9 dic 2007
Diferencia de cubos
27x3 - 8y3
= (3x)3 – (2y)3
= (3x – 2y) (9x2 + 6xy + 4y2)
x9 – y12
= (x3)3 – (y4)3
= (x3- y4)(x6+x3y4 + y8)
y3-23
= (y – 2)(y2 + 2y + y2)
x3y3 – 1
= (xy - 1)(x2y2 + xy + 1)
m3 – 8n3
= (m- 2n)(m2 + 2nm + 4n2)
Suma de cubo
27x3 + 8y3
= (3x)3 + (2y)3
= (3x + 2y) (9x2 - 6xy + 4y2)
x9 + y12
= (x3)3 + (y4)3
= (x3+ y4)(x6 - x3y4 + y8)
y3+ 23
= (y + 2)(y2 - 2y + y2)
x3y3 + 1
= (xy + 1)(x2y2 - xy + 1)
m3 + 8n3
= (m+ 2n)(m2 - 2nm + 4n2)
= (3x)3 – (2y)3
= (3x – 2y) (9x2 + 6xy + 4y2)
x9 – y12
= (x3)3 – (y4)3
= (x3- y4)(x6+x3y4 + y8)
y3-23
= (y – 2)(y2 + 2y + y2)
x3y3 – 1
= (xy - 1)(x2y2 + xy + 1)
m3 – 8n3
= (m- 2n)(m2 + 2nm + 4n2)
Suma de cubo
27x3 + 8y3
= (3x)3 + (2y)3
= (3x + 2y) (9x2 - 6xy + 4y2)
x9 + y12
= (x3)3 + (y4)3
= (x3+ y4)(x6 - x3y4 + y8)
y3+ 23
= (y + 2)(y2 - 2y + y2)
x3y3 + 1
= (xy + 1)(x2y2 - xy + 1)
m3 + 8n3
= (m+ 2n)(m2 - 2nm + 4n2)
factorizacion de trinomio por la forma ax2 + bx + c
4x2 + 8x + 3
= 4x2 + 6x + 2x + 3
= (4x2 + 6x) + (2x + 3)
= 2x(2x + 3) + (2x + 3)
= (2x + 1)(2x + 3)
2x2 + 5x + 3
= 2x2 + 3x + 2x + 3
= x(2x+3) + (2x + 3)
= (2x + 3) ( x + 1)
6x2 + 7x + 2
= 6x2 + 4x + 3x + 2
=2x(3x + 2)+(3x + 2)
=(2x + 1)(3x + 2)
6x2 + 5x - 4
= 6x2 + 8x - 3x - 4
= (6x2 + 8x) - (3x + 4)
= 2x(3x + 4) - (3x + 4)
= (2x - 1)(3x + 4)
6x2+17x+12
= 6x2 + 8x + 9x + 12
= (6x2 + 8x) + (9x + 12)
= 2x(3x + 4) + 3(3x + 4)
= (2x + 3)(3x + 4)
= 4x2 + 6x + 2x + 3
= (4x2 + 6x) + (2x + 3)
= 2x(2x + 3) + (2x + 3)
= (2x + 1)(2x + 3)
2x2 + 5x + 3
= 2x2 + 3x + 2x + 3
= x(2x+3) + (2x + 3)
= (2x + 3) ( x + 1)
6x2 + 7x + 2
= 6x2 + 4x + 3x + 2
=2x(3x + 2)+(3x + 2)
=(2x + 1)(3x + 2)
6x2 + 5x - 4
= 6x2 + 8x - 3x - 4
= (6x2 + 8x) - (3x + 4)
= 2x(3x + 4) - (3x + 4)
= (2x - 1)(3x + 4)
6x2+17x+12
= 6x2 + 8x + 9x + 12
= (6x2 + 8x) + (9x + 12)
= 2x(3x + 4) + 3(3x + 4)
= (2x + 3)(3x + 4)
factorizacion de trinomio cubico perfecto
(a+b)3 = a3 + 3a2b + 3ab2 = b3
x3 +6x2 + 12x + 8
= x3 +3(x2)(2)+ 3(2)2x + (2)3
= (x+2)3
-27x3 + 54x2y - 36xy2 + 8y3
= (-3x)3 + 3(-3x)2(2y) + 3(-3x)(2y)2 + (2y)3
= (-3x + 2y)3
-125x3 – 300x2y – 240xy2 - 64y2
= (-5x)3 + 3(-5x)2(-4y) + 3(-5x)(4y)2 + (4y)3
= (-3x + 2y)3
8x3 – 36x2 – 54x - 27
= (-2x)3 + 3(-2x)2(-3) + 3(-2x)(-3)2 + (-3)3
= (-2x - 3)3
64a3 + 48a + 12a + 1
= (4a)3 + 3(4a)2(1) + 3(4a)(1)2 + (4a)3
= (4a + 1)3
x3 +6x2 + 12x + 8
= x3 +3(x2)(2)+ 3(2)2x + (2)3
= (x+2)3
-27x3 + 54x2y - 36xy2 + 8y3
= (-3x)3 + 3(-3x)2(2y) + 3(-3x)(2y)2 + (2y)3
= (-3x + 2y)3
-125x3 – 300x2y – 240xy2 - 64y2
