Algebra classe 10

Exercice 2.1

L'expression A peut être développée en x² + 6x + 9 - 5x - 15 ou factorisée en (x + 3)((x + 3) - 5).
L'expression x² + x - 6 peut être factorisée en (x + 3)(x - 2).

Exercice 2.2

La factorisation d'expressions algébriques est démontrée à travers plusieurs exemples, y compris:
- 10x + 15y = 5(2x + 3y)
- xy + 4yz = y(x + 4z)
- a⁸b⁴x² = a³b³x²(ab - x²)
- 8x⁵ - 6x² = 2x²(4x³ - 3)
- 3a³b⁴ - 12a²b³ = 3a²b³(ab - 4)
- 6a²bc² - 15abc³ = 3abc²(2a - 5c)
- -30x⁴y³z² - 15x³y³z³ = -15x³y³z(2x + z²)
- 23x⁶y³ - 23x⁵y³ + 46x²y⁶ = 23x⁴y²(x² - x²y + 2y³)
- 7a²x³y - 21ax²y - 28x²y² = 7x²y(a²x - 3a - 4y²)
- 2a³b² + 8ab³ - 6ab² = 2ab(ab + 4b² - 3a)
- 3ab(bc)³ - ab(bc)² = ab(bc)²(3(bc) - 1) = ab³c²(3bc - 1)
- 4xⁿym + 2x²yⁿoz = 2xⁿym(2 + x²y²)

Exercice 2.3

L'exercice demande de simplifier une expression impliquant des symboles mathématiques.

Exercice 2.4

L'exercice utilise la factorisation par regroupement pour simplifier des expressions algébriques, y compris :
- (2a + 3b)(2x + y) + (3a + 5b)(2x + y) = (2x + y)(5a + 8b)
- 2(3 + x) - 3(3 + x) + 4(3 + x) = (3 + x)(2 - 3 + 4)
- 2a(a - b) - (a - b)² = (a - b)(2a - (a - b)) = (a - b)(2a - a + b) = (a - b)(a + b)
- (x - 3)(x + 1) - 2(x - 3) + 2 (x- 3)² = (x - 3)((x+1) - 2 + 2(x - 3)) = (x - 3)(x + 1 - 2 + 2x - 6) = (x - 3)(3x - 7)
- (x - 2)² - (x - 2)(x + 4) + (x + 3)(x - 2) = (x - 2)((x - 2) - (x + 4) + (x + 3)) = (x - 2)(x - 2 - x - 4 + x + 3) = (x - 2)(x - 3)
- x(x - 4)⁸ - (x - 4)⁷(2x + 1) = (x - 4)⁷(x(x - 4) - (2x + 1)) = (x - 4)⁷(x² - 4x - 2x - 1) = (x - 4)⁷(x² - 6x - 1)
- x(x - 7) - 5(7 - x) = x(x - 7) + 5(-7 + x) = x(x - 7) + 5(x - 7) = (x - 7)(x + 5)
- (x - 2y)(a - b) - (b - a)(2x + y) = (x - 2y)(a - b) + (-b + a)(2x + y) = (x - 2y)(a - b) + (a - b)(2x + y) = (a - b)((x - 2y) + (2x + y)) = (a - b)(3x - y)
- (4a - 2b)(2x - 3y) + (3y - 2x)(b - 2a) = (4a - 2b)(2x - 3y) - (-3y + 2x)(b - 2a) = (4a - 2b)(2x - 3y) - (2x - 3y)(b - 2a) = (2x - 3y)((4a - 2b) - (b - 2a)) = (2x - 3y)(4a - 2b - b + 2a) = (2x - 3y)(6a - 3b) = 3(2x - 3y)(2a - b)
- a²(x - 1)(a + b) + a³(1 - x) = a²(x - 1)(a + b) - a³(-1 + x) = a²(x - 1)(a + b) - a³(x - 1) = a²(x - 1)((a + b) - a) = a²(x - 1)b

Exercice 2.5

L'exercice montre comment développer le carré d'une somme ou d'une différence, par exemple:
- (2x + y)² = 4x² + 4xy + y²
- (4x - 3y)² = 9y² - 24xy + 16x²
- (3ab²)² = 9a²b⁴
- (x - 4y)² = x² + 16y² - 8xy
- (x + 2y)² = x² + 4xy + 4y²
- ( x - 3)² = x² - 6x + 9

