a maths一問(10分!急!)

(a) (1 + px - x^2)^4
= [1 + x(p - x)]^4
= 1 + 4x(p - x) + 6x^2(p - x)^2 + ...
= 1 + 4px - 4x^2 + 6x^2(p^2 - ...) + ...
= 1 + 4px + (6p^2 - 4)x^2 + ...
(b) (i) 20 = 6p^2 - 4
6p^2 = 24
p^2 = 4
p = 2 or -2
(ii) When p = 2, coefficient of x = 4(2) = 8.
When p = -2, coefficient of x = 4(-2) = -8.

(a):

[1 + px - (x ^ 2)] ^ 4
= {[1 + px - (x ^ 2)] ^ 2} ^ 2
= {[1 + px - (x ^ 2)] + [px + (p ^ 2)(x ^ 2) + ... ] + [(x ^ 2) + ... ]} ^ 2
= {1 + 2px + [(p ^ 2) - 2](x ^ 2) + A(x ^ 3) + B(x ^ 4)} ^ 2
= {1 + 2px + [(p ^ 2) - 2](x ^ 2)} + [2px + 4(p ^ 2)(x ^ 2)] + {[(p ^ 2) - 2](x ^ 2)} + ...
= 1 + 4px + 2(p ^ 2)(x ^ 2) + C(x ^ 3) + D(x ^ 4) + E(x ^ 5) + F(x ^ 6) + G(x ^ 7) + H(x ^ 8)
= 1 + 4px + 2(p ^ 2)(x ^ 2) + ......

In here, the art is we don' t need to expand all the term during each multiplication (square). We don't need to work up to (x ^ 2). Please note my bracket, then you will know the trick there.

(bi):

2(p ^ 2) = 20
p = sqrt(10) = 3.1623

(bii):

4p = 4sqrt(10) = 12.6491