martes, 31 de mayo de 2016

CUBO DEL BINOMIO

Corresponde al cubo del primer término, más (o menos) el triple del cuadrado del primer término multiplicado por el segundo, más el triple del primer término multiplicado por el cuadrado del segundo y más (o menos) el cubo del segundo, así:


CUBO DE UN BINOMIO.
(a +- b)³ = a³ +- 3a²b + 3ab² +- b³


Ejercicios: CUBO DEL BINOMIO

    1. (a + 3)³= a³ + 3(a)²(3) + 3(a)(3)² + (3)³
    = 27 + 9a² + 27a + 27

  1. 2. (p – q)³ = p³ – 3(p)²(q) + 3(p)(q)² – q³
    = p³ – 3p²q + 3pq² - q³

    3. (x + 2)³ = x³ + 3(x)²(2) + 3(x)(2)² + 2³
    = x³ + 6x² + 12x + 8

  1. 4. (a – 3)³ = a³ + 3(a)²(3) + 3(a)(3)²+ (3)³
    = a³ + 9a² + 27a + 27

  1. 5. (t + 4)³ = t³ + 3(t)²(4) + 3(t)(4)² + (4)³ =
    = t³ + 3(t)²(4) + 3(t)(4)² + (4)³
    = t³ + 12t² + 48t + 64

  1. 6. (2 – a)³ = 2³ – 3(2)²(a) + 3(2)(a)² – a³
    = 8 – 12a + 6a² - a³

  1. 7. (2a – b)³ = (2a)³ -3(2a)²(b)+3(2a)(b)²- b³ =8a³ – 3(4a²)b + 6ab² – b³
    = 8a³ – 12a²b + 6ab² – b³

  1. 8. (3a - 5b)³ = (3a)³-3(3a)²(5b)+3(3a)(5b)²-(5b)³
    = 27a³ - 135a²b + 225ab² - 125b³


9. (2x + 3y)³=(2x)³+3(2x)²(3y)+3(2x)(3y)²+(3y)³ = 8x³ + 36 x²y + 54xy² + 27y³


10. (1 – 3y)³ = (1)³ – 3(1)²(3y)+3(1)(3y)²- (3y)³ = 1 – 9y + 27y² - 27y³


11. (2 + 3t)³ = 2³ + 3 (2)²(3t) + 3(2)(3t)² + (3t)³ = 8 + 36t + 54t² +27t³

  1. 12. (3a –2x)³=(3a)³–3(3a)²(2x)+3(3a)(2x)²-(2x)³ = 27a³ – 54a²x + 36ax² – 8x³


13. (5a – 1)³= (5 a)³–3(5a)²(1)+3(5a)(1)² - (1)³ =125 a³ – 75a² + 15a - 1


14. (3a²-2a)³=(3a²)³–3(3a²)²(2a)+3(3a²)(2a)²-(2a)³ 
= (27x6) - 3(9x4) ²(2a)+(9a²)(4a²(8a³)
 = (27x6) - (54x5) +(36a4) (8a³)


  1. 15. (t² + t³)³ =(t²)³+ 3(t²)²(t³) + 3(t²)(t³)² + (t³)³
    = t + 3tt³ + 3t²t + t
    = t⁶ + 3t⁷ + 3t⁸ + t⁹

  1. 16. ( 1 + x)³= (1)³ + 3(1)²(x) + 3(1)(x)² + (x
    = 1 + 3x + 3x + x¹²

  1. 17. (2t – 3a²)³=(2t)³–3(2t)²(3a²)+3(2t)(3a²)²– (3a²)³ =
    = 8t³ – 36t²a² + 54ta - 27a

18. (u² +5v)³=(u²)³+3(u²)²(5v)+3(u²)(5v)²+ (5v)³
= u + 15u⁴5v + 75u²v² - 125v³

  1. 19. ( ½ - a)³ = (½ - 3(1/2)²(a) +3(1/2)(a)² – a³
= 1/8 – 3/4 + 3/2 a² – a³


20. (1/2x + 2y)³=(1/2x)³+3(1/2x)²(2y)+ 3(1/2x)(2y)² + (2y)³
    = x³/8 + 6x²y/4 – 12y²x/2 + 8y³
    = 1/8 + 3/2 x²y  – 6 x y²+ 8 y³

    21. (2/3a – 1/3b)³ =
    =(2/3a)³–3(2/3a)²(1/3b)+3(2/3a)(1/3b)²+(1/3b)³
= 8a³/27 – 12a²b/27 + 6ab²/27 – 1/27b³
= 8/27 a³  – 4/9  a²+  2/9 ab²  – 1/27 b³


22. (5p/2 + 3q/2)³ =
=(5p/2)³+3(5p/2)²(3q/2)+3(5p/2)(3q/2)²+(3q/2)³
= 125p³/8 + 225p²q/8 + 135pq²/8 + 27q³/8


  1. 23. ( 1 m/10 – 1 n /5)³ =
= (1m/10)³–3(1m/10)²(1n/5)+3(1m/10)(1n/5)² – (1n/5)³
= m³/1000 – 3mn/500 + 3mn²/250 – n³/125

  1. 23. (a – a/3)³ = a³–3(a)²(a/3)+3(a)(a/3)² – (a/3)³
= a³ – 3a³/3 + 3a³/9 – a³/27
= a³ – a³ + a³/– a³/27
=  a³/– a³/27


  1. 24. (1t/2 + 2t²)³ =
= (1t/2)³ – 3(1t/2)²(2t²) + 3(1t/2)(2t²)² - (2t²)³ =t³/8 – 6t³/4 + 12t/2 – 8t
=t³/8 – 3t³/2 + 6t⁵ - 8t⁶


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