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Popular Álgebra >

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

Solución

desarrollar

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

Solución

x^{20} - 10 x^{17} + 45 x^{14} - 120 x^{11} + 210 x^{8} - 252 x^{5} + 210 x^{2} - \frac{120}{x} + \frac{45}{x ^{4}} - \frac{10}{x ^{7}} + \frac{1}{x ^{10}}

Pasos de solución

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

Aplicar el teorema del binomio:

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

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

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

Expandir sumatorio

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

Simplificar

\frac{10 !}{0 ! ( 10 - 0 ) !} (x^{2})^{10} (- \frac{1}{x})^{0} : x^{20}

Simplificar

\frac{10 !}{1 ! ( 10 - 1 ) !} (x^{2})^{9} (- \frac{1}{x})^{1} : - 10 x^{17}

Simplificar

\frac{10 !}{2 ! ( 10 - 2 ) !} (x^{2})^{8} (- \frac{1}{x})^{2} : 45 x^{14}

Simplificar

\frac{10 !}{3 ! ( 10 - 3 ) !} (x^{2})^{7} (- \frac{1}{x})^{3} : - 120 x^{11}

Simplificar

\frac{10 !}{4 ! ( 10 - 4 ) !} (x^{2})^{6} (- \frac{1}{x})^{4} : 210 x^{8}

Simplificar

\frac{10 !}{5 ! ( 10 - 5 ) !} (x^{2})^{5} (- \frac{1}{x})^{5} : - 252 x^{5}

Simplificar

\frac{10 !}{6 ! ( 10 - 6 ) !} (x^{2})^{4} (- \frac{1}{x})^{6} : 210 x^{2}

Simplificar

\frac{10 !}{7 ! ( 10 - 7 ) !} (x^{2})^{3} (- \frac{1}{x})^{7} : - \frac{120}{x}

Simplificar

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

Simplificar

\frac{10 !}{9 ! ( 10 - 9 ) !} (x^{2})^{1} (- \frac{1}{x})^{9} : - \frac{10}{x ^{7}}

Simplificar

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

= x^{20} - 10 x^{17} + 45 x^{14} - 120 x^{11} + 210 x^{8} - 252 x^{5} + 210 x^{2} - \frac{120}{x} + \frac{45}{x ^{4}} - \frac{10}{x ^{7}} + \frac{1}{x ^{10}}

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