What is expand (x^2-1/(4x))^{12} ?

The solution to expand (x^2-1/(4x))^{12} is

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expand (x^2-1/(4x))^{12}

expand

(x^{2} - \frac{1}{4 x})^{12}

x^{24} - 3 x^{21} + \frac{33 x ^{18}}{8} - \frac{55 x ^{15}}{16} + \frac{495 x ^{12}}{256} - \frac{99 x ^{9}}{128} + \frac{231 x ^{6}}{1024} - \frac{99 x ^{3}}{2048} + \frac{495}{65536} - \frac{55}{65536 x ^{3}} + \frac{33}{524288 x ^{6}} - \frac{3}{1048576 x ^{9}} + \frac{1}{16777216 x ^{12}}

Solution steps

(x^{2} - \frac{1}{4 x})^{12}

Apply binomial theorem:

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

a = x^{2}, b = - \frac{1}{4 x}

= i = 0 \sum 12 (i 12) (x^{2})^{(12 - i)} (- \frac{1}{4 x})^{i}

Expand summation

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

Simplify

\frac{12 !}{0 ! ( 12 - 0 ) !} (x^{2})^{12} (- \frac{1}{4 x})^{0} : x^{24}

Simplify

\frac{12 !}{1 ! ( 12 - 1 ) !} (x^{2})^{11} (- \frac{1}{4 x})^{1} : - 3 x^{21}

Simplify

\frac{12 !}{2 ! ( 12 - 2 ) !} (x^{2})^{10} (- \frac{1}{4 x})^{2} : \frac{33 x ^{18}}{8}

Simplify

\frac{12 !}{3 ! ( 12 - 3 ) !} (x^{2})^{9} (- \frac{1}{4 x})^{3} : - \frac{55 x ^{15}}{16}

Simplify

\frac{12 !}{4 ! ( 12 - 4 ) !} (x^{2})^{8} (- \frac{1}{4 x})^{4} : \frac{495 x ^{12}}{256}

Simplify

\frac{12 !}{5 ! ( 12 - 5 ) !} (x^{2})^{7} (- \frac{1}{4 x})^{5} : - \frac{99 x ^{9}}{128}

Simplify

\frac{12 !}{6 ! ( 12 - 6 ) !} (x^{2})^{6} (- \frac{1}{4 x})^{6} : \frac{231 x ^{6}}{1024}

Simplify

\frac{12 !}{7 ! ( 12 - 7 ) !} (x^{2})^{5} (- \frac{1}{4 x})^{7} : - \frac{99 x ^{3}}{2048}

Simplify

\frac{12 !}{8 ! ( 12 - 8 ) !} (x^{2})^{4} (- \frac{1}{4 x})^{8} : \frac{495}{65536}

Simplify

\frac{12 !}{9 ! ( 12 - 9 ) !} (x^{2})^{3} (- \frac{1}{4 x})^{9} : - \frac{55}{65536 x ^{3}}

Simplify

\frac{12 !}{10 ! ( 12 - 10 ) !} (x^{2})^{2} (- \frac{1}{4 x})^{10} : \frac{33}{524288 x ^{6}}

Simplify

\frac{12 !}{11 ! ( 12 - 11 ) !} (x^{2})^{1} (- \frac{1}{4 x})^{11} : - \frac{3}{1048576 x ^{9}}

Simplify

\frac{12 !}{12 ! ( 12 - 12 ) !} (x^{2})^{0} (- \frac{1}{4 x})^{12} : \frac{1}{16777216 x ^{12}}

= x^{24} - 3 x^{21} + \frac{33 x ^{18}}{8} - \frac{55 x ^{15}}{16} + \frac{495 x ^{12}}{256} - \frac{99 x ^{9}}{128} + \frac{231 x ^{6}}{1024} - \frac{99 x ^{3}}{2048} + \frac{495}{65536} - \frac{55}{65536 x ^{3}} + \frac{33}{524288 x ^{6}} - \frac{3}{1048576 x ^{9}} + \frac{1}{16777216 x ^{12}}

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