# How do you use the binomial series to expand  (x - 2)^9?

Jan 8, 2018

${x}^{9} - 18 {x}^{8} + 144 {x}^{7} - 672 {x}^{6} + 2016 {x}^{5} - 4032 {x}^{4} + 5376 {x}^{3} - {4608}^{2} + 2304 x - 512$

#### Explanation:

$9$th row of pascal's triangle:

1 , 9 , 36 , 84 , 126 , 126 , 84 , 36 , 9 , 1

the powers of each term in the binomial expression always add to $9 :$
$\left({x}^{9}\right) \left(- {2}^{0}\right) + \left({x}^{8}\right) \left(- {2}^{1}\right) + \left({x}^{7}\right) \left(- {2}^{2}\right) + \left({x}^{6}\right) \left(- {2}^{3}\right) + \left({x}^{5}\right) \left(- {2}^{4}\right) + \left({x}^{4}\right) \left(- {2}^{5}\right) + \left({x}^{3}\right) \left(- {2}^{6}\right) + \left({x}^{2}\right) \left(- {2}^{7}\right) + \left({x}^{1}\right) \left(- {2}^{8}\right) + \left({x}^{0}\right) \left(- {2}^{9}\right)$

when each term is multiplied by its corresponding coefficient, this becomes

$1 \left({x}^{9}\right) \left(- {2}^{0}\right) + 9 \left({x}^{8}\right) \left(- {2}^{1}\right) + 36 \left({x}^{7}\right) \left(- {2}^{2}\right) + 84 \left({x}^{6}\right) \left(- {2}^{3}\right) + 126 \left({x}^{5}\right) \left(- {2}^{4}\right) + 126 \left({x}^{4}\right) \left(- {2}^{5}\right) + 84 \left({x}^{3}\right) \left(- {2}^{6}\right) + 36 \left({x}^{2}\right) \left(- {2}^{7}\right) + 9 \left({x}^{1}\right) \left(- {2}^{8}\right) + 1 \left({x}^{0}\right) \left(- {2}^{9}\right)$

${n}^{0} = 1$

$\left({x}^{9}\right) + 9 \left({x}^{8}\right) \left({\left(- 2\right)}^{1}\right) + 36 \left({x}^{7}\right) \left({\left(- 2\right)}^{2}\right) + 84 \left({x}^{6}\right) \left({\left(- 2\right)}^{3}\right) + 126 \left({x}^{5}\right) \left({\left(- 2\right)}^{4}\right) + 126 \left({x}^{4}\right) \left({\left(- 2\right)}^{5}\right) + 84 \left({x}^{3}\right) \left({\left(- 2\right)}^{6}\right) + 36 \left({x}^{2}\right) \left({\left(- 2\right)}^{7}\right) + 9 \left({x}^{1}\right) \left({\left(- 2\right)}^{8}\right) + \left({\left(- 2\right)}^{9}\right)$

$= {x}^{9} + 9 {x}^{8} {\left(- 2\right)}^{1} + 36 \left({x}^{7}\right) \left({\left(- 2\right)}^{2}\right) + 84 \left({x}^{6}\right) \left({\left(- 2\right)}^{3}\right) + 126 \left({x}^{5}\right) \left({\left(- 2\right)}^{4}\right) + 126 \left({x}^{4}\right) {\left(- 2\right)}^{5} + 84 \left({x}^{3}\right) \left({\left(- 2\right)}^{6}\right) + 36 \left({x}^{2}\right) \left({\left(- 2\right)}^{7}\right) + 9 \left({x}^{1}\right) \left({\left(- 2\right)}^{8}\right) + \left({\left(- 2\right)}^{9}\right)$

$= {x}^{9} - 18 {x}^{8} + 144 {x}^{7} - 672 {x}^{6} + 2016 {x}^{5} - 4032 {x}^{4} + 5376 {x}^{3} - {4608}^{2} + 2304 x - 512$

$= {x}^{9} - 18 {x}^{8} + 144 {x}^{7} - 672 {x}^{6} + 2016 {x}^{5} - 4032 {x}^{4} + 5376 {x}^{3} - {4608}^{2} + 2304 x - 512$