F.4 ..m2 (Binomial Theorem)

1.
The coefficient of x^4 term:
(-1)^(2 + 1) + nC2 * (3)^2 = -189
(-1) * n!/(n - 2)!2! * 9 = -189
n!/(n - 2)!2! = 21
n!/(n - 2) = 42
n(n - 1) = 42
n^2 - n - 42 = 0
(n - 7)(n + 6) = 0
n = 7 or n = -6 (rejected)
Hence, n = 7


2.
(3 - x)^4
= 3^4 - 4C1*(3)^3*x + 4C2*­(3)^2*x^2 + ……
= 81 - 108x + 54x^2 + ……

(1 + ax)^3
= (1)^3 + 3*(1)^2*(ax) + 3*(1)*(ax)^2 + (ax)^3
= 1 + 3ax + 3a^2x^2 + ……

The coefficient of x^2 in the expansion of (3 - x)^4 (1+ ax)^3:
(81)(3a^2) + (-108)(3a) + 54 = 378
243a^2 - 324a - 324 = 0
3a^2 - 4a - 4 = 0
(3a + 2)(a - 2) = 0
a = -2/3 or a = 2


3.
(1 - 2x)^m
= (1)^m - mC1*(1)^(m - 1)*(2x) + mC2*(1)^(m - 2)*(2x)^2 + ……
= 1 - 2mx + 2m(m - 1)x^2 + ……

(1 + x)^n
= (1)^n + nC1*(1)^(n - 1)*(x) + nC2­*(1)^(n - 2)*(x)^2 + ……
= 1 + nx + [n(n - 1)/2]x^2 + ……

x term of the expansion of (1 - 2x )^m (1 + x)^n:
(1)(n) + (1)(-2m) = -8
n = 2m - 8 …… (1)

x^2 term of the expansion of (1 - 2x )^m (1 + x)^n:
(1)[n(n - 1)/2] + (-2m)(n) + (1)[2m(m - 1)] = 18
n(n - 1) - 4mn + 4m(m - 1) = 36 …… (2)

Put (1) into (2):
(2m - 8)(2m - 8 - 1) - 4m(2m - 8) + 4m(m - 1) = 36
4m^2 - 34 m + 72 - 8m^2 + 32m + 4m^2 - 4m = 36
-6m = -36
m = 6

Put m = 10 into (1):
n = 2(6) - 8
n = 4

Hence, m = 6 and n = 4