# How do you use the binomial series to expand (2x-3)^10?

Feb 23, 2017

${\left(2 x - 3\right)}^{10} = 1024 {x}^{10} - 15360 {x}^{9} + 103680 {x}^{8} - 414720 {x}^{7} + 1088640 {x}^{6} - 1959552 {x}^{5} + 2449440 {x}^{4} - 2099520 {x}^{3} + 1180980 {x}^{2} - 393660 x + 59049$

#### Explanation:

Remember that ((n),(k)) = _nC_k = (n!)/((n-k)!k!)

${\left(2 x - 3\right)}^{10} {=}_{10} {C}_{0} {\left(2 x\right)}^{10} {\left(- 3\right)}^{0} {+}_{10} {C}_{1} {\left(2 x\right)}^{9} {\left(- 3\right)}^{0} {+}_{10} {C}_{2} {\left(2 x\right)}^{8} {\left(- 3\right)}^{2} {+}_{10} {C}_{3} {\left(2 x\right)}^{7} {\left(- 3\right)}^{3} {+}_{10} {C}_{4} {\left(2 x\right)}^{6} {\left(- 3\right)}^{4} {+}_{10} {C}_{5} {\left(2 x\right)}^{5} {\left(- 3\right)}^{5} {+}_{10} {C}_{6} {\left(2 x\right)}^{4} {\left(- 3\right)}^{6} {+}_{10} {C}_{7} {\left(2 x\right)}^{3} {\left(- 3\right)}^{7} {+}_{10} {C}_{8} {\left(2 x\right)}^{2} {\left(- 3\right)}^{8} {+}_{10} {C}_{9} {\left(2 x\right)}^{1} {\left(- 3\right)}^{9} {+}_{10} {C}_{10} {\left(2 x\right)}^{0} {\left(- 3\right)}^{10}$

You can use Pascal's Triangle to find your combinations or the formula above. From Pascal's triangle : _10C_0 = _10C_10 = 1; _10C_1 = _10C_9 = 10; _10C_2 = _10C_8 = 45; _10C_3 = _10C_7 = 120; _10C_4 = _10C_6 = 210; _10C_5 = 252

Substitute into the equation and simplify:
${\left(2 x - 3\right)}^{10} = 1024 {x}^{10} + 10 \left(512\right) {x}^{9} \left(- 3\right) + 45 \left(256\right) {x}^{8} \left(9\right) + 120 \left(128\right) {x}^{7} \left(- 27\right) + 210 \left(64\right) {x}^{6} \left(81\right) + 252 \left(32\right) {x}^{5} \left(- 243\right) + 210 \left(16\right) {x}^{4} \left(729\right) + 120 \left(8\right) {x}^{3} \left(- 2187\right) + 45 \left(4\right) {x}^{2} \left(6561\right) + 10 \left(2\right) x \left(- 19683\right) + 59049$

Simplify by multiplying the constants:
${\left(2 x - 3\right)}^{10} = 1024 {x}^{10} - 15360 {x}^{9} + 103680 {x}^{8} - 414720 {x}^{7} + 1088640 {x}^{6} - 1959552 {x}^{5} + 2449440 {x}^{4} - 2099520 {x}^{3} + 1180980 {x}^{2} - 393660 x + 59049$