# How do you use the binomial series to expand (x – 2)^5?

Feb 1, 2016

Us the pascal's triangle to find the answer easily:

:

$\rightarrow {\left(a + b\right)}^{5} = 1 {a}^{5} + 5 {a}^{4} b + 10 {a}^{3} {b}^{2} + 10 {a}^{2} {b}^{3} + 5 a {b}^{4} + 1 {b}^{5}$

$\rightarrow {\left(x - 2\right)}^{5}$

$= 1 \left({x}^{5}\right) + 5 \left({x}^{4}\right) \left(- 2\right) + 10 \left({x}^{3}\right) \left(- {2}^{2}\right) + 10 \left({x}^{2}\right) \left(- {2}^{3}\right) + 5 x \left(- {2}^{4}\right) + \left(- {2}^{5}\right)$

$= {x}^{5} + 5 {x}^{4} \left(- 2\right) + 10 {x}^{3} \left(- {2}^{2}\right) + 10 {x}^{2} \left(- {2}^{3}\right) + 5 x \left(- {2}^{4}\right) + \left(- {2}^{5}\right)$

$= {x}^{5} + 5 {x}^{4} \left(- 2\right) + 10 {x}^{3} \left(4\right) + 10 {x}^{2} \left(- 8\right) + 5 x \left(16\right) + \left(- 32\right)$

$= {x}^{5} - 10 {x}^{4} + 40 {x}^{3} - 80 {x}^{2} + 80 x - 32$