F.4 AMaths Binomial Theorem

(a) (1+qx-3x^2)^n
=[1+x(q-3x)]^n
= 1+nx(q-3x) + n(n-1)/2 x^2(q-3x)^2 + n(n-1)(n-2)/6 x^3(q-3x)^3 + n(n-1)(n-2)(n-3)/24 x^4(q-3x)^4 + ...
= 1+nqx - 3nx^2 +n(n-1)/2 (q^2x^2 - 6qx^3 + 9x^4) + n(n-1)(n-2)/6 (q^3x^3 -9q^2x^4+...) + n(n-1)(n-2)(n-3)/24 q^4x^4 + ...
= 1+nqx + [n(n-1)q^2/2 - 3n]x^2 + [n(n-1)(n-2)q^3/6 -3n(n-1)q]x^3 + [9n(n-1)/2 - 2n(n-1)(n-2)q^2/3 + n(n-1)(n-2)(n-3)q^4/24]x^4 + ...

(b)
(i) nq = -12 and n(n-1)q^2/2 - 3n = 42
q = -12/n
and n(n-1)(144/n^2)/2 3n = 42
72-72/n - 3n = 42
n^2 - 10n + 24 = 0
(n-4)(n-6) = 0
n = 4 or 6
q = -3 or -2
When n = 4, q = -3 and when n = 6 q = -2

(ii)
When n = 4, q = -3, coefficient of x^4 is 9(4)(3)/2 - 2(4)(3)(2)(9)/3+4(3)(2)(1)(81)/24 = -9
When n = 6, q = -2, coefficient of x^4 is 9(6)(5)/2-2(6)(5)(4)(4)/3 + (6)(5)(4)(3)(16)/24 = 55