Fiche de révision 2e Corrigé
I. Développements.
1. Développer et réduire les expressions suivantes (distributivité) :
A = 3 × (x + 5) B = 3x × (– 4 + x) C = – w(– 1 + w) = 3x + 15 = - 12x + 3x2 = w – w2 D = – 2y(3y + 5) E = – 2(5x – 1) F = – 3a(6 – 5a) = – 6y2 – 10y = – 10x + 2 F = – 18a + 15a2 G = 3x + 5x(4 – 2x) – 2(x2 – 3x + 5) H = 8 + 2x – 2x(3x – 4) + 5x(3 – x) = 3x + 20x – 10x2 – 2x2 + 6x – 10 = 8 + 2x – 6x2 + 8x + 15x – 5x2 = – 12x2 + 29x – 10 = – 11x2 + 25x + 8
2. Développer et réduire les expressions suivantes (double distributivité) :
A = (x + 3)(4 + x) B = (2u + 5)(u + 3) C = (v – 4)(2v + 3) = 4x + x2 + 12 + 3x = 2u2 + 6u + 5u +15 = 2v2 + 3v – 8v – 12 = x2 + 7x + 12 = 2u2 + 11u +15 = 2v2 – 5v – 12 D = (n – 2)(5n – 6) E = (3x + 6)(2x – 1) F = (4x + 5)(2x + 6) = n2 – 6n –10n + 12 = 6x2 – 3x + 12x – 6 = 8x2 + 24x + 10x + 30 = n2 – 16n + 12 = 6x2 + 9x – 6 = 8x2 + 34x + 30 G = (5u + 1)(2 – 3u) H = (– 3 + n)(2n – 5) I = (5y – 2)(3 – 4y) = 10u – 15u2 + 2 – 3u = – 6n + 15 + 2n2 – 5n = 15y – 20y2 – 6 + 8y = – 15u2 + 7u + 2 = 2n2 – 11n + 15 = – 20y2 + 23y – 6
3. Développer et réduire les expressions suivantes (attention aux priorités) : A = 3(4x + 7) + 4(2x − 9) B = 7x(2x − 5) − x(2x − 5) = 12x + 21 + 8x − 36 = 14x2 − 35x − 2x2 + 5x
= 20x − 15 = 12x2 − 30x
C = (2x + 3)(5x − 8) − (2x − 4)(5x − 1) D = (5x − 2)(5x − 8) − (3x − 5)(x + 7)
= (10x2 − 16x + 15x − 24) − (10x2 − 2x − 20x + 4) = (25x2 − 40x − 10x +16) − (3x2 + 21x − 5x − 35) = (10x2 − x − 24) − (10x2 − 22x + 4) = (25x2 − 50x +16) − (3x2 + 16x − 35)
= 10x2 − x − 24 − 10x2 + 22x − 4 = 25x2 − 50x +16 − 3x2 − 16x + 35
= 21x − 28 = 22x2 − 66x + 51
E = (x + 7)(3 − 2x) + (5x − 2)(4x + 1)
= (3x − 2x2 + 21 − 14x) + (20x2 + 5x − 8x − 2) = − 2x2 − 11x + 21 + 20x2 − 3x − 2
= 18x2 − 14x + 19
4. Développer avec les identités remarquables.
A = (x + 5)2 B = (4x + 6)2 C = (4x − 6)2 D = (x − 5)2
= x2 + 10x + 25 = (4x)2 + 2 µ (4x) µ 6 + 62 = (4x)2 − 2 µ (4x) µ 6 + 62 = x2 − 10x + 25 = 16x2 + 48x + 36 = 16x2 − 48x + 36
E = (3x − 7)2 F = (1 − 6x)2
= (3x)2 − 2(3x) µ7 + 72 = 12 − 2 µ 1 µ 6x + (6x)2 = 9x2 − 42x + 49 = 1 − 12x + 36x2
= 36x2 − 12x + 1
G = (2t − 9)2 H = (y + 3)(y − 3)
= (2t)2 − 2 µ (2t) µ 9 + 92 = y2 − 32 = 4t2 − 36t + 81 = y2 − 9
I = (2x + 5)(2x − 5) J = (3 + 4x)(3 − 4x)
= (2x)2 − 52 = 32 − (4x)2
= 4x2 − 25 = 9 − 16x2
= − 16x2 + 9
5. Développer avec les identités remarquables.
K = (3x + 7)2 + (7x − 3)2 L = (x + 2)2 − (3x − 5)2
= (3x)2 + 2 µ (3x) µ 7 + 32 + (7x)2 − 2 µ (7x) µ 3 + 32 = x2 + 2 µ x µ 2 + 22 − [(3x)2 − 2 µ (3x) µ 5 + 52] = 9x2 + 42x + 9 + 49x2 − 42x + 9 = x2 + 4x + 4 − [9x2 − 30x + 25]
= 58x2 + 18 = x2 + 4x + 4 − [9x2 − 30x + 25]
= x2 + 4x + 4 − 9x2 + 30x − 25
= − 8x2 + 34x − 21
M = 3(x + 5) − (x − 8)2 N = (2x − 5)2 − (4x + 1)2 = 3x + 15 − (x2 – 2 µ x µ 8 + 82) = (2x − 5)2 − (4x + 1)2
= 3x + 15 − x2 – 16x + 64 = (2x)2 − 2 µ x µ 5 + 52 − [(4x)2 + 2 µ 1 µ 4x + 12] = − x2 – 13x + 79 = 4x2 − 10x + 25 − [16x2 + 8x + 1]
= 4x2 − 10x + 25 − 16x2 − 8x − 1 = − 12x2 − 18x + 24
II. Factorisations.
1. Factoriser.
A = 9y + 12 B = x² + 5x C = (x + 1)2 − 2(x + 1) = 3(3y + 4) = x(x + 5) = (x + 1)[ (x + 1) − 2]
= (x + 1)(x − 1) Attention, on ne développe pas !
