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Popular >

expand (x-1/(2x))^{10}

Solution

expand

(x - \frac{1}{2 x})^{10}

Solution

x^{10} - 5 x^{8} + \frac{45 x ^{6}}{4} - 15 x^{4} + \frac{105 x ^{2}}{8} - \frac{63}{8} + \frac{105}{32 x ^{2}} - \frac{15}{16 x ^{4}} + \frac{45}{256 x ^{6}} - \frac{5}{256 x ^{8}} + \frac{1}{1024 x ^{10}}

Solution steps

(x - \frac{1}{2 x})^{10}

Apply binomial theorem:

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

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

= i = 0 \sum 10 (i 10) x^{(10 - i)} (- \frac{1}{2 x})^{i}

Expand summation

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

Simplify

\frac{10 !}{0 ! ( 10 - 0 ) !} x^{10} (- \frac{1}{2 x})^{0} : x^{10}

Simplify

\frac{10 !}{1 ! ( 10 - 1 ) !} x^{9} (- \frac{1}{2 x})^{1} : - 5 x^{8}

Simplify

\frac{10 !}{2 ! ( 10 - 2 ) !} x^{8} (- \frac{1}{2 x})^{2} : \frac{45 x ^{6}}{4}

Simplify

\frac{10 !}{3 ! ( 10 - 3 ) !} x^{7} (- \frac{1}{2 x})^{3} : - 15 x^{4}

Simplify

\frac{10 !}{4 ! ( 10 - 4 ) !} x^{6} (- \frac{1}{2 x})^{4} : \frac{105 x ^{2}}{8}

Simplify

\frac{10 !}{5 ! ( 10 - 5 ) !} x^{5} (- \frac{1}{2 x})^{5} : - \frac{63}{8}

Simplify

\frac{10 !}{6 ! ( 10 - 6 ) !} x^{4} (- \frac{1}{2 x})^{6} : \frac{105}{32 x ^{2}}

Simplify

\frac{10 !}{7 ! ( 10 - 7 ) !} x^{3} (- \frac{1}{2 x})^{7} : - \frac{15}{16 x ^{4}}

Simplify

\frac{10 !}{8 ! ( 10 - 8 ) !} x^{2} (- \frac{1}{2 x})^{8} : \frac{45}{256 x ^{6}}

Simplify

\frac{10 !}{9 ! ( 10 - 9 ) !} x^{1} (- \frac{1}{2 x})^{9} : - \frac{5}{256 x ^{8}}

Simplify

\frac{10 !}{10 ! ( 10 - 10 ) !} x^{0} (- \frac{1}{2 x})^{10} : \frac{1}{1024 x ^{10}}

= x^{10} - 5 x^{8} + \frac{45 x ^{6}}{4} - 15 x^{4} + \frac{105 x ^{2}}{8} - \frac{63}{8} + \frac{105}{32 x ^{2}} - \frac{15}{16 x ^{4}} + \frac{45}{256 x ^{6}} - \frac{5}{256 x ^{8}} + \frac{1}{1024 x ^{10}}

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Frequently Asked Questions (FAQ)

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