 + x ) F = 7a + 7 = × 1 7 ( a + 1 ) G = 4x² + 4x = 4x (x + 1 ) H = 6y + 6y² = 6y (1 + y ) I = 3x² + 5x = x ( 3x + 5) 4x 1 × 4x x × J = 2ab + b² = b ( 2a + b )

x + 3y x – 6 5 + 3x 3x – 4y -x + 2 1 + 2x 3x + 1 7x – 6 y + 3x Exercice 2A.3 Compléter l’intérieur des parenthèses, comme dans l’exemple : x + 3y x – 6 5 + 3x 3x – 4y -x + 2 1 + 2x 3x + 1 7x – 6 y + 3x

5x + 5  2 5(x + 2) 6x – 6  4 6(x – 4) 4  9 – 4x 4(9 – x) 7x + 7  2 Exercice 2A.4 Écrire le terme souligné sous forme d’un produit puis factoriser l’expression : 5x + 5  2 5(x + 2) 6x – 6  4 6(x – 4) 4  9 – 4x 4(9 – x) 7x + 7  2 7(x + 2) 5  7 – 5x 5(7 – x) 8x – 8  3 8(x – 3) 6  2x + 6  3 6(2x + 3) 3  2 – 3  5x 3(2 – 5x) 6  5x – 6  7 6(5x – 7)

A = 13(x + 2) + 5(x + 2) B = 7(2x – 3) + 2(2x – 3) A = (x + 2)(13 + 5) Exercice 2A.5 Factoriser les expressions suivantes comme dans l’exemple : A = 13(x + 2) + 5(x + 2) B = 7(2x – 3) + 2(2x – 3) A = (x + 2)(13 + 5) B = (2x – 3)(7 + 2) A = 18(x + 2) B = 9(2x – 3) C = 3x(x + 2) – 5(x + 2) D = 4(x + 3) + 9x(x + 3) C = (x + 2)(3x – 5) D = (x + 3)(4 + 9x) E = 7x(3x + 1) – 10x(3x + 1) E = (3x + 1)(7x – 10x) E = – 3x(3x + 1)

EXERCICE 2B .1.a A = (x – 3)(2x + 1)+ 7(2x + 1) B = (x + 1)(x + 2) – 5(x + 2) A = (2x + 1)(x – 3+ 7) B = (x + 2)(x + 1 – 5) A = (x – 3)(x + 4) B = (x + 2)(x – 4) C = (3 – x)(4x + 1) – 8(4x + 1) C = (4x + 1)(3 – x – 8) D = 5(1 + 2x) – (x + 1)(1 + 2x) C = (4x + 1)(– x – 5) D = (1 + 2x) (5 – (x + 1)) D = (1 + 2x) (5 – x – 1) D = (1 + 2x) (4 – x) E = -6(3x – 2) – (3x – 2)(x – 4) E = (3x – 2)(-6 – (x – 4)) E = (3x – 2)(-6 – x + 4) E = (3x – 2)(-2 – x)

EXERCICE 2B .1.b A = (x + 1)( 3 – x) + (x + 1)(2 + 5x) A = (x + 1)( 3 – x + 2 + 5x) A = (x + 1) ( 4x + 5) B = (x + 2)(x + 1) + (x + 2)(7x – 5) B = (x + 2)(x + 1 + 7x – 5) B = (x + 2)( 8x – 4) B = 4(x +2)(2x - 1) C = (x + 3)(3 – 2x) – (x + 3)(5 + x) C = (x + 3)(3 – 2x – (5 + x)) C = (x + 3)(3 – 2x – 5 – x) C = (x + 3)( –3x – 2) D = (2x + 1)(x – 5) – (3x + 1)(2x + 1) D = (2x + 1)(x – 5 – (3x + 1)) D = (2x + 1)(x – 5 – 3x – 1) D = (2x + 1)(– 2x – 6) D = – 2(2x + 1)(x + 3)