Exercice 2.6

L'exercice porte sur la factorisation d'expressions algébriques à l'aide de différentes identités remarquables, y compris:
- 9x² - 12x + 4 = (3x - 2)²
- 16x² - 81 = (4x + 9)(4x - 9)
- x² - 1/4 = (x + 1/2)(x - 1/2)
- 8x³ - 12x² + 6x - 1 = (2x - 1)³
- 25x⁶ - 49 = (5x³ + 7)(5x³ - 7)
- 27x³ - 64 = (3x - 4)(9x² + 12x + 16)
- x³ - 9x = x(x² - 9) = x(x + 3)(x - 3)
- (3x - 1)² - (5x + 7)² = ((3x - 1) - (5x + 7))((3x - 1) + (5x + 7)) = (3x - 1 - 5x - 7)(3x - 1 + 5x + 7) = (-2x - 8)(8x + 6) = -2(x + 4)(4x + 3) = -4(x + 4)(4x + 3)
- -x³ + 9x² - 27x + 27 = (3 - x)³
- x⁸ - 1 = (x⁴ - 1)(x⁴ + 1) = (x² - 1)(x² + 1)(x⁴ + 1) = (x - 1)(x + 1)(x² + 1)(x⁴ + 1)
- 125x³ + 8y³ = (5x + 2y)(25x² - 10xy + 4y²)
- (x + 1)² - (2x - 1)² = ((x + 1) - (2x - 1))((x + 1) + (2x - 1)) = (x + 1 - 2x + 1)(x + 1 + 2x - 1) = (-x + 2)(3x) = 3x(2 - x)
- 8 - x²y³ = (2 - xy²)(4 + 2xy³ + x²y⁶)
- a²x⁶ - 25 = (ax³ + 5)(ax³ - 5)
- 12ax² - 36axy + 27ay² = 3a(4x² - 12xy + 9y²) = 3a(2x - 3y)²

Exercice 2.7

L'exercice utilise la factorisation par regroupement pour simplifier des expressions algébriques, y compris:
- 4x³ + 4x² + 7x + 7 = 4x²(x + 1) + 7(x + 1) = (x + 1)(4x² + 7)
- a² + ac + ab + bc = a(a + c) + b(a + c) = (a + c)(a + b)
- x²(3x - 1) - 3x + 1 = x² (3x - 1) -(3x - 1) = (3x - 1)(x² - 1) = (3x - 1)(x + 1)(x - 1)
- 20xy + 4y - 10x - 2 = (20xy + 4y) - (10x + 2) = 4y(5x + 1) - 2(5x + 1) = (5x + 1)(4y - 2) = 2(5x + 1)(2y - 1)
- a³ + 3a²b + 3ab² + b³ - a - b = (a + b)³ - a - b = (a + b)((a + b)² - 1) = (a + b)((a + b) + 1)(a + b - 1) = (a + b)(a + b + 1)(a + b - 1)
- 6x² + xy + 18xz + 3y² = x(6x + y) + 3z(6x + y) = (6x + y)(x + 3z)
- xy - zy + xu - zu - x² + z² = (xy - zy) + (xu - zu) - (x² - z²) = y(x - z) + u(x - z) - z(x - z) = (x - z)(y + u - z)
- x² - y² + xa + ya = (x² - y²) + (xa + ya) = (x + y)(x - y) + a(x + y) = (x + y)((x - y) + a)
- x⁵ + x⁴ + x³ + x² + x + 1 = (x⁵ + x⁴ + x³) + (x² + x + 1) = x³(x² + x + 1) + (x² + x + 1) = (x² + x + 1)(x³ + 1) = (x² + x + 1)(x + 1)(x² - x + 1) = (x + 1)(x² - x + 1)(x² + x + 1)

