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Popolare Algebra >

espandere (x+3)^{10}

Soluzione

espandere

(x + 3)^{10}

Soluzione

x^{10} + 30 x^{9} + 405 x^{8} + 3240 x^{7} + 17010 x^{6} + 61236 x^{5} + 153090 x^{4} + 262440 x^{3} + 295245 x^{2} + 196830 x + 59049

Fasi della soluzione

(x + 3)^{10}

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 10 (i 10) x^{(10 - i)} \cdot 3^{i}

Espandere la sommatoria

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

Semplificare

\frac{10 !}{0 ! ( 10 - 0 ) !} x^{10} \cdot 3^{0} : x^{10}

Semplificare

\frac{10 !}{1 ! ( 10 - 1 ) !} x^{9} \cdot 3^{1} : 30 x^{9}

Semplificare

\frac{10 !}{2 ! ( 10 - 2 ) !} x^{8} \cdot 3^{2} : 405 x^{8}

Semplificare

\frac{10 !}{3 ! ( 10 - 3 ) !} x^{7} \cdot 3^{3} : 3240 x^{7}

Semplificare

\frac{10 !}{4 ! ( 10 - 4 ) !} x^{6} \cdot 3^{4} : 17010 x^{6}

Semplificare

\frac{10 !}{5 ! ( 10 - 5 ) !} x^{5} \cdot 3^{5} : 61236 x^{5}

Semplificare

\frac{10 !}{6 ! ( 10 - 6 ) !} x^{4} \cdot 3^{6} : 153090 x^{4}

Semplificare

\frac{10 !}{7 ! ( 10 - 7 ) !} x^{3} \cdot 3^{7} : 262440 x^{3}

Semplificare

\frac{10 !}{8 ! ( 10 - 8 ) !} x^{2} \cdot 3^{8} : 295245 x^{2}

Semplificare

\frac{10 !}{9 ! ( 10 - 9 ) !} x^{1} \cdot 3^{9} : 196830 x

Semplificare

\frac{10 !}{10 ! ( 10 - 10 ) !} x^{0} \cdot 3^{10} : 59049

= x^{10} + 30 x^{9} + 405 x^{8} + 3240 x^{7} + 17010 x^{6} + 61236 x^{5} + 153090 x^{4} + 262440 x^{3} + 295245 x^{2} + 196830 x + 59049

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