D = (t − 7)(2t + 1) + (2t + 1)² E = (x + 2)(2x − 1) + (x + 2)(3x + 2) = (2t + 1)[(t − 7) + 1] = (x + 2)[(2x − 1) + (3x + 2)]
= (2t + 1)(t − 6) = (x + 2)(5x + 1)
F = (5x − 3)(x − 7) − (2x + 4)(x − 7) G = (2x − 1)(x − 5) + (3x + 7)(x − 5) = (x − 7)[ (5x − 3) − (2x + 4)] = (x − 5)[ (2x − 1) + (3x + 7)]
= (x − 7)( 5x − 3 − 2x − 4) = (x − 5)(2x − 1 + 3x + 7)
= (x − 7)( 3x − 7) = (x − 5)(5x + 6)
H = (2x + 5)(x − 3) + (2x + 5)(− 3x + 1) K = (3x + 7)(2x − 9) − (3x + 7)(5x − 7) = (2x + 5)[(x − 3) + (− 3x + 1)] = (3x + 7)[(2x − 9) − (5x − 7)]
= (2x + 5)(x − 3 − 3x + 1) = (3x + 7)(2x − 9 − 5x + 7) = (2x + 5)(− 2x − 2) = (3x + 7)(− 3x − 2)
L = (− 3x + 4)(3x − 8) − (− 3x + 4)(7x + 2) M = (8y + 3)(5y + 7) − 3(8y + 3)(2y − 1) = (− 3x + 4)[(3x − 8) − (7x + 2)] = (8y + 3)[(5y + 7) − 3 (2y − 1)]
= (− 3x + 4)(3x − 8 − 7x − 2) = (8y + 3)(5y + 7 − 6y + 3) = (− 3x + 4)( − 4x − 10) = (8y + 3)(− y + 10)
N = (x − 1)2 + (x − 1)(2x + 3) O = (2x + 3)(x − 5) − (x − 5)2 = (x − 1)[(x − 1) + (2x + 3)] = (x − 5) [(2x + 3) − (x − 5)]
= (x − 1)(x − 1 + 2x + 3) = (x − 5) (2x + 3 − x + 5)
= (x − 1)(3x + 2) = (x − 5) (x + 8)
P = (2t − 7) − (5t + 1)(2t − 7) Q = 2y2 − y(4y − 7) = (2t − 7)[1 − (5t + 1)] = y (2y − (4y − 7)]
= (2t − 7)(1 − 5t − 1) = y (2y − 4y + 7)
= (2t − 7)(− 5t) = y (− 2y + 7)
2. Factoriser, si possible, avec les identités remarquables.
A = x2 + 8x + 16
= x2 + 2 µ x µ 4 + 42
On reconnait la forme a2++++ 2ab ++++ b2 = (a + b)2 = (x + 4)2
B = 9x2 + 30x + 25 C = x2 + 10x + 25 = (3x)2 + 2 µ 3x µ 5 + 52 = x2 + 2 µ x µ 5 + 52
= (3x + 5)2 = (x + 5)2
D = 4t2 + 24t + 36 E = 9x2 + 64 + 48x
= (2t)2 + 2 µ 2t µ 6 + 62 = (3x)2 + 82 + 2 µ 3x µ 8 = (2t + 6) 2 = (3x + 8)2
F = x2 − 20x + 100 = x2 − 2 µ 3x µ 8 + 102
On reconnait la forme a2---- 2ab ++++ b2 = (a - b)2 = (x − 10)2
G = 9 + 4x2 − 12x H = x2 − 2x + 1
= 32 + (2x)2 − 2 µ 3 µ 2x = x2 − 2 µ x µ 1 + 12
= (3 − 2x)2 = (x − 1)2
I = y2 − 18y + 81 J = 16x2 + 25 − 40x
= y2 − 2 µ y µ 9 + 92 = (4x)2 + 52 − 2 µ 4x µ 5
= (y − 9)2 = (4x − 5)2
K = x2 − 16 = x2 − 42
On reconnait la forme a2---- 2ab ++++ b2 = (a - b)2 = (x − 4)(x + 4)
L = 25x2 − 36 M = x2 − 100 N = 25 − 4y
= (5x)2 − 62 = (x − 10)(x + 10) = 52 − 4y
= (5x − 6)(5x + 6) On ne peut pas factoriser
O = (x + 4)2 − 49 P = (3 − 2 x)2 − 4
= (x + 4)2 − 72 = [(3 − 2 x) − 2][(3 − 2 x) + 2]
= [(x + 4) − 7][(x + 4) + 7] = (− 2 x + 1)(− 2 x + 5) = (x − 3)(x + 11)
Q = 121 − (x − 7)2 R = (7x + 8)2 − (9 − 5x)2
= 112 − (x − 7)2 = [(7x + 8)− (9 − 5x)] [(7x + 8)+ (9 − 5x)]
= [11 − (x − 7)][11 + (x − 7)] = (2x − 1)(2x + 17) = (− x + 18)(x + 4)
