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Popolare Algebra >

espandere (3x+1)^{10}

Soluzione

espandere

(3 x + 1)^{10}

Soluzione

59049 x^{10} + 196830 x^{9} + 295245 x^{8} + 262440 x^{7} + 153090 x^{6} + 61236 x^{5} + 17010 x^{4} + 3240 x^{3} + 405 x^{2} + 30 x + 1

Fasi della soluzione

(3 x + 1)^{10}

Applicare il teorema binomiale:

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

a = 3 x, b = 1

= i = 0 \sum 10 (i 10) (3 x)^{(10 - i)} \cdot 1^{i}

Espandere la sommatoria

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

Semplificare

\frac{10 !}{0 ! ( 10 - 0 ) !} (3 x)^{10} \cdot 1^{0} : 59049 x^{10}

Semplificare

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

Semplificare

\frac{10 !}{2 ! ( 10 - 2 ) !} (3 x)^{8} \cdot 1^{2} : 295245 x^{8}

Semplificare

\frac{10 !}{3 ! ( 10 - 3 ) !} (3 x)^{7} \cdot 1^{3} : 262440 x^{7}

Semplificare

\frac{10 !}{4 ! ( 10 - 4 ) !} (3 x)^{6} \cdot 1^{4} : 153090 x^{6}

Semplificare

\frac{10 !}{5 ! ( 10 - 5 ) !} (3 x)^{5} \cdot 1^{5} : 61236 x^{5}

Semplificare

\frac{10 !}{6 ! ( 10 - 6 ) !} (3 x)^{4} \cdot 1^{6} : 17010 x^{4}

Semplificare

\frac{10 !}{7 ! ( 10 - 7 ) !} (3 x)^{3} \cdot 1^{7} : 3240 x^{3}

Semplificare

\frac{10 !}{8 ! ( 10 - 8 ) !} (3 x)^{2} \cdot 1^{8} : 405 x^{2}

Semplificare

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

Semplificare

\frac{10 !}{10 ! ( 10 - 10 ) !} (3 x)^{0} \cdot 1^{10} : 1

= 59049 x^{10} + 196830 x^{9} + 295245 x^{8} + 262440 x^{7} + 153090 x^{6} + 61236 x^{5} + 17010 x^{4} + 3240 x^{3} + 405 x^{2} + 30 x + 1

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