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Popolare Algebra >

espandere (x+3y)^9

Soluzione

espandere

(x + 3 y)^{9}

Soluzione

x^{9} + 27 x^{8} y + 324 x^{7} y^{2} + 2268 x^{6} y^{3} + 10206 x^{5} y^{4} + 30618 x^{4} y^{5} + 61236 x^{3} y^{6} + 78732 x^{2} y^{7} + 59049 x y^{8} + 19683 y^{9}

Fasi della soluzione

(x + 3 y)^{9}

Applicare il teorema binomiale:

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

a = x, b = 3 y

= i = 0 \sum 9 (i 9) x^{(9 - i)} (3 y)^{i}

Espandere la sommatoria

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

Semplificare

\frac{9 !}{0 ! ( 9 - 0 ) !} x^{9} (3 y)^{0} : x^{9}

Semplificare

\frac{9 !}{1 ! ( 9 - 1 ) !} x^{8} (3 y)^{1} : 27 x^{8} y

Semplificare

\frac{9 !}{2 ! ( 9 - 2 ) !} x^{7} (3 y)^{2} : 324 x^{7} y^{2}

Semplificare

\frac{9 !}{3 ! ( 9 - 3 ) !} x^{6} (3 y)^{3} : 2268 x^{6} y^{3}

Semplificare

\frac{9 !}{4 ! ( 9 - 4 ) !} x^{5} (3 y)^{4} : 10206 x^{5} y^{4}

Semplificare

\frac{9 !}{5 ! ( 9 - 5 ) !} x^{4} (3 y)^{5} : 30618 x^{4} y^{5}

Semplificare

\frac{9 !}{6 ! ( 9 - 6 ) !} x^{3} (3 y)^{6} : 61236 x^{3} y^{6}

Semplificare

\frac{9 !}{7 ! ( 9 - 7 ) !} x^{2} (3 y)^{7} : 78732 x^{2} y^{7}

Semplificare

\frac{9 !}{8 ! ( 9 - 8 ) !} x^{1} (3 y)^{8} : 59049 x y^{8}

Semplificare

\frac{9 !}{9 ! ( 9 - 9 ) !} x^{0} (3 y)^{9} : 19683 y^{9}

= x^{9} + 27 x^{8} y + 324 x^{7} y^{2} + 2268 x^{6} y^{3} + 10206 x^{5} y^{4} + 30618 x^{4} y^{5} + 61236 x^{3} y^{6} + 78732 x^{2} y^{7} + 59049 x y^{8} + 19683 y^{9}

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