# How do you use the binomial series to expand (x - 10z)^7?

Jan 28, 2018

${x}^{7} - 70 {x}^{6} z + 2100 {x}^{5} {z}^{2} - 35000 {x}^{4} {z}^{3} + 350000 {x}^{3} {z}^{4} - 21 \cdot {10}^{5} {x}^{2} {z}^{5} + 7 \cdot {10}^{6} x {z}^{6} - {\left(10 z\right)}^{7}$

#### Explanation:

$a = x , b = - 10 z$

Terms will have coefficients

1 7 21 35 35 21 7 1 in that order.

${\left(x - 10 z\right)}^{7} = {x}^{7} - 7 {x}^{6} \left(10 z\right) + 21 {x}^{5} \left(10 z\right) 2 - 35 {x}^{4} {\left(10 z\right)}^{3} + 35 {x}^{3} {\left(10 z\right)}^{4} - 21 {x}^{2} {\left(10 z\right)}^{5} + 7 x {\left(10 z\right)}^{6} - {\left(10 z\right)}^{7}$

${x}^{7} - 70 {x}^{6} z + 2100 {x}^{5} {z}^{2} - 35000 {x}^{4} {z}^{3} + 350000 {x}^{3} {z}^{4} - 21 \cdot {10}^{5} {x}^{2} {z}^{5} + 7 \cdot {10}^{6} x {z}^{6} - {\left(10 z\right)}^{7}$