How do you use the binomial series to expand (x+2)^7?

Dec 19, 2015

${x}^{7} + 14 {x}^{6} + 84 {x}^{5} + 280 {x}^{4} + 560 {x}^{3} + 672 {x}^{2} + 448 x + 128$

Explanation:

The seventh row of Pascal's triangle is

$1 , 7 , 21 , 35 , 35 , 21 , 7 , 1$

Thus,

${\left(a + b\right)}^{7} = {a}^{7} + 7 {a}^{6} b + 21 {a}^{5} {b}^{2} + 35 {a}^{4} {b}^{3} + 35 {a}^{3} {b}^{4} + 21 {a}^{2} {b}^{5} + 7 a {b}^{6} + {b}^{7}$

This can be applied:

${\left(x + 2\right)}^{7} = {x}^{7} + 7 {x}^{6} \left(2\right) + 21 {x}^{5} {\left(2\right)}^{2} + 35 {x}^{4} {\left(2\right)}^{3} + 35 {x}^{3} {\left(2\right)}^{4} + 21 {x}^{2} {\left(2\right)}^{5} + 7 x {\left(2\right)}^{6} + {\left(2\right)}^{7}$

$= {x}^{7} + 14 {x}^{6} + 84 {x}^{5} + 280 {x}^{4} + 560 {x}^{3} + 672 {x}^{2} + 448 x + 128$
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