math展式求係數問題..

(1 + x - 2x^2)^4=(2x^2-x-1)^4=(2x+1)^4(x-1)^4
There are 4 possible cases to be x^3
1. (x^3)(x^0)
2. (x^2)(x)
3. (x)(x^2)
4. (x^0)(x^3)

Therefore, coeff. of x^3 of (1 + x - 2x^2)^4
is 4C3(2^3)+4C2(2^2)4C1(-1)^3+4C1(2)4C2(-1)^2+4C3(-1)^3
=-20

(1 + x - 2x^2)^4
=(1 + x - 2x^2)(1 + x - 2x^2)(1 + x - 2x^2)(1 + x - 2x^2)
=(1 + x - 2x^2 + x + x^2 - 2x^3 - 2x^2 - 2x^3 + 4x^4)(1 + x - 2x^2)
(1 + x - 2x^2)
=(1 + 2x - 3x^2 - 4x^3 + 4x^4)(1 + x - 2x^2)(1 + x - 2x^2)
=(1 + 2x - 3x^2 - 4x^3 + 4x^4 + x + 2x^2 - 3x^3 - 4x^4 + 4x^5 - 2x^2
- 4x^2 + 5x^4 + 6x^5 - 6x^6)(1 + x - 2x^2)
=(1 + 3x - 7x^2 - 7x^3 + 5x^4 + 10x^5 - 6x^6)(1 + x - 2x^2)
=1 + 3x - 7x^2 - 7x^3 + 5x^4 + 10x^5 - 6x^6 + x + 3x^2 - 7x^3 - 7x^4
+ 5x^5 + 10x^6 - 6x^7 - 2x^2 + 9x^4 + 9x^5 - 7x^6 + 12x^7 + 8x^8
=1 + 4x - 6x^2 - 14x^3 + 7x^4 + 24x^5 - 3x^6 + 6x^7 + 8x^8
x^3 的係數 is - 14