EXERCICE 2B .1.b C = (x + 3)(3 – 2x) – (x + 3)(5 + x) C = (x + 3)(3 – 2x – (5 + x)) C = (x + 3)(3 – 2x – 5 – x) C = (x + 3)( –3x – 2) D = (2x + 1)(x – 5) – (3x + 1)(2x + 1) D = (2x + 1)(x – 5 – (3x + 1)) D = (2x + 1)(x – 5 – 3x – 1) D = (2x + 1)(– 2x – 6) D = – 2(2x + 1)(x + 3) E = (x – 6)(2 – x) – (2 – x)(3 + 4x) E = (2 – x)((x – 6) – (3 + 4x)) E = (2 – x)(x – 6 – 3 – 4x) E = (2 – x)(– 3x – 9) E = – 3(2 – x)(x + 3) E = 3(x – 2)(x + 3)

EXERCICE 2B .1.c A = (x + 1)² + (x + 1)(3x + 1) B = (2x + 1)² + (2x + 1)(x + 3) A = 2(x + 1) ( 2x + 1 ) B =(2x + 1) (2x + 1 + x + 3) B =(2x + 1) (3x + 4) C = (x – 3)² – (x – 3)(4x + 1) C = (x – 3)(x – 3 – (4x + 1)) D = (x + 1)(2x – 5) + (2x – 5)² C = (x – 3)(x – 3 – 4x – 1)) D = (2x – 5)(x + 1 + 2x – 5) C = (x – 3)( –3x – 4) D = (2x – 5)(3x – 4) E = (3x – 4)(2 – x) – (3x – 4)² E = (3x – 4)(2 – x – (3x – 4)) E = (3x – 4)(2 – x – 3x + 4) E = (3x – 4)( – 4x + 6) E = 2(3x – 4)( 3 – 2x )

EXERCICE 2B .2 A =(x + 1)(x + 2) +( 2x + 2)(3x - 4) A =(x + 1)(x + 2) +2( x + 1)(3x - 4) A =(x + 1)(x + 2 +2(3x - 4)) A =(x + 1)(x + 2 + 6x - 8) A =(x + 1)(7x - 6) B = (x - 1)(2x + 1) +( 6x + 3)(3 - x) B = (x - 1)(2x + 1) +3(2x + 1)(3 - x) B = (2x + 1)((x - 1) +3(3 - x)) B = (2x + 1)(x - 1 + 9 - 3x) B = (2x + 1)( -2x + 8) C =(10x - 5)(x + 2) + (1 - x)(2x - 1) C =5(2x - 1)(x + 2) + (1 - x)(2x - 1) C =(2x - 1)(5(x + 2) + (1 - x) ) C =(2x - 1)(5x + 10 +1 - x ) C =(2x -1)(4x + 11)

EXERCICE 2B .2 C =(10x - 5)(x + 2) + (1 - x)(2x - 1) C =5(2x - 1)(x + 2) + (1 - x)(2x - 1) C =(2x - 1)(5(x + 2) + (1 - x) ) C =(2x - 1)(5x + 10 +1 - x ) C =(2x -1)(4x + 11) D = ( 4x + 4)(1 - 2x) +(x + 1)² D = 4(x + 1)(1 - 2x) +(x + 1)² D = (x + 1)(4(1 - 2x) +(x + 1)) D = (x + 1)(4 - 8x + x + 1) D = (x + 1)(-7x +5) E = (2x + 1)² -(x + 3)(10x + 5) E = (2x + 1)² -5(x + 3)(2x + 1) E = (2x + 1)((2x + 1) -5( x + 3)) E = (2x + 1)(2x + 1 - 5x - 15) E =(2x + 1)(-3x - 14 )

Exercice 3A.1 - Retrouver l’expression dont on connaît le carré : 12b 4y

EXERCICE 3A.2 A = x² + 10x + 25 B = x² + 6x + 9 D = 4x² + 12x + 9 C = 36 + 12x + x² D = (2x)² + 2  2x  3 + 3² C = 6² + 2  x  6 + x² C = (x + 6)² D = (2x + 3)² E = 16x² + 40x + 25 E = (4x)² + 2  4x  5 + 5² E = (4x + 5)²

EXERCICE 3A.3 A = x² – 2x + 1 B = 4x² – 20x + 25 C = 9 – 6x + x² D = 36x² – 12x + 1 C = 3² – 2  3  x + x² D = (6x)² – 2  6x  1 + 1² C = (3 – x)² D = (6x – 1)² E = 100 – 40x + 4x² E = 10² – 2  10  2x + (2x)² E = (10 – 2x)² E =4 (5 – x)²