= (-5x)3 + 3(-5x)2(-4y) + 3(-5x)(4y)2 + (4y)3
= (-3x + 2y)3
8x3 – 36x2 – 54x - 27
= (-2x)3 + 3(-2x)2(-3) + 3(-2x)(-3)2 + (-3)3
= (-2x - 3)3
64a3 + 48a + 12a + 1
= (4a)3 + 3(4a)2(1) + 3(4a)(1)2 + (4a)3
= (4a + 1)3
factorizacion de trinomio perfecto
x2 + 2x + 1
= (x)2 + 2(x)(1) + (1)
= (x + 1)2
x2 – 2x + 1
= (x)2 – 2(x)(1) + (1)
= (x - 1)2
x2 + 4x + 4
= (x)2 + 2(x)(2) + (2)2
= (x + 2)2
x2 - 4x + 4
= (x)2 - 2(x)(2) + (2)2
=(x - 2)2
x2 + 8x + 16
= (x)2 + 2(x)(4) + (4)2
=(x + 4)2
= (x)2 + 2(x)(1) + (1)
= (x + 1)2
x2 – 2x + 1
= (x)2 – 2(x)(1) + (1)
= (x - 1)2
x2 + 4x + 4
= (x)2 + 2(x)(2) + (2)2
= (x + 2)2
x2 - 4x + 4
= (x)2 - 2(x)(2) + (2)2
=(x - 2)2
x2 + 8x + 16
= (x)2 + 2(x)(4) + (4)2
=(x + 4)2
8 dic 2007
6 dic 2007
operaciones fundamentales algebraicas
SUMAS
1.- (5x + 9y - 3m + 4z) + (7x - 2y + 6z + 11m) = 12x + 7y + 8m + 10z
2.- (9x² + 3yz + 5y²) + (-7x² + 3yz - 7y²) = 2x2 + 6yz - 2y2
3.- (7.3a - 5.1b - 0.3c) + (9.1a + 4.5b - 8.4c) = 16.4a - 0.6b - 8.7c
4.- (9.5x - 4.1y + 7.6z) + (5.9x + 9.4y - 1.7z) = 15.4x + 5.3y + 5.9z
5.- 2x/5y + 3x/4y = 23x/20y
RESTAS
1.- (5x + 3y - 2z) – (9x - 5y + 7z) = -4x + 8y – 9z
2.- (5x²y + 3x²y² + 8xy²) - (3x²y - 4x²y²) = 2x2y + 7x2y2 + 8xy2
3.- (4.6a - 2.7b + 8.3c) – (4.1a - 7.6b - 1.9c) = 0.5a + 4.9b +10.2c
4.- (2.3a+7.6b-8.1c) – (3.4a-8.1b+1.7c) = 1.1a + 15.7b – 9.8c
5.- 2a/5c – 9a/10c = -5a/10c = -a/2c
MULTIPLICACION
1.- (7a-2b+9c) (2a+5b) = 14a2 + 31ab + 18ac – 10b2 + 45bc
2.- (8x3y2 – 9x2y – 2x) (5x2y – 6x) = 40x5y2 – 93x4y2 + 44x3y + 12x2
3.- (4.5m3 – 3.2m2) (-2.4m2 – 5.1m) = -10.8m5 – 15.27m4 + 16.32m3
4.- (4.5a2 – 7.4a + 5.9b) (2.3b – 3.5c) =
10.35a2b – 17.02ab + 13.57b2 – 15.75a2c + 25.9ac – 20.65bc
5.- (5x/9b) (8x/2b) = 40x2/18b2 = 20x2/9b2
DIVISIONES
1.- (4x² - 2x -12) : (x-2) = 4x + 6
2.- (3x² + 14x + 8) : (x + 4) = 3x + 2
3.- (3.1x2 – 8.4x + 4.4) : (x - 2) = 3.1x – 2.2
4.- (6.3x² + 5.1x - 6.3) : (3x - 2) = 2.1x + 3.1
5.- (2a/5b+a/2b) : (4a/8b-3a/4b) =
9a/10b : -2a/8b = 72ab/-20ab = 18/-5
POTENCIACION
1.- (2x2y3z4)2 = 4x4y6z8
2.- (6x2y3)2 = 36x4y6
3.- (-2.4x)2 = 5.76x2
4.- (2x/3y)2 = 4x2/9y2
5.- (3x/4y)3 =27x3/64y3
OBTENER LA RAIZ CUADRADA
1.- 2500x4 = 50x2
2.- 676a6 = 26a3
3.- 256x2 = 16x
4.- 2209y8 = 47y4
5.- 1764a12 = 42a6
1.- (5x + 9y - 3m + 4z) + (7x - 2y + 6z + 11m) = 12x + 7y + 8m + 10z
2.- (9x² + 3yz + 5y²) + (-7x² + 3yz - 7y²) = 2x2 + 6yz - 2y2
3.- (7.3a - 5.1b - 0.3c) + (9.1a + 4.5b - 8.4c) = 16.4a - 0.6b - 8.7c
4.- (9.5x - 4.1y + 7.6z) + (5.9x + 9.4y - 1.7z) = 15.4x + 5.3y + 5.9z
