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Popolare Algebra >

espandere (x+sqrt(3))^{16}

Soluzione

espandere

(x + 3)^{16}

Soluzione

x^{16} + 163 x^{15} + 360 x^{14} + 16803 x^{13} + 16380 x^{12} + 393123 x^{11} + 216216 x^{10} + 3088803 x^{9} + 1042470 x^{8} + 9266403 x^{7} + 1945944 x^{6} + 10614243 x^{5} + 1326780 x^{4} + 4082403 x^{3} + 262440 x^{2} + 349923 x + 6561

Fasi della soluzione

(x + 3)^{16}

Applicare il teorema binomiale:

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

a = x, b = 3

= i = 0 \sum 16 (i 16) x^{(16 - i)} (3)^{i}

Espandere la sommatoria

= \frac{16 !}{0 ! ( 16 - 0 ) !} x^{16} (3)^{0} + \frac{16 !}{1 ! ( 16 - 1 ) !} x^{15} (3)^{1} + \frac{16 !}{2 ! ( 16 - 2 ) !} x^{14} (3)^{2} + \frac{16 !}{3 ! ( 16 - 3 ) !} x^{13} (3)^{3} + \frac{16 !}{4 ! ( 16 - 4 ) !} x^{12} (3)^{4} + \frac{16 !}{5 ! ( 16 - 5 ) !} x^{11} (3)^{5} + \frac{16 !}{6 ! ( 16 - 6 ) !} x^{10} (3)^{6} + \frac{16 !}{7 ! ( 16 - 7 ) !} x^{9} (3)^{7} + \frac{16 !}{8 ! ( 16 - 8 ) !} x^{8} (3)^{8} + \frac{16 !}{9 ! ( 16 - 9 ) !} x^{7} (3)^{9} + \frac{16 !}{10 ! ( 16 - 10 ) !} x^{6} (3)^{10} + \frac{16 !}{11 ! ( 16 - 11 ) !} x^{5} (3)^{11} + \frac{16 !}{12 ! ( 16 - 12 ) !} x^{4} (3)^{12} + \frac{16 !}{13 ! ( 16 - 13 ) !} x^{3} (3)^{13} + \frac{16 !}{14 ! ( 16 - 14 ) !} x^{2} (3)^{14} + \frac{16 !}{15 ! ( 16 - 15 ) !} x^{1} (3)^{15} + \frac{16 !}{16 ! ( 16 - 16 ) !} x^{0} (3)^{16}

Semplificare

\frac{16 !}{0 ! ( 16 - 0 ) !} x^{16} (3)^{0} : x^{16}

Semplificare

\frac{16 !}{1 ! ( 16 - 1 ) !} x^{15} (3)^{1} : 163 x^{15}

Semplificare

\frac{16 !}{2 ! ( 16 - 2 ) !} x^{14} (3)^{2} : 360 x^{14}

Semplificare

\frac{16 !}{3 ! ( 16 - 3 ) !} x^{13} (3)^{3} : 16803 x^{13}

Semplificare

\frac{16 !}{4 ! ( 16 - 4 ) !} x^{12} (3)^{4} : 16380 x^{12}

Semplificare

\frac{16 !}{5 ! ( 16 - 5 ) !} x^{11} (3)^{5} : 393123 x^{11}

Semplificare

\frac{16 !}{6 ! ( 16 - 6 ) !} x^{10} (3)^{6} : 216216 x^{10}

Semplificare

\frac{16 !}{7 ! ( 16 - 7 ) !} x^{9} (3)^{7} : 3088803 x^{9}

Semplificare

\frac{16 !}{8 ! ( 16 - 8 ) !} x^{8} (3)^{8} : 1042470 x^{8}

Semplificare

\frac{16 !}{9 ! ( 16 - 9 ) !} x^{7} (3)^{9} : 9266403 x^{7}

Semplificare

\frac{16 !}{10 ! ( 16 - 10 ) !} x^{6} (3)^{10} : 1945944 x^{6}

Semplificare

\frac{16 !}{11 ! ( 16 - 11 ) !} x^{5} (3)^{11} : 10614243 x^{5}

Semplificare

\frac{16 !}{12 ! ( 16 - 12 ) !} x^{4} (3)^{12} : 1326780 x^{4}

Semplificare

\frac{16 !}{13 ! ( 16 - 13 ) !} x^{3} (3)^{13} : 4082403 x^{3}

Semplificare

\frac{16 !}{14 ! ( 16 - 14 ) !} x^{2} (3)^{14} : 262440 x^{2}

Semplificare

\frac{16 !}{15 ! ( 16 - 15 ) !} x^{1} (3)^{15} : 349923 x

Semplificare

\frac{16 !}{16 ! ( 16 - 16 ) !} x^{0} (3)^{16} : 6561

= x^{16} + 163 x^{15} + 360 x^{14} + 16803 x^{13} + 16380 x^{12} + 393123 x^{11} + 216216 x^{10} + 3088803 x^{9} + 1042470 x^{8} + 9266403 x^{7} + 1945944 x^{6} + 10614243 x^{5} + 1326780 x^{4} + 4082403 x^{3} + 262440 x^{2} + 349923 x + 6561

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