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Popolare Algebra >

espandere (e+2f)^{10}

Soluzione

espandere

(e + 2 f)^{10}

Soluzione

1024 f^{10} + 5120 e f^{9} + 11520 e^{2} f^{8} + 15360 e^{3} f^{7} + 13440 e^{4} f^{6} + 8064 e^{5} f^{5} + 3360 e^{6} f^{4} + 960 e^{7} f^{3} + 180 e^{8} f^{2} + 20 e^{9} f + e^{10}

Fasi della soluzione

(e + 2 f)^{10}

Applicare il teorema binomiale:

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

a = e, b = 2 f

= i = 0 \sum 10 (i 10) e^{(10 - i)} (2 f)^{i}

Espandere la sommatoria

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

\frac{10 !}{0 ! ( 10 - 0 ) !} e^{10} (2 f)^{0} = e^{10}

Semplifica

\frac{10 !}{1 ! ( 10 - 1 ) !} e^{9} (2 f)^{1} : 20 e^{9} f

Semplifica

\frac{10 !}{2 ! ( 10 - 2 ) !} e^{8} (2 f)^{2} : 180 e^{8} f^{2}

Semplifica

\frac{10 !}{3 ! ( 10 - 3 ) !} e^{7} (2 f)^{3} : 960 e^{7} f^{3}

Semplifica

\frac{10 !}{4 ! ( 10 - 4 ) !} e^{6} (2 f)^{4} : 3360 e^{6} f^{4}

Semplifica

\frac{10 !}{5 ! ( 10 - 5 ) !} e^{5} (2 f)^{5} : 8064 e^{5} f^{5}

Semplifica

\frac{10 !}{6 ! ( 10 - 6 ) !} e^{4} (2 f)^{6} : 13440 e^{4} f^{6}

Semplifica

\frac{10 !}{7 ! ( 10 - 7 ) !} e^{3} (2 f)^{7} : 15360 e^{3} f^{7}

Semplifica

\frac{10 !}{8 ! ( 10 - 8 ) !} e^{2} (2 f)^{8} : 11520 e^{2} f^{8}

Semplifica

\frac{10 !}{9 ! ( 10 - 9 ) !} e^{1} (2 f)^{9} : 5120 e f^{9}

Semplifica

\frac{10 !}{10 ! ( 10 - 10 ) !} e^{0} (2 f)^{10} : 1024 f^{10}

= e^{10} + 20 e^{9} f + 180 e^{8} f^{2} + 960 e^{7} f^{3} + 3360 e^{6} f^{4} + 8064 e^{5} f^{5} + 13440 e^{4} f^{6} + 15360 e^{3} f^{7} + 11520 e^{2} f^{8} + 5120 e f^{9} + 1024 f^{10}

Riscrivere in forma standard

= 1024 f^{10} + 5120 e f^{9} + 11520 e^{2} f^{8} + 15360 e^{3} f^{7} + 13440 e^{4} f^{6} + 8064 e^{5} f^{5} + 3360 e^{6} f^{4} + 960 e^{7} f^{3} + 180 e^{8} f^{2} + 20 e^{9} f + e^{10}

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