2D8 fiche de révision développements et factorisations

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Corrigé de l’exercice 1

Développeretréduire les expressions suivantes.

A= (x−8) (x+ 8) A=x²−8²

A=x²−64 B = (5x+ 9)²

B = (5x)²+ 2×5x×9 + 9² B= 25x²+ 90x+ 81

C = (7x−5)²

C = (7x)²−2×7x×5 + 5²

C = 49x²−70x+ 25 D= (−7x−8) (−9x+ 1)

D= 63x²+ (−7x) + 72x+ (−8)

D= 63x²+ 65x−8

E= (3x+ 5)²−(10x+ 2) (10x−2)

E= (3x)²+ 2×3x×5 + 5²−((10x)²−2²) E= 9x²+ 30x+ 25−(100x² −4)

E= 9x²+ 30x+ 25−100x²+ 4 E=−91x²+ 30x+ 29

F = (−9x+ 5) (−6x+ 1) + (2x−9)²

F = 54x²+ (−9x) + (−30x) + 5 + (2x)²−2×2x×9 + 9² F = 54x²−39x+ 5 + 4x²−36x+ 81

F = 58x²−75x+ 86

A= (4x−8) (−4x+ 10)

A=−16x²+ 40x+ 32x+ (−80)

A=−16x²+ 72x−80 B = (9x−9)²

B = (9x)²−2×9x×9 + 9² B= 81x²−162x+ 81

C = (2x+ 5) (2x−5) C = (2x)²−5²

C = 4x²−25 D= (2x+ 2)²

D= (2x)²+ 2×2x×2 + 2²

D= 4x²+ 8x+ 4

E= (−10x−3) (−5x+ 10)−(5x+ 2)²

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E= 50x²+ (−100x) + 15x+ (−30)−((5x)²+ 2×5x×2 + 2²) E= 50x²−85x−30−(25x²+ 20x+ 4)

E= 50x²−85x−30−25x²−20x−4 E= 25x²−105x−34

F =−(3x−5) (3x+ 5)−(8x−1)²

F =−((3x)²−5²)−((8x)²−2×8x×1 + 1²) F =−(9x²−25)−(64x²−16x+ 1)

F =−9x²+ 25−64x²+ 16x−1 F =−73x²+ 16x+ 24

A= (3x+ 5)²

A= (3x)²+ 2×3x×5 + 5²

A= 9x²+ 30x+ 25 B = (2x+ 2) (2x−2) B = (2x)²−2²

B= 4x²−4

C = (−8x+ 8) (−4x+ 9)

C = 32x²+ (−72x) + (−32x) + 72

C = 32x²−104x+ 72 D= (x−6)²

D=x²−2×x×6 + 6² D=x²−12x+ 36

E= (−5x−5) (6x−9) + (3x+ 1)²

E=−30x²+ 45x+ (−30x) + 45 + (3x)²+ 2×3x×1 + 1² E=−30x²+ 15x+ 45 + 9x²+ 6x+ 1

E=−21x²+ 21x+ 46

F = (3x+ 2) (3x−2)−(3x−3)²

F = (3x)²−2²−((3x)²−2×3x×3 + 3²) F = 9x²−4−(9x²−18x+ 9)

F = 9x²−4−9x²+ 18x−9 F = 18x−13

Factoriser les expressions suivantes.

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A= (−4x+ 1) (2x+ 8) + 4x²−64 A= (−4x+ 1) (2x+ 8) + (2x)²−8² A= (−4x+ 1) (2x+ 8) + (2x+ 8) (2x−8) A= (2x+ 8) (−4x+ 1 + 2x−8)

A= (2x+ 8) (−2x−7)

B = (5x−2)²−(10x−3) (5x−2) B = (5x−2) 5x−2−(10x−3) B = (5x−2) (5x−2−10x+ 3)

B= (5x−2) (−5x+ 1)

C= 4x²−64 C= (2x)²−8²

C= (2x+ 8) (2x−8)

