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Popolare Algebra >

espandere (x^2+3/x)^{12}

Soluzione

espandere

(x^{2} + \frac{3}{x})^{12}

Soluzione

x^{24} + 36 x^{21} + 594 x^{18} + 5940 x^{15} + 40095 x^{12} + 192456 x^{9} + 673596 x^{6} + 1732104 x^{3} + 3247695 + \frac{4330260}{x ^{3}} + \frac{3897234}{x ^{6}} + \frac{2125764}{x ^{9}} + \frac{531441}{x ^{12}}

Fasi della soluzione

(x^{2} + \frac{3}{x})^{12}

Applicare il teorema binomiale:

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

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

= i = 0 \sum 12 (i 12) (x^{2})^{(12 - i)} (\frac{3}{x})^{i}

Espandere la sommatoria

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

Semplificare

\frac{12 !}{0 ! ( 12 - 0 ) !} (x^{2})^{12} (\frac{3}{x})^{0} : x^{24}

Semplificare

\frac{12 !}{1 ! ( 12 - 1 ) !} (x^{2})^{11} (\frac{3}{x})^{1} : 36 x^{21}

Semplificare

\frac{12 !}{2 ! ( 12 - 2 ) !} (x^{2})^{10} (\frac{3}{x})^{2} : 594 x^{18}

Semplificare

\frac{12 !}{3 ! ( 12 - 3 ) !} (x^{2})^{9} (\frac{3}{x})^{3} : 5940 x^{15}

Semplificare

\frac{12 !}{4 ! ( 12 - 4 ) !} (x^{2})^{8} (\frac{3}{x})^{4} : 40095 x^{12}

Semplificare

\frac{12 !}{5 ! ( 12 - 5 ) !} (x^{2})^{7} (\frac{3}{x})^{5} : 192456 x^{9}

Semplificare

\frac{12 !}{6 ! ( 12 - 6 ) !} (x^{2})^{6} (\frac{3}{x})^{6} : 673596 x^{6}

Semplificare

\frac{12 !}{7 ! ( 12 - 7 ) !} (x^{2})^{5} (\frac{3}{x})^{7} : 1732104 x^{3}

Semplificare

\frac{12 !}{8 ! ( 12 - 8 ) !} (x^{2})^{4} (\frac{3}{x})^{8} : 3247695

Semplificare

\frac{12 !}{9 ! ( 12 - 9 ) !} (x^{2})^{3} (\frac{3}{x})^{9} : \frac{4330260}{x ^{3}}

Semplificare

\frac{12 !}{10 ! ( 12 - 10 ) !} (x^{2})^{2} (\frac{3}{x})^{10} : \frac{3897234}{x ^{6}}

Semplificare

\frac{12 !}{11 ! ( 12 - 11 ) !} (x^{2})^{1} (\frac{3}{x})^{11} : \frac{2125764}{x ^{9}}

Semplificare

\frac{12 !}{12 ! ( 12 - 12 ) !} (x^{2})^{0} (\frac{3}{x})^{12} : \frac{531441}{x ^{12}}

= x^{24} + 36 x^{21} + 594 x^{18} + 5940 x^{15} + 40095 x^{12} + 192456 x^{9} + 673596 x^{6} + 1732104 x^{3} + 3247695 + \frac{4330260}{x ^{3}} + \frac{3897234}{x ^{6}} + \frac{2125764}{x ^{9}} + \frac{531441}{x ^{12}}

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