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Popolare Algebra >

espandere (x+1)^{19}

Soluzione

espandere

(x + 1)^{19}

Soluzione

x^{19} + 19 x^{18} + 171 x^{17} + 969 x^{16} + 3876 x^{15} + 11628 x^{14} + 27132 x^{13} + 50388 x^{12} + 75582 x^{11} + 92378 x^{10} + 92378 x^{9} + 75582 x^{8} + 50388 x^{7} + 27132 x^{6} + 11628 x^{5} + 3876 x^{4} + 969 x^{3} + 171 x^{2} + 19 x + 1

Fasi della soluzione

(x + 1)^{19}

Applicare il teorema binomiale:

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

a = x, b = 1

= i = 0 \sum 19 (i 19) x^{(19 - i)} \cdot 1^{i}

Espandere la sommatoria

= \frac{19 !}{0 ! ( 19 - 0 ) !} x^{19} \cdot 1^{0} + \frac{19 !}{1 ! ( 19 - 1 ) !} x^{18} \cdot 1^{1} + \frac{19 !}{2 ! ( 19 - 2 ) !} x^{17} \cdot 1^{2} + \frac{19 !}{3 ! ( 19 - 3 ) !} x^{16} \cdot 1^{3} + \frac{19 !}{4 ! ( 19 - 4 ) !} x^{15} \cdot 1^{4} + \frac{19 !}{5 ! ( 19 - 5 ) !} x^{14} \cdot 1^{5} + \frac{19 !}{6 ! ( 19 - 6 ) !} x^{13} \cdot 1^{6} + \frac{19 !}{7 ! ( 19 - 7 ) !} x^{12} \cdot 1^{7} + \frac{19 !}{8 ! ( 19 - 8 ) !} x^{11} \cdot 1^{8} + \frac{19 !}{9 ! ( 19 - 9 ) !} x^{10} \cdot 1^{9} + \frac{19 !}{10 ! ( 19 - 10 ) !} x^{9} \cdot 1^{10} + \frac{19 !}{11 ! ( 19 - 11 ) !} x^{8} \cdot 1^{11} + \frac{19 !}{12 ! ( 19 - 12 ) !} x^{7} \cdot 1^{12} + \frac{19 !}{13 ! ( 19 - 13 ) !} x^{6} \cdot 1^{13} + \frac{19 !}{14 ! ( 19 - 14 ) !} x^{5} \cdot 1^{14} + \frac{19 !}{15 ! ( 19 - 15 ) !} x^{4} \cdot 1^{15} + \frac{19 !}{16 ! ( 19 - 16 ) !} x^{3} \cdot 1^{16} + \frac{19 !}{17 ! ( 19 - 17 ) !} x^{2} \cdot 1^{17} + \frac{19 !}{18 ! ( 19 - 18 ) !} x^{1} \cdot 1^{18} + \frac{19 !}{19 ! ( 19 - 19 ) !} x^{0} \cdot 1^{19}

Semplificare

\frac{19 !}{0 ! ( 19 - 0 ) !} x^{19} \cdot 1^{0} : x^{19}

Semplificare

\frac{19 !}{1 ! ( 19 - 1 ) !} x^{18} \cdot 1^{1} : 19 x^{18}

Semplificare

\frac{19 !}{2 ! ( 19 - 2 ) !} x^{17} \cdot 1^{2} : 171 x^{17}

Semplificare

\frac{19 !}{3 ! ( 19 - 3 ) !} x^{16} \cdot 1^{3} : 969 x^{16}

Semplificare

\frac{19 !}{4 ! ( 19 - 4 ) !} x^{15} \cdot 1^{4} : 3876 x^{15}

Semplificare

\frac{19 !}{5 ! ( 19 - 5 ) !} x^{14} \cdot 1^{5} : 11628 x^{14}

Semplificare

\frac{19 !}{6 ! ( 19 - 6 ) !} x^{13} \cdot 1^{6} : 27132 x^{13}

Semplificare

\frac{19 !}{7 ! ( 19 - 7 ) !} x^{12} \cdot 1^{7} : 50388 x^{12}

Semplificare

\frac{19 !}{8 ! ( 19 - 8 ) !} x^{11} \cdot 1^{8} : 75582 x^{11}

Semplificare

\frac{19 !}{9 ! ( 19 - 9 ) !} x^{10} \cdot 1^{9} : 92378 x^{10}

Semplificare

\frac{19 !}{10 ! ( 19 - 10 ) !} x^{9} \cdot 1^{10} : 92378 x^{9}

Semplificare

\frac{19 !}{11 ! ( 19 - 11 ) !} x^{8} \cdot 1^{11} : 75582 x^{8}

Semplificare

\frac{19 !}{12 ! ( 19 - 12 ) !} x^{7} \cdot 1^{12} : 50388 x^{7}

Semplificare

\frac{19 !}{13 ! ( 19 - 13 ) !} x^{6} \cdot 1^{13} : 27132 x^{6}

Semplificare

\frac{19 !}{14 ! ( 19 - 14 ) !} x^{5} \cdot 1^{14} : 11628 x^{5}

Semplificare

\frac{19 !}{15 ! ( 19 - 15 ) !} x^{4} \cdot 1^{15} : 3876 x^{4}

Semplificare

\frac{19 !}{16 ! ( 19 - 16 ) !} x^{3} \cdot 1^{16} : 969 x^{3}

Semplificare

\frac{19 !}{17 ! ( 19 - 17 ) !} x^{2} \cdot 1^{17} : 171 x^{2}

Semplificare

\frac{19 !}{18 ! ( 19 - 18 ) !} x^{1} \cdot 1^{18} : 19 x

Semplificare

\frac{19 !}{19 ! ( 19 - 19 ) !} x^{0} \cdot 1^{19} : 1

= x^{19} + 19 x^{18} + 171 x^{17} + 969 x^{16} + 3876 x^{15} + 11628 x^{14} + 27132 x^{13} + 50388 x^{12} + 75582 x^{11} + 92378 x^{10} + 92378 x^{9} + 75582 x^{8} + 50388 x^{7} + 27132 x^{6} + 11628 x^{5} + 3876 x^{4} + 969 x^{3} + 171 x^{2} + 19 x + 1

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