Exercices Résolus - ChiII PDF

# Exercice 2.1 - A:= (x+3)^2 -5(x+3) - forme développic de A: x^2 + 6x + 9 - 5x - 15 - fame factaisie de A: (x + 3)((x + 3) - 5) - a) x^2 + x - 6 - b) (x + 3)(x - 2) - au ㅂ x ER - non # Exercice 2.2 - 10 x + 15 y = 5(2x + 3y) - xy + 4 y z = y(x + 4z) - a^8 b^4 x ^2 = a^3 b^3 x^2 (a...

# Exercice 2.1 - A:= (x+3)^2 -5(x+3) - forme développic de A: x^2 + 6x + 9 - 5x - 15 - fame factaisie de A: (x + 3)((x + 3) - 5) - a) x^2 + x - 6 - b) (x + 3)(x - 2) - au ㅂ x ER - non # Exercice 2.2 - 10 x + 15 y = 5(2x + 3y) - xy + 4 y z = y(x + 4z) - a^8 b^4 x ^2 = a^3 b^3 x^2 (ab - x^2) - 8x^5 - 6x^2 = 2x^2 (4x^3 - 3) - 3a^3 b^4 - 12 a^2 b^3 = 3a^2 b^3 (ab - 4) - 6 a^2 b c^2 - 15abc^3 = 3abc^2 (2a - 5c) - -30 x^4 y^3 z^2 - 15 x^3 y^3 z^3 = -15 x^3 y^3 z (2x + z^2) - 23 x^6 y^3 - 23 x^5 y^3 + 46 x^2 y^6 = 23 x^4 y^2 (x^2 - x^2 y + 2y^3) - 7 a^2 x^3 y - 21 ax^2 y - 28 x^2 y^2 = 7 x^2 y (a^2 x - 3a - 4y^2) - 2a^3 b^2 + 8ab^3 - 6arb = 2 ab (ab + 4b^2 - 3a) - 3ab(bc)^3 - ab(bc)^2 = ab(bc)^2 (3(bc) - 1) = ab^3 c^2 (3bc - 1) - 4x^nym + 2x^2 y^moz = 2x^nym (2 + x^2 y^2) # Exercice 2.3 - (-)= ♡+ ♡ # Exercice 2.4 - (2a + 3b)(2x + y) + (3a + 5b)(2x + y) = (2x + y)((2a + 3b) + (3a + 5b)) = (2x + y)(5a + 8b) - 2(3 +) - 3(3 +) + 4(3 +) = (3 +)(2 - 3 + 4) - 2a(a - l) - (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)(x(x - 4) - (2x + 1)) = (x - 4)^3 (x^2 - 4x - 2x - 1) = (x - 4)^7 (x^2 - 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) - a^2 (x - 1)(a + b) + a^3(1-x) = a^2 (x - 1)(a + b) - a^3(-1 + x) = a^2 (x - 1)(a + b) - a^3(x - 1) = a^2 (x - 1)((a + b) -a) = a^2 (x - 1) # Exercice 2.5 - (2x + y)^2 = 4x^2 + 4xy + y^2 - (4x - 3y)^2 = 9y^2 - 24xy + 16x^2 - (3ab^2)^2 = 9a^2 b^4 - (x - 4y)^2 = x^2 + 16y^2 - 8xy - (x + 2g)^2 = x^2 + 4xy + 4y^2 - (x - 3)^2 = x^2 - 6x + 9 # Exercice 2.6 - 9x^2 - 12x + 4 = (3x - 2)^2 - 16x^2 - 81 = (4x + 9)(4x - 9) - x^2 - 1/4 = (x + 1/2)(x - 1/2) - 8x^3 - 12x^2 + 6x - 1 = (2x - 1)^3 - 25x^6 - 49 = (5x^3 + 7)(5x^3 - 7) - 27x^3 - 64 = (3x - 4)(9x^2 + 12x + 16) - x^3 - 9x = x(x^2 - 9) = x(x + 3)(x - 3) - (3x - 1)^2 - 15x + 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) - -x^3 + 9x^2 - 27x + 27 = (3 - x)^3 - x^8 - 1 = (x^4 - 1)( + 1) = (x^2 - 1)(x^2 + 1)(x^4 + 1) = (x - 1)(x + 1)(x^2 + 1)(x^4 +1) - 125x^3 + 8y^3 = (5x + 2y)(25x^2 - 10xy + 4y^2) - (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 - x^2 y^3 = (2 - xy^2)(4 + 2xy^3 + x^2 y^6) - a^2 x^6 - 25 = (ax^3 + 5)(ax^3 - 5) - 12ax^2 - 36axy + 27ay^2 = 3a(4x^2 - 12xy + 9y^2) = 3a(2x - 3y)^2 # Exercice 2.7 - 4x^3 + 4x^2 + 7x + 7 = 4x^2(x + 1) + 7(x + 1) = (x + 1)(4x^2 + 7) - a^2 + ac + ab + bc = a(a + c) + b(a + c) = (a + c)(a + b) - x^2(3x - 1) - 3x + 1 = x^2 (3x - 1) -(3x - 1) = (3x - 1)(x^2 - 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) - a^3 + 3a^2 b + 3ab^2 + b^3 - 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) - 6x^2 + xy + 18xz + 3y^2 = x(6x + y) + 3z(6x + y) = (6x + y)(x + 3z) - xy - zy + xu - zu - x^2 + z^2 = (xy - zy) + (xu - zu) - (x^2 - z^2) = y(x - z) + u(x - z) - z(x - z) = (x - z)(y + u - z) - x^2 - y^2 + xa + ya = (x^2 - y^2) + (xa + ya) = (x + y)(x - y) + a(x + y) = (x + y)((x - y) + a) - x^5 + x^4 + x^3 + x^2 + x + 1 = (x^5 + x^4 + x^3) + (x^2 + x + 1) = x^3(x^2 + x + 1) + (x^2 + x + 1) = (x^2 + x + 1)(x^3 + 1) = (x^2 + x + 1)(x + 1)(x^2 - x + 1) = (x + 1)(x^2 - x + 1)(x^2 + x + 1) # Exercice 2.8 - x^2 + 5x + 6 - △ = b^2 - 4ac = 5^2 - 4 - 1 - 6 = 25 - 24 = 1 - x_1,2 = (-b ± √△)/2a = (-5 ± √1)/2 = (-5 ±1)/2 - x_1 = (-5 + 1)/2 = -2 - x_2 = (-5 - 1)/2 = -3 - x^2 + 5x + 6 = (x - (-2))(x - (-3)) = (x + 2)(x + 3) - 2x^2 - 2x - 24 - △= b^2 - 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 - 2x^2 - 2x - 24 = 2(x - 4)(x - (-3)) = 2(x - 4)(x + 3) - 2x^2 + 7x + 10 - △ = b^2 - 4ac = 7^2 - 4 * 2 * 10 = -31 = - 4 * 7.75 < 0 - 2x^2 + 7x + 10 se factaise pas. - 2x^2 + 9x + 7 - △ = b^2 - 4ac = 9^2 - 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 - 2x^2 + 9x + 7 = 2(x - (-1))(x - (-7/2)) = 2(x + 1)(x + 7/2) = (x + 1)(2x + 7) - 6x^2 + 15x + 6 - △ = b^2 - 4ac = 15^2 - 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 - 6x^2 + 15x + 6 = 6(x - (-1/2))(x - (-2)) = 3(2x + 1)(x + 2) - x^2 - 26x + 169 - △= b^2 - 4ac = (-26)^2 - 4 * 1 * 169 = 0 - x_1,2 = (-b ± √△)/2a = (-(-26) ± √0)/2 = 26/2 = 13 - x^2 - 26x + 169 = (x - 13)(x - 13) = (x - 13)^2 - 27x^2 - 75x + 48 - △ = b^2 - 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 - 27x^2 - 75x + 48 = 27(x - 16/9)(x - 5/9) = 3(9x - 16)(x - 1) - 4x^2 + x - 5 - △ = b^2 - 4ac = 1^2 - 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 - 4x^2 + x - 5 = 4(x - 1)(x - (-5/4)) = (x - 1)(4x + 5) - 11x^2 + 28x - 15 - △ = b^2 - 4ac = 28^2 - 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 - 11x^2 + 28x - 15 = 11 x(x - (-3)) = (11x - 5)(x + 3) - 3x^2 + 26x - 9 - △= b^2 - 4ac = 26^2 - 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 - 3x^2 + 26x - 9 = 3 (x - 1/3)(x - (-9)) = (3x - 1)(x + 9) - 4x^2 + 12x + 9 - △ = b^2 - 4ac = 12^2 - 4 * 4 * 9 = 0 - x_1,2 = (-b ± √△)/2a = (-12 ± √0)/8 = -12/8 = -3/2 - 4x^2 + 12x + 9 = 4 (x - (-3/2))(x - (-3/2)) = (2x + 3)(2x + 3) = (2x + 3)^2 - 15x^2 + 225x + 600 - △ = 225^2 - 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 - 15x^2 + 225x + 600 = 15(x - (-677/10))(x - (-823/10)) = 15(x + 677/10)(x + 823/10) # Exercice 2.9 - x^5 + x^3 + x^2 + 1 = (x^5 + x^3) + (x^2 + 1) = x^3(x^2 + 1) + (x^2 + 1) = (x^2 + 1)(x^3 + 1) = (x^2 + 1)(x + 1)(x^2 - x + 1) - 16x^4 - 1 = (4x^2 - 1)(4x^2 + 1) = (2x + 1)(2x - 1)(4x^2 + 1) - 2x^3 + 3x^2 - 8x - 12 = (2x^3 + 3x^2) - (8x + 12) = x^2(2x + 3) - 4(2x + 3) = (2x + 3)(x^2 - 4) = (2x + 3)(x + 2)(x - 2) - a^3 - a + 2a^2 - 2 = (a^3 - a) + (2a^2 - 2) = a(a^2 - 1) + 2(a^2 - 1) = (a^2 - 1)(a + 2) = (a + 1)(a - 1)(a + 2) - (x^2 - 1)^2 - 3(x^2 - 1) = (x^2 - 1)((x^2 - 1) - 3) = (x^2 - 1)(x^2 - 4) = (x + 1)(x - 1)(x + 2)(x - 2) - a^2 x + b^2 z - a^2 z - b^2 x = (a^2 x - a^2 z) + (b^2 z - b^2 x) = a^2(x - z) + b^2(z - x) = a^2(x - z) - b^2(x - z) = (x - z)(a^2 - b^2) = (x - z)(a + b)(a - b) - 2x^2 - 8x - 10 = 2(x^2 - 4x - 5) - △ = b^2 - 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(x^2 - 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) - x^2 - 8x + 16 - 100y^2 = (x^2 - 8x + 16) - 100y^2 = (x - 4)^2 - 100y^2 = (x - 4)^2 - (10y)^2 = (x - 4 + 10y)(x - 4 - 10y) - 27 - 54x + 36x^2 - 8x^3 = (3 - 2x)^3 - x^5 - 5x^3 + x^2 - 1 = (x^5 - x^3) + (x^2 - 1) = x^3(x^2 - 1) + (x^2 - 1) = (x^2 - 1)(x^3 + 1) = (x + 1) (x - 1)(x + 1)(x^2 - x + 1) = (x - 1)(x + 1)^2 (x^2 - x + 1) - 36x^2 - 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 - 2x^3 - 3x^2 + x = x(2x^2 - 3x + 1) - △ = b^2 - 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 - 2x^2 - 3x + 1 = 2(x - 1)(x - 1/2) = (x - 1)(2x - 1) - 2x^3 - 3x^2 + x = x(x - 1)(2x - 1) - x^4 - 4 = (x^2 + 2)(x^2 - 2) = (x^2 + 2)(x + 2)(x - 2) - 8 x^2 y - 4xy - 12y = 4y(2 x^2 - x - 3) - △ = b^2 - 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 - 2x^2 - x - 3 = 2(x - 3/2)(x - (-1)) = (2x - 3)(x + 1) - 8x^2 y - 4xy - 12 y = 4y(2x - 3)(x + 1) - (x + 1)(x^2 + 16) - 8x^2 - 8x = (x + 1)(x^2 + 16) -(8x^2 + 8x) = (x + 1)(x^2 + 16) - 8x(x + 1) = (x + 1)((x^2 + 16) - 8x) = (x + 1)(x^2 - 8x + 16) = (x + 1)(x - 4)^2 - 4x^2 (x - 7) - (4x + 1)(7 - x) = 4x^2(x - 7) + (4x+ 1)(x- 7) = (x - 7)(4x^2 + 4x + 1) = (x - 7)(2x + 1)^2 - 30x^3 + 55x^2 - 10x = 5x(6x^2 + 11x - 2) - △= b^2 - 4ac = 11^2 - 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 - 6x^2 + 11x - 2 = 6(x - 1/6)(x - (-2)) = (6x - 1)(x + 2) - 30x^3 + 55x^2 - 10x = 5x(6x - 1)(x + 2) - 4x^4 - 36x^3 + 108x^2 - 108x = 4x(x^3 - 9x^2 + 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 - x^2) - x + 3 = (x - 3)(26 - x^2) - (x - 3) = (x - 3)(26 - x^2 - 1) = (x - 3)(25 - x^2) = (x - 3)(5 + x)(5 - x) # Exercice 2.10 - x^4 / x^3 = x^(4 - 3) = x - (-x)^-6 / (-x)^3 = (-x)^(-6 - 3)= (-x)^-9 - (x^2)^-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^