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

Aug 19, 2016

This is a lot of work! I have demonstrated the method. I will let you complete the process.

#### Explanation:

Using Pascal's triangle: So for reference type ${\left(p + q\right)}^{9}$ we have:

${p}^{9} \textcolor{m a \ge n t a}{+ 9 {p}^{8} q} + 36 {p}^{7} {q}^{2} + 84 {p}^{6} {q}^{3} + 126 {p}^{5} {q}^{4} + 126 {p}^{4} {q}^{5} + 84 {p}^{3} {q}^{6} + 36 {p}^{2} {q}^{7} + 9 p {q}^{8} + {q}^{9}$

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The coefficients as shown in Pascal's triangle are derived by the binomial expansion:

Note that: ""_""^nC_r-> (n!)/((n-r)!r!)

${\text{_""^9C_0p^9q^0color(magenta)(+""_""^9C_1p^8q^1)+""_""^9C_2p^7q^2+....+""_}}^{9} {C}_{9} {p}^{0} {q}^{9}$

Thus for example: color(magenta)(""_""^9C_1p^8q^1) -> (9 !)/((9-1)!1! )p^8q^1

However, $p = 2 x \text{ and } q = - y$ giving

(n !)/((n-1)!1! )p^8q^1 " "->color(magenta)(" "(9xx cancel(8!))/(cancel(8!))(2x)^8(-y))

$= \left(9 \times 256\right) {x}^{8} \left(- y\right) = \textcolor{m a \ge n t a}{- 2304 {x}^{8} y}$

$\textcolor{g r e e n}{\text{I will let you fill in all the numbers. Good luck! A lot of work!}}$

Using software I get the final solution of:

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