EXERCICE 3A.4 a. B = 9 – x² A = x² – 4 B = 3² – x² A = x² – 2² B = (3 – x)(3 + x) A = (x – 2)(x + 2) D = x² – 49 C = x² – 16 D = x² – 7² C = x² – 4² D = (x – 7)(x + 7) C = (x – 4)(x + 4) E = 25 – x² E = 5² – x² E = (5 – x)(5 + x)

EXERCICE 3A.4 b. B = 16 – 9x² A = 4x² – 9 A = (2x)² – 3² A = (2x – 3)(2x + 3) B = (4 – 3x)(4 + 3x) C = 16x² – 25 D = 49x² – 36 C = (4x)² – 5² D = (7x)² – 6² D = (7x – 6)(7x + 6) C = (4x – 5)(4x + 5) E = 4 – 64x² E = 2² – (8x)² E = (2 – 8x)(2 + 8x)

Voyons ce qui vous attends au brevet.... ET MAINTENANT, Voyons ce qui vous attends au brevet.... Exercice 3B.1 Exercice 3B.2

EXERCICE 3B.1.a. a² – b² (a – b) ( a + b) A = (x + 1)² – 4 FACILE, NON ? A = (x + 1)² – 4 Je reconnais la différence de deux carrés et je l'écris a² – b² A = (x + 1)² – 2² (a – b) ( a + b) A = ((x +1) – 2)((x+1) + 2) Je réduis chaque facteur Je supprime les parenthèses intérieures J'applique l'égalité remarquable A = (x +1 – 2)(x+1 + 2) A = (x – 1)(x+3)

EXERCICE 3B.1.a. B = (x + 2)² – 9 B = (x + 2)² – 3² B = ((x +2) – 3)((x+2) +3) B = (x +2 – 3)(x+2 + 3) B = (x – 1)(x+5) C = (2x + 1)² – 25 C = (2x + 1)² –5² C = ((2x +1) – 5)((2x+1) +5) C = (2x +1 – 5)(2x+1 + 5) C = (2x – 4)(2x+6)

EXERCICE 3B.1.a. C = (2x + 1)² – 25 C = (2x + 1)² –5² C = ((2x +1) – 5)((2x+1) +5) C = (2x +1 – 5)(2x+1 + 5) C = (2x – 4)(2x+6) D = 16 – (3x + 2)² D = 4² – (3x + 2 )² D = (4 – (3x + 2 ))(4 + (3x + 2 )) D = (4 – (3x + 2 ))(4 + (3x + 2 )) D = (4 – 3x – 2 )(4 + 3x + 2 ) D = (2 – 3x )( 3x + 6 )

EXERCICE 3B.1.a. D = 16 – (3x + 2)² D = 4² – (3x + 2 )² D = (4 – (3x + 2 ))(4 + (3x + 2 )) D = (4 – (3x + 2 ))(4 + (3x + 2 )) D = (4 – 3x – 2 )(4 + 3x + 2 ) D = (2 – 3x )( 3x + 6 ) E = 36 – (4 – 3x)² E =6² – (4 – 3x )² E = (6 – (4 – 3x )(6 + (4 – 3x )) E = (6 – (4 – 3x )(6 + (4 – 3x )) E = (6 – 4 + 3x )(6 + 4 – 3x ) E = (2 + 3x )( 10 – 3x )

La différence de deux carrés est déjà écrite. EXERCICE 3B.1.b. a² – b² A = (x + 1)² – (2x + 3)² (a – b ) ( a + b ) A = ((x +1) – (2x + 3 ))((x+1) + (2x +3 )) A = ((x +1) – (2x + 3 ))((x+1) + (2x +3 )) La différence de deux carrés est déjà écrite. A = (x + 1 – 2x – 3 )( x + 1 + 2x + 3 ) A = ( – x – 2 )( 3x + 4)