Exercice 2.8

L'exercice utilise la formule quadratique pour factoriser des expressions polynomiales de degré deux:
- x² + 5x + 6
  - △ = b² - 4ac = 5² - 4 * 1 * 6 = 1
  - x_1,2 = (-b ± √△)/2a = (-5 ± √1)/2 = (-5 ±1)/2
  - x_1 = (-5 + 1)/2 = -2
  - x_2 = (-5 - 1)/2 = -3
  - x² + 5x + 6 = (x - (-2))(x - (-3)) = (x + 2)(x + 3)
2x² - 2x - 24
- △= b² - 4ac = (-2)² - 4 * 2 * (-24) = 196
- x_1,2 = (-b ± √△)/2a = (-(-2) ± √196)/4 = (2 ± 14)/4
- x_1 = (2 + 14)/4 = 4
- x_2 = (2 - 14)/4 = -3
- 2x² - 2x - 24 = 2(x - 4)(x - (-3)) = 2(x - 4)(x + 3)
2x² + 7x + 10
- △ = b² - 4ac = 7² - 4 * 2 * 10 = -31 = - 4 * 7.75 < 0
- 2x² + 7x + 10 se factorise pas.
2x² + 9x + 7
- △ = b² - 4ac = 9² - 4 * 2 * 7 = 25
- x_1,2 = (-b ± √△)/2a = (-9 ± √25)/4 = (-9 ± 5)/4
- x_1 = (-9 + 5)/4 = -1
- x_2 = (-9 - 5)/4 = - 7/2
- 2x² + 9x + 7 = 2(x - (-1))(x - (-7/2)) = 2(x + 1)(x + 7/2) = (x + 1)(2x + 7)
6x² + 15x + 6
- △ = b² - 4ac = 15² - 4 * 6 * 6 = 81
- x_1,2 = (-b ± √△)/2a = (-15 ± √81)/12 = (-15 ± 9)/12
- x_1 = (-15 + 9)/12 = -1/2
- x_2 = (-15 - 9)/12 = -2
- 6x² + 15x + 6 = 6(x - (-1/2))(x - (-2)) = 3(2x + 1)(x + 2)
x² - 26x + 169
- △= b² - 4ac = (-26)² - 4 * 1 * 169 = 0
- x_1,2 = (-b ± √△)/2a = (-(-26) ± √0)/2 = 26/2 = 13
- x² - 26x + 169 = (x - 13)(x - 13) = (x - 13)²
27x² - 75x + 48
- △ = b² - 4ac = (-75)² - 4 * 27 * 48 = 441
- x_1,2 = (-b ± √△)/2a = (-(-75) ± √441)/54 = (75 ± 21)/54
- x_1 = (75 + 21)/54 = 16/9
- x_2 = (75 - 21)/54 = 5/9
- 27x² - 75x + 48 = 27(x - 16/9)(x - 5/9) = 3(9x - 16)(x - 1)
4x² + x - 5
- △ = b² - 4ac = 1² - 4 * 4 * (-5) = 81
- x_1,2 = (-b ± √△)/2a = (-1 ± √81)/8 = (-1 ± 9)/8
- x_1 = (-1 + 9)/8 = 1
- x_2 = (-1 - 9)/8 = -5/4
- 4x² + x - 5 = 4(x - 1)(x - (-5/4)) = (x - 1)(4x + 5)
11x² + 28x - 15
- △ = b² - 4ac = 28² - 4 * 11 * (-15) = 1444
- x_1,2 = (-b ± √△)/2a = (-28 ± √1444)/22 = (-28 ± 38) / 22
- x_1 = (-28 + 38)/22 = 5/11
- x_2 = (-28 - 38)/22 = -3
- 11x² + 28x - 15 = 11 x(x - (-3)) = (11x - 5)(x + 3)
3x² + 26x - 9
- △= b² - 4ac = 26² - 4 * 3 * (-9) = 784
- x_1,2 = (-b ± √△)/2a = (-26 ± √784)/6 = (-26 ± 28)/6
- x_1 = (-26 + 28)/6 = 1/3
- x_2 = (-26 - 28)/6 = -9
- 3x² + 26x - 9 = 3 (x - 1/3)(x - (-9)) = (3x - 1)(x + 9)
4x² + 12x + 9
- △ = b² - 4ac = 12² - 4 * 4 * 9 = 0
- x_1,2 = (-b ± √△)/2a = (-12 ± √0)/8 = -12/8 = -3/2
- 4x² + 12x + 9 = 4 (x - (-3/2))(x - (-3/2)) = (2x + 3)(2x + 3) = (2x + 3)²
15x² + 225x + 600
- △ = 225² - 4 * 15 * 600 = 5044516
- x_1,2 = (-b ± √△)/2a = (-225 ± √5044516)/30 = (-225 ± 2246)/30
- x_1 = (-225 + 2246)/30 = 677/10
- x_2 = (-225 - 2246)/30 = -823/10
- 15x² + 225x + 600 = 15(x - (-677/10))(x - (-823/10)) = 15(x + 677/10)(x + 823/10)