5.- 2x/5y + 3x/4y = 23x/20y
RESTAS
1.- (5x + 3y - 2z) – (9x - 5y + 7z) = -4x + 8y – 9z
2.- (5x²y + 3x²y² + 8xy²) - (3x²y - 4x²y²) = 2x2y + 7x2y2 + 8xy2
3.- (4.6a - 2.7b + 8.3c) – (4.1a - 7.6b - 1.9c) = 0.5a + 4.9b +10.2c
4.- (2.3a+7.6b-8.1c) – (3.4a-8.1b+1.7c) = 1.1a + 15.7b – 9.8c
5.- 2a/5c – 9a/10c = -5a/10c = -a/2c
MULTIPLICACION
1.- (7a-2b+9c) (2a+5b) = 14a2 + 31ab + 18ac – 10b2 + 45bc
2.- (8x3y2 – 9x2y – 2x) (5x2y – 6x) = 40x5y2 – 93x4y2 + 44x3y + 12x2
3.- (4.5m3 – 3.2m2) (-2.4m2 – 5.1m) = -10.8m5 – 15.27m4 + 16.32m3
4.- (4.5a2 – 7.4a + 5.9b) (2.3b – 3.5c) =
10.35a2b – 17.02ab + 13.57b2 – 15.75a2c + 25.9ac – 20.65bc
5.- (5x/9b) (8x/2b) = 40x2/18b2 = 20x2/9b2
DIVISIONES
1.- (4x² - 2x -12) : (x-2) = 4x + 6
2.- (3x² + 14x + 8) : (x + 4) = 3x + 2
3.- (3.1x2 – 8.4x + 4.4) : (x - 2) = 3.1x – 2.2
4.- (6.3x² + 5.1x - 6.3) : (3x - 2) = 2.1x + 3.1
5.- (2a/5b+a/2b) : (4a/8b-3a/4b) =
9a/10b : -2a/8b = 72ab/-20ab = 18/-5
POTENCIACION
1.- (2x2y3z4)2 = 4x4y6z8
2.- (6x2y3)2 = 36x4y6
3.- (-2.4x)2 = 5.76x2
4.- (2x/3y)2 = 4x2/9y2
5.- (3x/4y)3 =27x3/64y3
OBTENER LA RAIZ CUADRADA
1.- 2500x4 = 50x2
2.- 676a6 = 26a3
3.- 256x2 = 16x
4.- 2209y8 = 47y4
5.- 1764a12 = 42a6
productos notables
1.- (2x - 7y)² =4x2 – 28xy + 49y2
2.- (2m + 3n)³ = 8m3 + 36m2n + 54mn2 + 27n3
3.-- (x + y)7 =x7 + 7x6y + 21x5y2 + 35x4y3 + 35x3y4 + 21x2y5 + 7xy6 + y7
4.- (6x² + m²x) (6x² - m²x) = 36x4 – m4x2
5.- (6a + 7) (6a + 8) = 36a2 + 90a + 56
6.- (2a - 5b) (4x - 9y + 14z) =
8ax – 18ay + 28az – 20bx + 45by – 70bz
2.- (2m + 3n)³ = 8m3 + 36m2n + 54mn2 + 27n3
3.-- (x + y)7 =x7 + 7x6y + 21x5y2 + 35x4y3 + 35x3y4 + 21x2y5 + 7xy6 + y7
4.- (6x² + m²x) (6x² - m²x) = 36x4 – m4x2
5.- (6a + 7) (6a + 8) = 36a2 + 90a + 56
6.- (2a - 5b) (4x - 9y + 14z) =
8ax – 18ay + 28az – 20bx + 45by – 70bz
5 dic 2007
ecuacione de primer grado con una incognita
1)2x+3y=12 2)6x+8y=32 3)3x+6y=12
2x=15-3 6x=32-8 3x=12-6
2x=12 6x=24 3x=6
x=12/2 x=24/6 x=6/3
x=6 x=4 x=2
4)4x+6=14 5)2x+3=15
4x=14-6 2x=15-3
4x=8 2x=12
x=8/4 x=12/2
x=2 x=6
2x=15-3 6x=32-8 3x=12-6
2x=12 6x=24 3x=6
x=12/2 x=24/6 x=6/3
x=6 x=4 x=2
4)4x+6=14 5)2x+3=15
4x=14-6 2x=15-3
4x=8 2x=12
x=8/4 x=12/2
x=2 x=6
factorizacion
1. - x²-x=x (x+1)
2. - x+xy=x (1+y)
3. - 6x²+ 18x³=6x² (1+3x)
4.- x³y²-x³y=x³y (y-1)
5. - 10a²b-15ab²+5ab=5ab (2ª-3b+1)
2. - x+xy=x (1+y)
3. - 6x²+ 18x³=6x² (1+3x)
4.- x³y²-x³y=x³y (y-1)
5. - 10a²b-15ab²+5ab=5ab (2ª-3b+1)
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