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

May 24, 2018

$1 {x}^{8} - 8 {x}^{7} + 28 {x}^{6} - 56 {x}^{5} + 70 {x}^{4} - 56 {x}^{3} + 28 {x}^{2} - 8 x + 1$

#### Explanation:

${\left(a - b\right)}^{n} = {\sum}_{i = 0}^{n} \left(\begin{matrix}n \\ i\end{matrix}\right) {a}^{n - i} {\left(- b\right)}^{i}$

${\left(x - 1\right)}^{8} = {\sum}_{i = 0}^{8} \left(\begin{matrix}8 \\ i\end{matrix}\right) {x}^{8 - i} {\left(- 1\right)}^{i}$

$= \left(8 , 0\right) {x}^{8} - \left(8 , 1\right) {x}^{7} + \left(8 , 2\right) {x}^{6} - \left(8 , 3\right) {x}^{5} + \left(8 , 4\right) {x}^{4} - \left(8 , 5\right) {x}^{3} + \left(8 , 6\right) {x}^{2} - \left(8 , 7\right) x + \left(8 , 8\right)$

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