Exercice 2.9

L'exercice met en évidence la factorisation d'expressions polynomiales en utilisant différentes techniques, y compris le regroupement et la factorisation par différences de carrés:
- x⁵ + x³ + x² + 1 = (x⁵ + x³) + (x² + 1) = x³(x² + 1) + (x² + 1) = (x² + 1)(x³ + 1) = (x² + 1)(x + 1)(x² - x + 1)
- 16x⁴ - 1 = (4x² - 1)(4x² + 1) = (2x + 1)(2x - 1)(4x² + 1)
- 2x³ + 3x² - 8x - 12 = (2x³ + 3x²) - (8x + 12) = x²(2x + 3) - 4(2x + 3) = (2x + 3)(x² - 4) = (2x + 3)(x + 2)(x - 2)
- a³ - a + 2a² - 2 = (a³ - a) + (2a² - 2) = a(a² - 1) + 2(a² - 1) = (a² - 1)(a + 2) = (a + 1)(a - 1)(a + 2)
- (x² - 1)² - 3(x² - 1) = (x² - 1)((x² - 1) - 3) = (x² - 1)(x² - 4) = (x + 1)(x - 1)(x + 2)(x - 2)
- a²x + b²z - a²z - b²x = (a²x - a²z) + (b²z - b²x) = a²(x - z) + b²(z - x) = a²(x - z) - b²(x - z) = (x - z)(a² - b²) = (x - z)(a + b)(a - b)
- 2x² - 8x - 10 = 2(x² - 4x - 5)
  - △ = b² - 4ac = (-4)² - 4 * 1 * (-5) = 36
  - x_1,2 = (-b ± √△)/2a = (-(-4) ± √36)/2 = (4 ±6)/2
  - x_1 = (4 + 6)/2 = 5
  - x_2 = (4 - 6)/2 = -1
  - 2(x² - 4x - 5) = 2( x - 5)(x - (-1)) = 2(x - 5)(x + 1)
- 5ab - sa^2 = 5a(b^6 - a^6) = 5a(b^3 + a^3)(b^3 - a^3) = 5a(b + a)(b^2 - ab +a^2)(b - a)(b^2 + ab + a^2) = 5a(b + a)(b - a)(b^2 - ab + a^2)(b^2 + ab + a^2)
- (b - a)x + (a - b)y - 2b + 2a = (b - a)x - (b - a)y -(2b - 2a) = (b - a)x - (b - a)y - 2(b - a) = (b - a)(x - y - 2)
- xy - 2x + 5y - 10 = (xy - 2x) + (5y - 10) = x(y - 2) + 5(y - 2) = (y - 2)(x + 5)
- x² - 8x + 16 - 100y² = (x² - 8x + 16) - 100y² = (x - 4)² - 100y² = (x - 4)² - (10y)² = (x - 4 + 10y)(x - 4 - 10y)
- 27 - 54x + 36x² - 8x³ = (3 - 2x)³
- x⁵ - 5x³ + x² - 1 = (x⁵ - x^3) + (x² - 1) = x^3(x^2 - 1) + (x^2 - 1) = (x^2 - 1)(x³ + 1) = (x + 1) (x - 1)(x + 1)(x^2 - x + 1) = (x - 1)(x + 1)² (x^2 - x + 1)
- 36x² - 84x + 49 = (6x - 7)²
- (2x - 5)(4x - 7) - 3(5 - 2x) = (2x - 5)(4x - 7) + 3(2x - 5) = (2x - 5)((4x - 7) + 3) = (2x - 5)(4x - 4) = 4(x - 1)(2x - 5)
- 4 + (xy)/x + x/4 = 4 + y + x/4
- 2x³ - 3x² + x = x(2x² - 3x + 1)
  - △ = b² - 4ac = (-3)² - 4 * 2 * 1 = 1
  - x_1,2 = (-b ± √△)/2a = (-(-3) ± √1)/4 = (3 ± 1)/4
  - x_1 = (3 + 1)/4 = 1
  - x_2 = (3 - 1)/4 = 1/2
  - 2x² - 3x + 1 = 2(x - 1)(x - 1/2) = (x - 1)(2x - 1)
  - 2x³ - 3x² + x = x(x - 1)(2x - 1)
- x⁴ - 4 = (x² + 2)(x² - 2) = (x² + 2)(x + 2)(x - 2)
- 8 x² y - 4xy - 12y = 4y(2x² - x - 3)
  - △ = b² - 4ac = (-1)² - 4 * 2 * (-3) = 25
  - x_1,2 = (-b ± √△)/2a = (-(-1) ± √25)/4 = (1 ± 5)/4
  - x_1 = (1 + 5)/4 = 3/2
  - x_2 = (1 - 5)/4 = -1
  - 2x² - x - 3 = 2(x - 3/2)(x - (-1)) = (2x - 3)(x + 1)
  - 8x² y - 4xy - 12 y = 4y(2x - 3)(x + 1)
- (x + 1)(x² + 16) - 8x² - 8x = (x + 1)(x² + 16) -(8x² + 8x) = (x + 1)(x² + 16) - 8x(x + 1) = (x + 1)((x² + 16) - 8x) = (x + 1)(x² - 8x + 16) = (x + 1)(x - 4)²
- 4x² (x - 7) - (4x + 1)(7 - x) = 4x²(x - 7) + (4x+ 1)(x- 7) = (x - 7)(4x^2 + 4x + 1) = (x - 7)(2x + 1)²
- 30x³ + 55x² - 10x = 5x(6x² + 11x - 2)
  - △= b² - 4ac = 11² - 4 * 6 * (-2) = 169
  - x_1,2 = (-b ± √△)/2a = (-11 ± √169)/12 = (-11 ± 13)/12
  - x_1 = (-11 + 13)/12 = 1/6
  - x_2 = (-11 - 13)/12 = -2
  - 6x² + 11x - 2 = 6(x - 1/6)(x - (-2)) = (6x - 1)(x + 2)
  - 30x³ + 55x² - 10x = 5x(6x - 1)(x + 2)
- 4x⁴ - 36x³ + 108x² - 108x = 4x(x³ - 9x² + 27x - 27) = 4x(x - 3)^3
- 16(x + 3) - x^5 - 3x^4 = 16(x + 3) - (x^5 + 3x^4) = 16(x + 3) - x^4(x + 3) = (x + 3)(16 - x^4) = (x + 3)(4^2 - (x^2)^2) = (x + 3)(4 + x^2)(4 - x^2) = (x + 3)(4 + x^2)(2 +x)(2 - x)
- 4(x - 1)^3 - (2x + 2)(x - 1)^2 = (x - 1)^2(4(x - 1) - (2x + 2)) = (x - 1)^2(4x - 4 - 2x - 2) = (x - 1)^2 (2x - 6) = 2(x - 3)(x - 1)^2
- 12 + 4x - 3x^2 - x^3 = (12 + 4x) -(3x^2 + x^3) = 4(3 + x) - x^2(3 + x) = (3 + x)(4 - x^2) = (3 + x)(2 + x)(2 - x)
- (x - 3)(26 - x²) - x + 3 = (x - 3)(26 - x²) - (x - 3) = (x - 3)(26 - x² - 1) = (x - 3)(25 - x^2) = (x - 3)(5 + x)(5 - x)

