Enseignement soutien scolaire aide aux devoirs

Factoriser

EX. 1

3x + 7x ... x
4y + 3xy ...
y
-2x + 2y ...
2
3 x2 + 5x ...
x
6x2 + 12x ...
6x
12x - 6xy ... 6x
(x + 2)(3 - x) + (3 - x)(2x + 5) ...
3 - x
2(x + 7) - (x + 7)2 ...
x + 7
(x - 3)(4x +7) + 7(x - 3) ...
x - 3
3xy2 + 9x2y ...
3xy

EX. 2
4x + 8 ....
4(x + 2)
(3x + 4)(x + 2) + ( x +2)(2x - 3) ...
(x + 2)[(3x + 4) + (2x - 3)] = (x + 2)(3x + 4 + 2x - 3) = (x + 2)(5x + 1)
(x + 2)(5x - 2) + 5x + 10 ...
(x + 2)(5x - 2) + 5(x + 2) = (x + 2)(5x - 2 + 5) = (x + 2)(5x + 3)
(x + 2)(2x - 3) - (4x + 3)(x + 2) ...
(x + 2)[(2x - 3) - (4x + 3)] = (x + 2)(2x - 3 - 4x - 3) = (x + 2)(-2x - 6)
(3x - 9)(x + 2) - 6x - 12 ...
(3x - 9)(x + 2) - 6(x + 2) = (x + 2)[(3x - 9) - 6] = (x + 2)(3x - 15)
3(x + 2)(x - 3) + 2(x - 7)(x + 2) ...
(x + 2)[3(x - 3) + 2(x - 7)] = (x + 2)(3x - 9 + 2x - 14) = (x + 2)(5x -23)

EX. 3

a) (x + 2)(x + 1) + (3x - 4)(x + 2)
(x + 2)[(x + 1) + (3x - 4)]
(x + 2)(x + 1 + 3x -4) = (x + 2)(4x -3)

b) (2x +3)(3 - x) - (3 -x)2
(3 - x)[(2x + 3) - (3 - x)]
(3 - x)(2x + 3 - 3 + x) = 3x(3 -x)

c) (7 - x)(5x + 2) - 3(7 - x)
(7 - x)[(5x + 2) - 3]
(7 - x)(5x + 2 - 3) = (7 - x)(5x - 1)

d) 3x - 9 + (3x + 9)(x - 3)
3(x - 3) + (3x + 9)(x - 3)
(x - 3)[3 + (3x + 9)] = (x - 3)(3x + 9)

EX. 4

a) (3x - 1)(3x + 2) - (4x + 3)(2 + 3x)
(3x + 2)[(3x - 1) - (4x + 3)]
(3x + 2)(3x - 1 - 4x -3) = (3x + 2)(-x - 4) = -(3x + 2)(x + 4)

b) x5 + 5x3 - x2y - 3x2
x2 ( x3 + 5x - y - 3)

c) 9 - (2x - 3)2
Soit a2 - b2 = (a + b)(a - b)
9 - (2x - 3)2 = 32 - (2x - 3)2 = [3 + (2x - 3)][3 - (2x - 3] = 2x(6 -2x)

d) 9x2 + 24x + 16
Soit a2 + 2ab + b2 = (a + b)(a + b)
9x2 + 24x + 16 = (3x + 4)(3x + 4)

EX. 5

a) 9 - 30x + 25x2
Soit a2 - 2ab + b2 = (a - b)(a - b)
9 - 30x + 25x2 = (3 - 5x)(3 - 5x)

b) (x - 3)2 - ( x2 - 9)
Soit a2 - b2 = (a + b)(a - b)
(x - 3)(x - 3) - (x + 3)(x - 3)
(x -3)[(x - 3) - (x + 3)] = (x -3)(-6) = -6(x -3)

c) 4 - x2 / 25
Soit a2 - b2 = (a + b)(a - b)
4 - x2 / 25 = 22 - x2 / 25 = (2 + x / 5)(2 - x / 5)

d) (5x - 1)(3x - 7) + (1 - 5x)(5x - 3)
(5x - 1)(3x - 7) - (5x - 1)(5x - 3)
(5x -1)[(3x - 7) - (5x -3)]
(5x -1)(3x - 7 -5x +3) = (5x -1)(-2x - 4) = -(5x -1)(2x + 4)