EXERCICE 3B.1.b. A = (x + 1)² – (2x + 3)² A = ((x +1) – (2x + 3 ))((x+1) + (2x +3 )) A = (x + 1 – 2x – 3 )( x + 1 + 2x + 3 ) A = ( – x – 2 )( 3x + 4) B = (2x – 1)² – (5 + x)² B = ((2x – 1) – (5 + x))((2x – 1) + (5 + x)) B = (2x – 1 – 5 – x )(2x – 1 + 5 + x ) B = (x – 6 )( 3x + 4 )

EXERCICE 3B.1.b. B = (2x – 1)² – (5 + x)² B = ((2x – 1) – (5 + x))((2x – 1) + (5 + x)) B = (2x – 1 – 5 – x )(2x – 1 + 5 + x ) B = (x – 6 )( 3x + 4 ) C = (4x – 1)² – (3x + 4)² C = ((4x – 1) – (3x + 4))((4x – 1)+(3x + 4)) C = (4x – 1 – 3x – 4 )(4x – 1 + 3x + 4 ) C = (x – 5 )( 7x + 3 ) D = (3x – 4)² – (6x + 1)² D = ((3x – 4) – (6x + 1))((3x – 4) + (6x + 1)) D = (3x – 4 – 6x – 1)(3x – 4 + 6x + 1) D = (– 3x – 5) (9x – 3)

EXERCICE 3B.1.b. D = (3x – 4)² – (6x + 1)² D = ((3x – 4) – (6x + 1))((3x – 4) + (6x + 1)) D = (3x – 4 – 6x – 1)(3x – 4 + 6x + 1) D = (– 3x – 5) (9x – 3) E = (x + 6)² – (3x – 1)² E = ((x + 6) – (3x – 1))((x + 6) + (3x – 1)) E = (x + 6 – 3x + 1)(x + 6 + 3x – 1) E = (– 2x + 7) (4x + 5)

k a + b k k ( a + b ) EXERCICE 3B.2 A = (x + 2)(3x – 1) + x² – 4 Je reconnais la différence de deux carrés A = (x + 2)(3x – 1) + (x – 2)(x + 2) k ( a + b ) A =(x + 2)(3x – 1 + x – 2) Je réduis le second facteur J'applique l'égalité remarquable et le facteur commun apparait A = (x + 2)( 4x– 3) J'applique la distributivité. Pas de problème de signe : c'est une somme

EXERCICE 3B.2 A = (x + 2)(3x – 1) + x² – 4 A = (x + 2)(3x – 1) + (x – 2)(x + 2) A =(x + 2)(3x – 1 + x – 2) A = (x + 2)( 4x– 3) B = (x + 4)(2x – 1) + x² – 16 B = (x + 4)(2x – 1) + (x – 4)(x + 4) B =(x + 4)(2x – 1 + x – 4) B = (x + 4)( 3x– 5)

EXERCICE 3B.2 B = (x + 4)(2x – 1) + x² – 16 B = (x + 4)(2x – 1) + (x – 4)(x + 4) B =(x + 4)(2x – 1 + x – 4) B = (x + 4)( 3x– 5) C = (x – 3)(x + 1) – (x² – 9) C = (x – 3)(x + 1) – (x – 3)(x + 3) C =(x – 3)((x + 1) – (x + 3 )) Par souci pratique (et esthétique), on préfèrera cette écriture C =(x – 3)(x + 1 – x – 3 ) C = (x – 3)( – 2) Les x disparaissent de ce facteur C = – 2 (x – 3)

EXERCICE 3B.2 C = (x – 3)(x + 1) – (x² – 9) C = (x – 3)(x + 1) – (x – 3)(x + 3) C =(x – 3)((x + 1) – (x + 3 )) C =(x – 3)(x + 1 – x – 3 ) C = (x – 3)( – 2) C = – 2 (x – 3) D = (2x + 1)(x – 2) – (x² – 4) D =(2x + 1)(x – 2) – (x –2)(x + 2) D =(x – 2)(2x + 1 – x – 2) D = (x – 2)( x – 1)

