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Popular Álgebra >

expandir (x^3-2/x)^8

Solução

expandir

(x^{3} - \frac{2}{x})^{8}

Solução

x^{24} - 16 x^{20} + 112 x^{16} - 448 x^{12} + 1120 x^{8} - 1792 x^{4} + 1792 - \frac{1024}{x ^{4}} + \frac{256}{x ^{8}}

Passos da solução

(x^{3} - \frac{2}{x})^{8}

Aplicar o teorema do binômio:

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

a = x^{3}, b = - \frac{2}{x}

= i = 0 \sum 8 (i 8) (x^{3})^{(8 - i)} (- \frac{2}{x})^{i}

Expandir a soma

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

Simplificar

\frac{8 !}{0 ! ( 8 - 0 ) !} (x^{3})^{8} (- \frac{2}{x})^{0} : x^{24}

Simplificar

\frac{8 !}{1 ! ( 8 - 1 ) !} (x^{3})^{7} (- \frac{2}{x})^{1} : - 16 x^{20}

Simplificar

\frac{8 !}{2 ! ( 8 - 2 ) !} (x^{3})^{6} (- \frac{2}{x})^{2} : 112 x^{16}

Simplificar

\frac{8 !}{3 ! ( 8 - 3 ) !} (x^{3})^{5} (- \frac{2}{x})^{3} : - 448 x^{12}

Simplificar

\frac{8 !}{4 ! ( 8 - 4 ) !} (x^{3})^{4} (- \frac{2}{x})^{4} : 1120 x^{8}

Simplificar

\frac{8 !}{5 ! ( 8 - 5 ) !} (x^{3})^{3} (- \frac{2}{x})^{5} : - 1792 x^{4}

Simplificar

\frac{8 !}{6 ! ( 8 - 6 ) !} (x^{3})^{2} (- \frac{2}{x})^{6} : 1792

Simplificar

\frac{8 !}{7 ! ( 8 - 7 ) !} (x^{3})^{1} (- \frac{2}{x})^{7} : - \frac{1024}{x ^{4}}

Simplificar

\frac{8 !}{8 ! ( 8 - 8 ) !} (x^{3})^{0} (- \frac{2}{x})^{8} : \frac{256}{x ^{8}}

= x^{24} - 16 x^{20} + 112 x^{16} - 448 x^{12} + 1120 x^{8} - 1792 x^{4} + 1792 - \frac{1024}{x ^{4}} + \frac{256}{x ^{8}}

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