Exercice 2.10

L'exercice utilise les règles des exposants pour simplifier des expressions algébriques, par exemple:
- x⁴/x³ = x^(4 - 3) = x
- (-x)^-6 / (-x)^3 = (-x)^(-6 - 3)= (-x)^-9
- (x²)^-2 = x^(-2 * 2) = x^-4
- (x^1/2)^4 = x^(1/2 * 4) = x^2
- √x ^3 = x^(3/2)
- √x^5 / √x^3 = x^(5 / 2) / x^(3 / 2) = x^((5/2) - (3/2)) = x
- √x^6 / √x^4 = x^(6 / 2) / x^(4 / 2) = x^(3 - 2) = x
- √x^(9 / 2) / √x^(11 / 2) = x^(9 / 4) / x^(11 / 4) = x^((9 / 4) - (11 / 4)) = x^(-1 / 2) = 1/√x
- 22^2 a^2 b^6 c^3 / 77^7 a^5 b^3 c^3 = (2^2 / 7^7)(a^2 / a^5)(b^6 / b^3) = 4/823543 a^(-3)b^3 = 4b^3 / 823543 a^3
- 64x^3 y^3 z^2 / 88x^4 y^7 z^9 = (64 / 88)(x^3 / x^4)(y^3 / y^7)(z^2 / z^9) = 8/11 x^(-1) y^(-4) z^(-7) = 8 / 11xy^4 z^7
- 54a^2 xy / 54a^2 xy = 1
- 18a^2b^2 * x^-4 * y^-3 / 128a^4 * b^-5 * u^-7 = (18 / 128) (a^2 / a^4)(b^2 / b^-5)(x^-4)(y^-3)(u^-7) = 9/64 a^-2 b^7 x^-4 y^-3 u^-7