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Popolare Algebra >

espandere (x+y)^{13}

Soluzione

espandere

(x + y)^{13}

Soluzione

x^{13} + 13 x^{12} y + 78 x^{11} y^{2} + 286 x^{10} y^{3} + 715 x^{9} y^{4} + 1287 x^{8} y^{5} + 1716 x^{7} y^{6} + 1716 x^{6} y^{7} + 1287 x^{5} y^{8} + 715 x^{4} y^{9} + 286 x^{3} y^{10} + 78 x^{2} y^{11} + 13 x y^{12} + y^{13}

Fasi della soluzione

(x + y)^{13}

Applicare il teorema binomiale:

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

a = x, b = y

= i = 0 \sum 13 (i 13) x^{(13 - i)} y^{i}

Espandere la sommatoria

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

Semplificare

\frac{13 !}{0 ! ( 13 - 0 ) !} x^{13} y^{0} : x^{13}

Semplificare

\frac{13 !}{1 ! ( 13 - 1 ) !} x^{12} y^{1} : 13 x^{12} y

Semplificare

\frac{13 !}{2 ! ( 13 - 2 ) !} x^{11} y^{2} : 78 x^{11} y^{2}

Semplificare

\frac{13 !}{3 ! ( 13 - 3 ) !} x^{10} y^{3} : 286 x^{10} y^{3}

Semplificare

\frac{13 !}{4 ! ( 13 - 4 ) !} x^{9} y^{4} : 715 x^{9} y^{4}

Semplificare

\frac{13 !}{5 ! ( 13 - 5 ) !} x^{8} y^{5} : 1287 x^{8} y^{5}

Semplificare

\frac{13 !}{6 ! ( 13 - 6 ) !} x^{7} y^{6} : 1716 x^{7} y^{6}

Semplificare

\frac{13 !}{7 ! ( 13 - 7 ) !} x^{6} y^{7} : 1716 x^{6} y^{7}

Semplificare

\frac{13 !}{8 ! ( 13 - 8 ) !} x^{5} y^{8} : 1287 x^{5} y^{8}

Semplificare

\frac{13 !}{9 ! ( 13 - 9 ) !} x^{4} y^{9} : 715 x^{4} y^{9}

Semplificare

\frac{13 !}{10 ! ( 13 - 10 ) !} x^{3} y^{10} : 286 x^{3} y^{10}

Semplificare

\frac{13 !}{11 ! ( 13 - 11 ) !} x^{2} y^{11} : 78 x^{2} y^{11}

Semplificare

\frac{13 !}{12 ! ( 13 - 12 ) !} x^{1} y^{12} : 13 x y^{12}

Semplificare

\frac{13 !}{13 ! ( 13 - 13 ) !} x^{0} y^{13} : y^{13}

= x^{13} + 13 x^{12} y + 78 x^{11} y^{2} + 286 x^{10} y^{3} + 715 x^{9} y^{4} + 1287 x^{8} y^{5} + 1716 x^{7} y^{6} + 1716 x^{6} y^{7} + 1287 x^{5} y^{8} + 715 x^{4} y^{9} + 286 x^{3} y^{10} + 78 x^{2} y^{11} + 13 x y^{12} + y^{13}

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