f4 Binomial Expansion

Larry

2009-12-17 1:43 am

a) If m is a positive integer, prove that(1-3x)^m=1-3mx+9/2m(m-1)x^2+terms
involving higher powers of x.
b) It is given that the coefficient of x^2 in the expansion of (1-3x)^m(1+2x+x^2)
is 61.Find the value of m.

唔該詳細步驟唔好跳步.........
b題ans係m=5

更新1:

(1 - 3mx + 9m(m - 1)x2/2 + ...) ( +<2x> +<1>)點解可以就咁乘埋???????? (1)( ) (-3mx)(<2x>) [9m(m - 1)x2/2](<1>) 唔係好明!!

回答 (1)

六呎將軍

2009-12-17 1:55 am

✔ 最佳答案

(a) (1 - 3x)m = [1 + (-3x)]m

= 1 + mC1 (-3x) + mC2 (-3x)2 + ...

= 1 - 3mx + [m(m - 1)/2] (9x2) + ...

= 1 - 3mx + 9m(m - 1)x2/2 + ...

(b) (1 - 3x)m (1 + 2x + x2)

= (1 - 3mx + 9m(m - 1)x2/2 + ...) (1 + 2x + x2)

Multiplications giving term x2:

(1)(x2) = x2

(-3mx)(2x) = - 6mx2

[9m(m - 1)x2/2] (1) = 9m(m - 1)x2/2

So coefficient is:

1 - 6m + 9m(m - 1)/2 = 61

2 - 12m + 9m2 - 9m = 122

9m2 - 21m - 120 = 0

3m2 - 7m - 40 = 0

(3m + 8)(m - 5) = 0

m = -8/3 (rej.) or 5

2009-12-16 21:00:38 補充：
選取組合:
兩者都有 constant, x 和 x^2 的項, 所以, 能在 result 中造成 x^2 的可能乘法有:

(1) 第一個的 constant 乘第二個的 x^2
(2) 第一個的 x 乘第二個的 x
(3) 第一個的 x^2 乘第二個的 constant

參考: Myself

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