F.4 MATHS

1.Find the coefficent of y^8 in theexpansion of (1－3y)(1+2y)^8
Sol
A=coefficent of y^8 in the expansion of(1+2y)^8
B=coefficent of y^7 in the expansion of(1+2y)^8
A=2^8=256
B=C(8，1)*2^7=1024
256－3*1024=-2816

2.Find the constant term in the (x+2)^6*(1/x^2－1)^2
Sol
(x+2)^6*(1/x^2－1)^2
=(1/x^4－2/x^2+1)(x+2)^6
A= the coefficent of x^4 in the expansion of (x+2)^6
B= the coefficent of x^2 in the expansion of (x+2)^6
C= the coefficent of constant term in the expansion of (x+2)^6
A=C(6，4)*1^4*2^2=60
B=C(6，2)*1^2*2^4=240
C=2^6=64
A－2B+C=60－480+64=－356

1
(1-3y)(1+2y)^8
= (1-3y)(2y+1)^8
= (1-3y) { (8C0) [(2y)^8] (1^0) + (8C1) [(2y)^7] (1^1) + ... }
= (1-3y) ( 256y^8 + 8‧128y^7 + ... )
= (1-3y) ( 256y^8 + 1024y^7 + ... )
= (1)(256y^8) + (-3y)(1024y^7) + ...
= 256y^8 - 3072y^8
= -2816y^8 + ...
∴The coefficient of y^8 is -2816.

2
(x+2)^6 (1/x^2 -1)^2
= (x+2)^6 [1/x^4 - 2 (1/x^2) + 1]
= (x+2)^6 [x^(-4) - 2x^(-2)+ 1]
= [(6C0)(x^6)(2^0) + ... + (6C2)(x^4)(2^2) + ... + (6C4)(x^2)(2^4) + ... + (6C0)(x^0)(2^6) ] [x^(-4) - 2x^(-2)+ 1]
= ( x^6 + ... + 60x^4 + ... + 240x^2 + ... + 64 ) [x^(-4) - 2x^(-2)+ 1]
= (60x^4)[x^(-4)] + (240x^2)[-2x^(-2)] + 64 + ...
= 60 - 480 + 64 + ...
= -356 + ...
∴The constant term is -356.

2014-03-24 12:53:17 補充：
Another method

1.
(1-3y)(1+2y)^8
= (1+2y)^8 + (-3y)(1+2y)^8

(1+2y)^8 's y^8：
(8C8)(1^0)(2y)^8 = 256y^8

(-3y)(1+2y)^8 's y^8：
(-3y) (8C7) [1^(8-7)] [(2y)^7] = -3072y^8

∴The coefficient of y^8 is 256-3072 = -2816.

2014-03-24 12:53:56 補充：
2.
(x+2)^6 (1/x^2 -1)^2
= (x+2)^6 [x^(-2) -1]^2
= [x^(-4) - 2x^(-2) + 1]
= x^(-4) (x+2)^6 - 2x^(-2) (x+2)^6 + (x+2)^6

x^(-4) (x+2)^6 's constant term：
x^(-4) (6C2)[x^(6-2)](2^2) = 60

-2x^(-2) (x+2)^6 's constant term：
-2x^(-2) (6C4)[x^(6-4)](2)^4 = -2*240 = -480

2014-03-24 12:54:06 補充：
(x+2)^6 's constant term：
2^6 = 64

∴The constant term is 60-480+64 = -356.

2014-03-24 15:35:01 補充：
更改：

在
2014-03-24 12:53:56 補充
2.
(x+2)^6 (1/x^2 -1)^2
= (x+2)^6 [x^(-2) -1]^2
= [x^(-4) - 2x^(-2) + 1]

「= [x^(-4) - 2x^(-2) + 1] 」→ 「 = (x+2)^6 [x^(-4) - 2x^(-2) + 1] 」