D=−(−x+ 9) (−6x+ 5) + (−6x+ 5) (2x−4)

D= (−6x+ 5) −(−x+ 9) + 2x−4 D= (−6x+ 5) (x−9 + 2x−4)

D= (−6x+ 5) (3x−13)

E = 81−(8x−10)² E = 9²−(8x−10)²

E = (9 + 8x−10) 9−(8x−10) E = (9 + 8x−10) (9−8x+ 10)

E = (8x−1) (−8x+ 19)

F = (x−1) (9x−5)−(9x−5) F = (x−1) (9x−5)−(9x−5)×1 F = (9x−5) x−1−1

F = (9x−5) (x−2)

A=−(−x+ 8) (−4x+ 8)−(−4x+ 8) (5x+ 1) A= (−4x+ 8) −(−x+ 8)−(5x+ 1)

A= (−4x+ 8) (x−8−5x−1) A= (−4x+ 8) (−4x−9)

B = 81x²−25 B = (9x)²−5²

B= (9x−5) (9x+ 5)

C= 25x²−9 + (x−4) (5x−3) C= (5x)²−3²+ (x−4) (5x−3) C= (5x−3) (5x+ 3) + (x−4) (5x−3) C= (5x−3) (5x+ 3 +x−4)

C= (5x−3) (6x−1)

D= (−9x+ 1) (8x−10) + (8x−10)²

D= (8x−10) (−9x+ 1 + 8x−10) D= (8x−10) (−x−9)

E = (x−9)²−100 E = (x−9)²−10²

E = (x−9 + 10) (x−9−10) E =

E = (x+ 1) (x−19)

F = (−10x+ 3)−(−10x+ 3) (2x+ 5) F = (−10x+ 3)×1−(−10x+ 3) (2x+ 5) F = (−10x+ 3) 1−(2x+ 5)

F = (−10x+ 3) (1−2x−5) F = (−10x+ 3) (−2x−4)

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A=−(2x−6) (x+ 3) + (2x−6)² A= (2x−6) −(x+ 3) + 2x−6 A= (2x−6) (−x−3 + 2x−6)

A= (2x−6) (x−9)

B =−(3x−1) (8x+ 5) + (−5x−10) (8x+ 5) B = (8x+ 5) −(3x−1)−5x−10

B = (8x+ 5) (−3x+ 1−5x−10) B= (8x+ 5) (−8x−9)

C=−(3x−10) (−x+ 2) + 9x²−100 C=−(3x−10) (−x+ 2) + (3x)²−10²

C=−(3x−10) (−x+ 2) + (3x−10) (3x+ 10) C= (3x−10) −(−x+ 2) + 3x+ 10

C= (3x−10) (x−2 + 3x+ 10) C= (3x−10) (4x+ 8)

D= (−7x−2)²−9 D= (−7x−2)²−3²

D= (−7x−2 + 3) (−7x−2−3) D=

D= (−7x+ 1) (−7x−5)

E = (−4x+ 4)−(−4x+ 4) (−10x+ 7) E = (−4x+ 4)×1−(−4x+ 4) (−10x+ 7) E = (−4x+ 4) 1−(−10x+ 7)

E = (−4x+ 4) (1 + 10x−7) E = (−4x+ 4) (10x−6)

F = 81x²−1 F = (9x)²−1²

F = (9x+ 1) (9x−1)

Ondonne A=−54x+ 9x²+ 81 + (−x+ 5) (−3x+ 9)^. I1. DévelopperetréduireA.

A=−54x+ 81 + 9x²+ (−x+ 5) (−3x+ 9) A= 9x²−54x+ 813x²+ (−9x) + (−15x) + 45 A= 9x²−54x+ 81 + 3x²−24x+ 45

A= 12x²−78x+ 126

I2. Factoriser A.

