The document discusses the binomial theorem and expanding binomial expressions. It provides examples of expanding (x + y)n, (2x + y)3, (x - 2y)3, and (a - b)n for various values of n. It then applies the binomial theorem to expand (x3 + 2)15 and calculates the coefficient of the x45 term, which is 5,005.

1. !
"# x , y ' ( )* n ' ( ' ,! ก )*#
n n n n
( x + y) n =   ⋅ x n +   ⋅ x n −1 y + K +   ⋅ x n − r y r + K +   ⋅ y n
0 1 r n
       
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 ก
1. ( 2 x + y)3
 3  3  3  3
( 2 x + y ) 3 =  ( 2 x ) 3 +  ( 2 x ) 2 ⋅ y +  ( 2 x ) ⋅ y 2 +   y 3
 0 1  2  3
       
= 1 ⋅ 23 ⋅ x 3 + 3 ⋅ 2 2 ⋅ x 2 ⋅ y + 3 ⋅ 2 ⋅ x ⋅ y 2 + 1 ⋅ y 3
= 8x 3 + 12 x 2 y + 6 xy 2 + y 3
3  3  3  3
2. ( x − 2 y ) 3 =   x 3 +   x 2 ( −2 y ) +   x ⋅ ( −2 y ) 2 +   ( −2 y ) 3
0 1  2  3
       
= x 3 − 6 x 2 y + 12 xy 2 − 8 y 3
 3  3  3  3
3. (a − 2 b ) 3 =  a 3 −  a 2 ( 2 b ) +  a ⋅ ( 2 b ) 2 −  ( 2 b ) 2
0 1  2  3
       
= a 3 − 6a 2 b + 12ab 2 − 8b 3
4. (a − b) 4
= a 4 − 4a 3b + 6a 2 b 2 − 4ab3 + b 4
5. (a − b)5 = a 5 − 5a 4 b + 10a 3b 2 − 10a 2 b 3 + 5ab 4 − b 5
(a + b)6 = a 6 + 6a 5b + 15a 4 b 2 + 20a 3b3 + 15a 2 b 4 + 6ab5 + b 6
( 2a + b ) 4 = ( 2a ) 4 + 4( 2a ) 3 b + 6( 2a ) 2 b 2 + 4( 2a ) b 3 + b 4
= 16a 4 + 32a 3b + 24a 2 b 2 + 8ab3 + b 4

2. 15
 3 1 
2 - . / 0 !! x กก ก x + 2 
 x 
n
1 ก23 Tr +1 =   ⋅ a n − r ⋅ b r
r
 
1
) a = x 3 , b = 2 )* n = 15 4 23
x
r
5# Tr +1 =   ⋅ ( x 3 )15−r ⋅  12 
15
 
r  
  x 
15   x 45−3r 
=   ⋅  2r
r  x 

   
15 
=   ⋅ x 45−5r
 
r
' 60 ก 7 / # ก - . / 0 !! x ก8 3
)25 45 − 5r = 0
5r = 45
r=9
15 
5# T9+1 =   ⋅ x 45−5×9
9
 
15!
T10 =
(15 − 9)!⋅9!
15!
=
6!⋅9!
15 ⋅ 14 ⋅ 13 ⋅ 12 ⋅ 11 ⋅ 9
=
6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅1
= 5,005
. / 0 !! x ก8 3 96 . / 0 10 !9 ' ก: 5,005