EX. 6

a) (3x - 2) + (7x +1)(3x - 2)
Mettons (3x -2) en facteur.
(3x - 2)(1 + 7x + 1) = (3x - 2)(7x + 2)

b) (3x +4)(5 -x) - (5 - x)
Mettons (5 - x) en facteur.
(5 - x)(3x + 4 - 1) = (5 - x)(3x + 3)

c) (4x + 1)2 - (3x - 2)(4x + 1)
Mettons (4x + 1) en facteur.
(4x + 1)[(4x + 1) - (3x - 2)]
((4x + 1)(4x + 1 - 3x + 2) = (4x + 1)(x + 3)

d) (x - 3)(3x - 1) - (3x - 1)2
Mettons (3x - 1) en facteur.
(3x - 1)[(x - 3) - (3x - 1)]
((3x - 1)(x - 3 - 3x + 1) = (3x - 1)(-2x - 2)
-2(3x - 1)(x + 1)

EX. 7
(2x + 3)(3x + 5) + (2x + 3)(5x + 3) = (2x + 3)(8x + 8)
(3x - 5)2 + (2x - 4)(3x - 5) =
(3x - 5)(5x - 9)
(2x + 5)2 + 2x + 5 =
(2x + 5)(2x + 6)
(7x + 5)(7x - 3) - (7x + 5)(5x +2) =
(7x + 5)(2x - 5)
(6x + 5)2 - (6x + 5)(4x - 3) =
(6x + 5)(2x + 8)
(x + 5)2 - (x + 5) =
(x + 5)(x + 4)
(7x + 8)2 - (3x + 4)2 =
(10x + 12)(4x + 4)
(3x + 4)2 - 36 =
(3x + 10)(3x - 2)
81 - (2x - 3)2 =
2x(6 - 2x)

EX. 8
4x2 - 9 = (2x +3)(2x -3) ... Vrai
x2 - 1 = (x -1)2
... Faux
9x2 - 25 = (3x - 5)2
... Faux
x2 + 16 = (x + 4)2
... Faux
9x2 + 12x + 4 = (3x + 2)2 ...
Vrai
x2 - 2x + 1 = (x- 1)(x -1) ...
Vrai
x2 - 1 / 9 = ( x + 1 / 3)(x - 1 / 3) ...
Vrai
x2 / 4 + 1 / 16 = ( x / 2 + 1 / 4)2
... Faux

EX. 9
A = 9x2 - 16. Utilisons l'identité remarquable a2 - b2 = (a + b)(a - b).
A = 9x2 - 16 = (3x + 4)(3x - 4)

B = 9x2 - 16 + (3x +4)(2x -3). Nous savons que 9x2 - 16 = (3x + 4)(3x - 4) d'où:
B = (3x + 4)(3x - 4) + (3x +4)(2x -3) = (3x + 4)(3x - 4 + 2x - 3) = (3x + 4)(5x - 7)

(3x + 4)(5x - 7) = 0. Un produit de facteurs est nul si l'un des facteurs au moins est nul. Il existe donc 2 solutions:
3x + 4 = 0 donc x = -4/3 et 5x - 7= 0 donc x = 7/5
-4/3 et 7/5 ne sont pas des nombres entiers.
7/5 = 1,4 est une solution décimale.

EX. 10

A = 4(6 - 3x)(2x + 1) - (4 - 2x)
A = (24 - 12x)(2x +1) - (4 - 2x)
A = 6(4 - 2x)(2x +1) - (4 - 2x)
A = (4 - 2x)(12x + 6) - (4 - 2x)
A = (4 - 2x)[(12x + 6) - 1]
A = (4 - 2x)(12x + 5) ou (2 - x)(24x + 10)

B = 3(8 - 4x)(3x - 5) - (12 - 6x)
B = (24 - 12x)(3x - 5) - (12 - 6x)
B = 2(12 - 6x)(3x - 5) - (12 - 6x)
B = (12 - 6x)(6x - 10) - (12 - 6x)
B = (12 - 6x)[(6x - 10) - 1]
B = (12 - 6x)(6x - 11)

EX. 11

A = (2 + 3x)(2x - 3) - (3 - 2x)(5 + 12x)
A = (2 + 3x)(2x - 3) + [-1(3 - 2x)(5 + 12x)]
A = (2 + 3x)(2x - 3) + (2x - 3)(5 + 12x)
A = (2x - 3)[(2 + 3x) + (5 + 12x)]
A = (2x - 3) (7 + 15x)

B = (2x - 1)(8x + 4) - (14x + 7)(5 - 2x)
B = 4(2x - 1)(2x + 1) - 7(2x + 1)(5 - 2x)
B = (8x - 4)(2x + 1) - (2x + 1)(35 - 14x)
B = (2x + 1)[(8x - 4) - (35 - 14x)]
B = (2x + 1)(22x - 39)

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