The document provides examples of expanding binomial expressions using the binomial theorem. It gives the expansions of expressions such as (x2 + y3)6, (3-y7)7, (1-x4)8, (2/3x - 3/2y)7, (x3 + mn)8, and (2m2-5m5)6. Each expression is expanded out to show the terms of the resulting polynomial.

1. POTENCIAS
EJERCICIOS DE BLADOR “211”

2. 3. (x2 + y3 )6 = (x2)6 + 6 (x2)5(y3) + 15(x2)4 (y3)2 + 20 (x2)3 (y3)3 + 15 (x2)2 (y3)4 + 6(x2) (y3)5 + (y3)6
R= x12 + 6x10 y3 + 15x8 y6 + 20x6 y9 + 15x4 y12 + 6x2 y 15 + y18
4.- (3-y7)7= ( 3)7 + 7 (3 )6 (-y 7) + 21( 3)5 (-y 7)2 + 35( 3)4(y7)3+35(3)3 ( -y7)4 + 21( 3)2 ( -y7)5 + 7(3)
(-y 7)6 + ( -y7)7
R= 2187 – 5103y14 + 5103y14 -2835y21 + 945y20 -189y35 21y42 –y49

3. 7
( - )6 = ( )6 -6 ( )5 ( ) +15 ( )4 ( )2 -20 ( )3 ( )3 +15 ( )2 ( )4 -6( ) ( )5
+( )6
6 5
= - + - + - +
8
(1-x4)8 = (1)8 - 8(1)7(x4) + 28(1)6(x4)2 -56(1)5(x4)3 +70 (1)4(x4)4 –
56(1)3(x4)5 + 28 (1)2(x4)6 – 8(1)(x4)7 + (x4)8
= 1-8x4+28x8-56x12+70x16-56x20 + 28x24 – 8x28 – x32
9. (2/3x - 3/2 y)7=
(2/3x)7 – 7(2/3x)6(3/2y) + 21(2/3x)5(3/2y)2-35(2/3x)4(3/2y)3+35(2/3x)3(3/2y)4 –
21(2/3x)2(3/2y)5+7(2/3y)6 – (3/2y)7=
128/2187x7 – 224/243x6y + 56/9x5y2- 70/3x4y3 +105/2x3y4 – 567/8x2y5 + 1701/32xy6- 2187/128y7
R=128/2187x7 – 224/243x6y + 56/9x5y2- 70/3x4y3 +105/2x3y4 – 567/8x2y5 + 1701/32xy6-
2187/128y7

4. 10.- =
11.(x3+mn)8=
(x3)8+ 8(x3)7mn+ 28(x3)6(mn)2 + 56 (x3)5(mn)3+70 (x3)4(mn)4+ 56(x3)3(mn)5 + 28(x3)2(mn)6+8
(x3)(mn)7+(mn)8=
X24+8x21mn+28x18m2n2 + 56x15m3n3 + 70x12m4n4+ 56x9m5n5+ 28x6m6n6+ 8x3m7n7+m8n8
R= X24+8x21mn+28x18m2n2 + 56x15m3n3 + 70x12m4n4+ 56x9m5n5+ 28x6m6n6+
8x3m7n7+m8n8
12.- =
14.
(2m₂-5m₅)₆=(2m₂)₆
(2m₂-5n₅)₆= (2m₂)₆- 6(2m₂) 5 (5n₅)+15(2m₂)₄ (5n₅)₂ -20(2m₂)₃ (5n₅)₃+15(2m₂)₂(5n₅)₄-6(2m₂)
(5n₅)₅+(5n₅)₆
=64m₁₂-960m₁₀n₅+6000m₈n₁₀-20000m₆n₁₅+37.500m₄n₃₀-37500m₂n₂+15625n₁₀

5. 15.
(4-x5/4)₇
=(4)₇-7(4)₆(x₅/4)+21(4)₅(x₅/4)₂-35(4)₄(x₅/4)₃+35(4)3(x₅/4)₄-21(4)₂(x₅/4)₅+7(4)(x₅/4)₆-(x₅/4)₇
=16384+7168x₅+1344x₁₀-140x₁₅+35x₂₀/4-21x₂₅/64+7x₃₀/1024-1 /16384 x₃₅