Question

How do you use the binomial series to expand `(x-1)^8`?

Answer 1

`1 x^8 - 8 x^7 + 28 x^6 - 56 x^5 + 70 x^4 - 56x^3 + 28 x^2 - 8x + 1`

Explanation:

`(a - b)^n = sum_{i = 0}^n ((n), (i))a^{n - i} (-b)^i`

`(x - 1)^8 = sum_{i = 0}^8 ((8), (i))x^{8 - i} (-1)^i`

`= (8, 0) x^8 - (8, 1) x^7 + (8, 2) x^6 - (8, 3) x^5 + (8, 4) x^4 - (8, 5) x^3 + (8, 6) x^2 - (8, 7) x + (8, 8)`

Pascal's triangle

1
11
121
1331
14641
15,10,10,51
16,15,20,15,61
17,21,35,35,21,71
18,28,56,70,56,28,81