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

2. 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

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

5. 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

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

8. 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

8. 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

12. 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)

15. 15. (t² + t³)³ =(t²)³+ 3(t²)²(t³) + 3(t²)(t³)² + (t³)³
    = t⁶ + 3t⁴t³ + 3t²t⁶ + t⁹
    = t⁶ + 3t⁷ + 3t⁸ + t⁹

16. 16. ( 1 + x⁴)³= (1)³ + 3(1)²(x⁴) + 3(1)(x⁴)² + (x⁴)³
    = 1 + 3x⁴ + 3x⁸ + x¹²

17. 17. (2t – 3a²)³=(2t)³–3(2t)²(3a²)+3(2t)(3a²)²– (3a²)³ =
    = 8t³ – 36t²a² + 54ta⁴ - 27a⁶

18. (u² +5v)³=(u²)³+3(u²)²(5v)+3(u²)(5v)²+ (5v)³

= u⁶ + 15u⁴5v + 75u²v² - 125v³

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

= 1/8 – 3/4 a + 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 x³+ 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  a²b +  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

23. 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

23. 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 + a³/3 – a³/27
=  a³/3 – a³/27

23. 24. (1t/2 + 2t²)³ =

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

=t³/8 – 3t³/2 + 6t⁵ - 8t⁶