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Popular >

expand (x-1/x)^{12}

Solution

expand

(x - \frac{1}{x})^{12}

Solution

x^{12} - 12 x^{10} + 66 x^{8} - 220 x^{6} + 495 x^{4} - 792 x^{2} + 924 - \frac{792}{x ^{2}} + \frac{495}{x ^{4}} - \frac{220}{x ^{6}} + \frac{66}{x ^{8}} - \frac{12}{x ^{10}} + \frac{1}{x ^{12}}

Solution steps

(x - \frac{1}{x})^{12}

Apply binomial theorem:

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

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

= i = 0 \sum 12 (i 12) x^{(12 - i)} (- \frac{1}{x})^{i}

Expand summation

= \frac{12 !}{0 ! ( 12 - 0 ) !} x^{12} (- \frac{1}{x})^{0} + \frac{12 !}{1 ! ( 12 - 1 ) !} x^{11} (- \frac{1}{x})^{1} + \frac{12 !}{2 ! ( 12 - 2 ) !} x^{10} (- \frac{1}{x})^{2} + \frac{12 !}{3 ! ( 12 - 3 ) !} x^{9} (- \frac{1}{x})^{3} + \frac{12 !}{4 ! ( 12 - 4 ) !} x^{8} (- \frac{1}{x})^{4} + \frac{12 !}{5 ! ( 12 - 5 ) !} x^{7} (- \frac{1}{x})^{5} + \frac{12 !}{6 ! ( 12 - 6 ) !} x^{6} (- \frac{1}{x})^{6} + \frac{12 !}{7 ! ( 12 - 7 ) !} x^{5} (- \frac{1}{x})^{7} + \frac{12 !}{8 ! ( 12 - 8 ) !} x^{4} (- \frac{1}{x})^{8} + \frac{12 !}{9 ! ( 12 - 9 ) !} x^{3} (- \frac{1}{x})^{9} + \frac{12 !}{10 ! ( 12 - 10 ) !} x^{2} (- \frac{1}{x})^{10} + \frac{12 !}{11 ! ( 12 - 11 ) !} x^{1} (- \frac{1}{x})^{11} + \frac{12 !}{12 ! ( 12 - 12 ) !} x^{0} (- \frac{1}{x})^{12}

Simplify

\frac{12 !}{0 ! ( 12 - 0 ) !} x^{12} (- \frac{1}{x})^{0} : x^{12}

Simplify

\frac{12 !}{1 ! ( 12 - 1 ) !} x^{11} (- \frac{1}{x})^{1} : - 12 x^{10}

Simplify

\frac{12 !}{2 ! ( 12 - 2 ) !} x^{10} (- \frac{1}{x})^{2} : 66 x^{8}

Simplify

\frac{12 !}{3 ! ( 12 - 3 ) !} x^{9} (- \frac{1}{x})^{3} : - 220 x^{6}

Simplify

\frac{12 !}{4 ! ( 12 - 4 ) !} x^{8} (- \frac{1}{x})^{4} : 495 x^{4}

Simplify

\frac{12 !}{5 ! ( 12 - 5 ) !} x^{7} (- \frac{1}{x})^{5} : - 792 x^{2}

Simplify

\frac{12 !}{6 ! ( 12 - 6 ) !} x^{6} (- \frac{1}{x})^{6} : 924

Simplify

\frac{12 !}{7 ! ( 12 - 7 ) !} x^{5} (- \frac{1}{x})^{7} : - \frac{792}{x ^{2}}

Simplify

\frac{12 !}{8 ! ( 12 - 8 ) !} x^{4} (- \frac{1}{x})^{8} : \frac{495}{x ^{4}}

Simplify

\frac{12 !}{9 ! ( 12 - 9 ) !} x^{3} (- \frac{1}{x})^{9} : - \frac{220}{x ^{6}}

Simplify

\frac{12 !}{10 ! ( 12 - 10 ) !} x^{2} (- \frac{1}{x})^{10} : \frac{66}{x ^{8}}

Simplify

\frac{12 !}{11 ! ( 12 - 11 ) !} x^{1} (- \frac{1}{x})^{11} : - \frac{12}{x ^{10}}

Simplify

\frac{12 !}{12 ! ( 12 - 12 ) !} x^{0} (- \frac{1}{x})^{12} : \frac{1}{x ^{12}}

= x^{12} - 12 x^{10} + 66 x^{8} - 220 x^{6} + 495 x^{4} - 792 x^{2} + 924 - \frac{792}{x ^{2}} + \frac{495}{x ^{4}} - \frac{220}{x ^{6}} + \frac{66}{x ^{8}} - \frac{12}{x ^{10}} + \frac{1}{x ^{12}}

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Frequently Asked Questions (FAQ)

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