A= 81 + 9x²−54x+ (−x+ 5) (−3x+ 9) A= 9x²−54x+ 81 + (−x+ 5) (−3x+ 9) A= (−3x+ 9)²+ (−x+ 5) (−3x+ 9) A= (−3x+ 9) (−3x+ 9−x+ 5)

A= (−3x+ 9) (−4x+ 14)

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I3. CalculerA pour x= −3 10 ^.

Noussavonsque A= 12x²−78x+ 126^.^Donc^pourx= −3 10 ^:

A= 12× −3 10

!₂

−78 × −3 10

! + 126

A= 3×4

1 ×

9

25×4+ −39×2

−1×^HH−1×

3×^HH−1 5×2 + 126 A= 27

25+585

25 +3 150 25

A= 3 762 25

I4. Résoudre l'équationA= 0^.

Noussavonsque A= (−3x+ 9) (−4x+ 14)^.^Nous^devons^donc ^résoudre(−3x+ 9) (−4x+ 14) = 0^.

Unproduit defacteurs estnulsignie qu'un desfacteursestnul. Donc:

−3x+ 9 = 0 ^ou −4x+ 14 = 0

−3x=−9 ^ou −4x=−14 x= 9

3 ^ou x= 14

4

Lessolutions decette équation sont 3 ^et 7 2^.

Ondonne A= (−2x+ 4)²+ (−2x+ 4) (5x+ 4)^. I1. DévelopperetréduireA.

A= (−2x+ 4)²+ (−2x+ 4) (5x+ 4)

A= (2x)²−2×2x×4 + 4²+−10x²+ (−8x) + 20x+ 16 A= 4x²−16x+ 16−10x²+ 12x+ 16

A=−6x²−4x+ 32

I2. Factoriser A.

A= (−2x+ 4)²+ (−2x+ 4) (5x+ 4) A= (−2x+ 4) (−2x+ 4 + 5x+ 4)

A= (−2x+ 4) (3x+ 8)

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Noussavonsque A=−6x²−4x+ 32^.^Donc^pour x= −2 9 ^:

A=−6 × −2 9

!₂

−4 × −2 9

! + 32

A= −2×3

1 ×

4

27×3+ −4

−1×^HH−1×

2×^HH−1 9 + 32 A= −8

27 + 24 27+864

27

A= 880 27

Noussavonsque A= (−2x+ 4) (3x+ 8)^.^Nous ^devons^donc ^résoudre(−2x+ 4) (3x+ 8) = 0^.

−2x+ 4 = 0 ^ou 3x+ 8 = 0

−2x=−4 ^ou 3x=−8 x= 4

2 ^ou x= −8

3

Lessolutions decette équation sont 2 ^et −8 3 ^.

Ondonne A= (9x+ 9) (9x−8) + (9x+ 9)²^. I1. DévelopperetréduireA.

A= (9x+ 9) (9x−8) + (9x+ 9)²

A= 81x²+ (−72x) + 81x+ (−72) + (9x)²+ 2×9x×9 + 9² A= 81x²+ 9x−72 + 81x²+ 162x+ 81

A= 162x²+ 171x+ 9

I2. Factoriser A.

A= (9x+ 9) (9x−8) + (9x+ 9)² A= (9x+ 9) (9x−8 + 9x+ 9)

A= (9x+ 9) (18x+ 1)

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Noussavonsque A= 162x²+ 171x+ 9^.^Donc^pourx= −3 5 ^:

A= 162 × −3 5

!₂

+ 171 × −3 5

! + 9

A= 1 458

25 + 171

−1×^HH−1×

3×^HH−1 5 + 9 A= 1 458

25 + −2 565 25 +225

25

A= −882 25

Noussavonsque A= (9x+ 9) (18x+ 1)^.^Nous^devons^donc ^résoudre(9x+ 9) (18x+ 1) = 0^.

9x+ 9 = 0 ^ou 18x+ 1 = 0 9x=−9 ^ou 18x=−1 x= −9

9 ^ou x= −1

18

Lessolutions decette équation sont −1 ^et −1 18 ^.