(5 - x) est l'opposé de ( x - 5) EXERCICE 3B.2 D = (2x + 1)(x – 2) – (x² – 4) D =(2x + 1)(x – 2) – (x –2)(x + 2) D =(x – 2)(2x + 1 – x – 2) D = (x – 2)( x – 1) E = 25 – x² – (x – 5)(2x + 3) E = (5 – x)(5 + x) – (x – 5)(2x + 3) E = (5 – x)(5 + x) + (5 – x)(2x + 3) E = (5 – x)(5 + x + 2x + 3) (5 - x) est l'opposé de ( x - 5) E = (5 – x)( 3x + 8 )

Exercice 4.1 Écrire chaque nombre comme une somme puis utiliser l’identité remarquable (a+b)²=a²+2ab+b² pour calculer : B = 102² B = (100 + 2)² E = 201² B = 100² + 2  100  2 + 2² E = (200 + 1)² B = 10 000 + 400 + 4 E = 200² + 2  200  1 + 1² B = 10 404 E = 40 000 + 400 + 1 C = 51² E = 40 401 C = (50 + 1)² C = 50² + 2  50  1 + 1² C = 2 500 + 100 + 1 F = 109² C = 2 601 F = (100 + 9)² F = 100² + 2  100  9 + 9² D = 1 005² F = 10 000 + 1 800 + 81 D = (1 000 + 5)² F = 11 881 D = 1 000² + 2  1 000  5 + 5² D = 1 000 000 + 10 000 + 25 D = 1 010 025

Exercice 4.2 Écrire chaque nombre comme une somme puis utiliser l’identité remarquable (a+b)²=a²-2ab+b² pour calculer : B = 98² B = (100 - 2)² E = 199² B = 100² - 2  100  2 + 2² E = (200 - 1)² B = 10 000 - 400 + 4 E = 200² - 2  200  1 + 1² B = 9 604 E = 40 000 - 400 + 1 C = 49² E = 39 601 C = (50 - 1)² C = 50² - 2  50  1 + 1² C = 2 500 - 100 + 1 F = 91² C = 2 401 F = (100 - 9)² F = 100² - 2  100  9 + 9² D = 990² F = 10 000 - 1 800 + 81 D = (1 000 - 10)² F = 9 281 D = 1 000² - 2  1 000  10 + 10² D = 1 000 000 - 20 000 + 100 D = 980 100

Exercice 4.3 Écrire chaque nombre comme le produit d’une somme par une différence puis utiliser l’identité remarquable (a+b)(a–b)=a²–b² pour calculer : B = 105  95 B = (100 + 5)(100 - 5) B = 100² - 5² E = 498  502 B = 10 000 - 25 E = (500 - 2)(500 + 2) B = 9 975 E = 500² - 2² E = 250 000 - 4 C = 51  49 E = 249 996 C = (50 + 1)(50 - 1) C = 50² - 1² C = 2 500 - 1 E = 1 007  993 C = 2 499 E = (1 000 + 7)(1 000 - 7) E = 1 000² - 7² D = 107  93 E = 1 000 000 - 49 D = (100 + 7)(100 - 7) E = 999 951 D = 100² - 7² D = 10 000 - 49 D = 9 951

Exercice 4.4 Utiliser l’identité remarquable a²–b²=(a+b)(a–b) pour factoriser puis calculer : B = 105² - 95² B = (105 - 95)(105 + 95) E = 9 876² – 9 875² B = 10  200 E = (9 876 - 9 875)(9 876 + 9 875) E = 1  19 751 B = 2 000 E = 19 751 C = 235² - 234² C = (235 - 234)(235 + 234) C = 1  469 F = 93² – 107² C = 469 F = (93 - 107)(93 + 107) F = - 14  200 D = 47² - 53² F = - 2 800 D = (47 - 53)(47 + 53) D = - 6  100 D = - 600

Exercice 5.1 – National 2007 Cet exercice est un questionnaire à choix multiples (QCM). Aucune justification n'est demandée. Pour chacune des questions, trois réponses sont proposées, une seule est exacte.Pour chacune des deux questions,indiquer sur la copie le numéro de la question et recopier la réponse exacte. a² + 2ab + b² (a + b)² 20 Question n°1: 9x2+30x + 25 4(4+1)= 20 (4+1)(4-2)= 10 Question n°2: (x + 1)2 (4+1)²= 25

Et le reste ? Au tableau !!