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Popular >

expand (1-x/2)^{11}

Solution

expand

(1 - \frac{x}{2})^{11}

Solution

1 - \frac{11 x}{2} + \frac{55 x ^{2}}{4} - \frac{165 x ^{3}}{8} + \frac{165 x ^{4}}{8} - \frac{231 x ^{5}}{16} + \frac{231 x ^{6}}{32} - \frac{165 x ^{7}}{64} + \frac{165 x ^{8}}{256} - \frac{55 x ^{9}}{512} + \frac{11 x ^{10}}{1024} - \frac{x ^{11}}{2048}

Solution steps

(1 - \frac{x}{2})^{11}

Apply binomial theorem:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = 1, b = - \frac{x}{2}

= i = 0 \sum 11 (i 11) \cdot 1^{(11 - i)} (- \frac{x}{2})^{i}

Expand summation

= \frac{11 !}{0 ! ( 11 - 0 ) !} \cdot 1^{11} (- \frac{x}{2})^{0} + \frac{11 !}{1 ! ( 11 - 1 ) !} \cdot 1^{10} (- \frac{x}{2})^{1} + \frac{11 !}{2 ! ( 11 - 2 ) !} \cdot 1^{9} (- \frac{x}{2})^{2} + \frac{11 !}{3 ! ( 11 - 3 ) !} \cdot 1^{8} (- \frac{x}{2})^{3} + \frac{11 !}{4 ! ( 11 - 4 ) !} \cdot 1^{7} (- \frac{x}{2})^{4} + \frac{11 !}{5 ! ( 11 - 5 ) !} \cdot 1^{6} (- \frac{x}{2})^{5} + \frac{11 !}{6 ! ( 11 - 6 ) !} \cdot 1^{5} (- \frac{x}{2})^{6} + \frac{11 !}{7 ! ( 11 - 7 ) !} \cdot 1^{4} (- \frac{x}{2})^{7} + \frac{11 !}{8 ! ( 11 - 8 ) !} \cdot 1^{3} (- \frac{x}{2})^{8} + \frac{11 !}{9 ! ( 11 - 9 ) !} \cdot 1^{2} (- \frac{x}{2})^{9} + \frac{11 !}{10 ! ( 11 - 10 ) !} \cdot 1^{1} (- \frac{x}{2})^{10} + \frac{11 !}{11 ! ( 11 - 11 ) !} \cdot 1^{0} (- \frac{x}{2})^{11}

Simplify

\frac{11 !}{0 ! ( 11 - 0 ) !} \cdot 1^{11} (- \frac{x}{2})^{0} : 1

Simplify

\frac{11 !}{1 ! ( 11 - 1 ) !} \cdot 1^{10} (- \frac{x}{2})^{1} : - \frac{11 x}{2}

Simplify

\frac{11 !}{2 ! ( 11 - 2 ) !} \cdot 1^{9} (- \frac{x}{2})^{2} : \frac{55 x ^{2}}{4}

Simplify

\frac{11 !}{3 ! ( 11 - 3 ) !} \cdot 1^{8} (- \frac{x}{2})^{3} : - \frac{165 x ^{3}}{8}

Simplify

\frac{11 !}{4 ! ( 11 - 4 ) !} \cdot 1^{7} (- \frac{x}{2})^{4} : \frac{165 x ^{4}}{8}

Simplify

\frac{11 !}{5 ! ( 11 - 5 ) !} \cdot 1^{6} (- \frac{x}{2})^{5} : - \frac{231 x ^{5}}{16}

Simplify

\frac{11 !}{6 ! ( 11 - 6 ) !} \cdot 1^{5} (- \frac{x}{2})^{6} : \frac{231 x ^{6}}{32}

Simplify

\frac{11 !}{7 ! ( 11 - 7 ) !} \cdot 1^{4} (- \frac{x}{2})^{7} : - \frac{165 x ^{7}}{64}

Simplify

\frac{11 !}{8 ! ( 11 - 8 ) !} \cdot 1^{3} (- \frac{x}{2})^{8} : \frac{165 x ^{8}}{256}

Simplify

\frac{11 !}{9 ! ( 11 - 9 ) !} \cdot 1^{2} (- \frac{x}{2})^{9} : - \frac{55 x ^{9}}{512}

Simplify

\frac{11 !}{10 ! ( 11 - 10 ) !} \cdot 1^{1} (- \frac{x}{2})^{10} : \frac{11 x ^{10}}{1024}

Simplify

\frac{11 !}{11 ! ( 11 - 11 ) !} \cdot 1^{0} (- \frac{x}{2})^{11} : - \frac{x ^{11}}{2048}

= 1 - \frac{11 x}{2} + \frac{55 x ^{2}}{4} - \frac{165 x ^{3}}{8} + \frac{165 x ^{4}}{8} - \frac{231 x ^{5}}{16} + \frac{231 x ^{6}}{32} - \frac{165 x ^{7}}{64} + \frac{165 x ^{8}}{256} - \frac{55 x ^{9}}{512} + \frac{11 x ^{10}}{1024} - \frac{x ^{11}}{2048}

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Frequently Asked Questions (FAQ)

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