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Popolare Algebra >

espandere (2x^3+b/x)^8

Soluzione

espandere

(2 x^{3} + \frac{b}{x})^{8}

Soluzione

256 x^{24} + 1024 b x^{20} + 1792 b^{2} x^{16} + 1792 b^{3} x^{12} + 1120 b^{4} x^{8} + 448 b^{5} x^{4} + 112 b^{6} + \frac{16 b ^{7}}{x ^{4}} + \frac{b ^{8}}{x ^{8}}

Fasi della soluzione

(2 x^{3} + \frac{b}{x})^{8}

Applicare il teorema binomiale:

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

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

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

Espandere la sommatoria

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

Semplificare

\frac{8 !}{0 ! ( 8 - 0 ) !} (2 x^{3})^{8} (\frac{b}{x})^{0} : 256 x^{24}

Semplificare

\frac{8 !}{1 ! ( 8 - 1 ) !} (2 x^{3})^{7} (\frac{b}{x})^{1} : 1024 b x^{20}

Semplificare

\frac{8 !}{2 ! ( 8 - 2 ) !} (2 x^{3})^{6} (\frac{b}{x})^{2} : 1792 b^{2} x^{16}

Semplificare

\frac{8 !}{3 ! ( 8 - 3 ) !} (2 x^{3})^{5} (\frac{b}{x})^{3} : 1792 b^{3} x^{12}

Semplificare

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

Semplificare

\frac{8 !}{5 ! ( 8 - 5 ) !} (2 x^{3})^{3} (\frac{b}{x})^{5} : 448 b^{5} x^{4}

Semplificare

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

Semplificare

\frac{8 !}{7 ! ( 8 - 7 ) !} (2 x^{3})^{1} (\frac{b}{x})^{7} : \frac{16 b ^{7}}{x ^{4}}

Semplificare

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

= 256 x^{24} + 1024 b x^{20} + 1792 b^{2} x^{16} + 1792 b^{3} x^{12} + 1120 b^{4} x^{8} + 448 b^{5} x^{4} + 112 b^{6} + \frac{16 b ^{7}}{x ^{4}} + \frac{b ^{8}}{x ^{8}}

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