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Popolare Algebra >

espandere (x^2+5)^{10}

Soluzione

espandere

(x^{2} + 5)^{10}

Soluzione

x^{20} + 50 x^{18} + 1125 x^{16} + 15000 x^{14} + 131250 x^{12} + 787500 x^{10} + 3281250 x^{8} + 9375000 x^{6} + 17578125 x^{4} + 19531250 x^{2} + 9765625

Fasi della soluzione

(x^{2} + 5)^{10}

Applicare il teorema binomiale:

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

a = x^{2}, b = 5

= i = 0 \sum 10 (i 10) (x^{2})^{(10 - i)} \cdot 5^{i}

Espandere la sommatoria

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

Semplificare

\frac{10 !}{0 ! ( 10 - 0 ) !} (x^{2})^{10} \cdot 5^{0} : x^{20}

Semplificare

\frac{10 !}{1 ! ( 10 - 1 ) !} (x^{2})^{9} \cdot 5^{1} : 50 x^{18}

Semplificare

\frac{10 !}{2 ! ( 10 - 2 ) !} (x^{2})^{8} \cdot 5^{2} : 1125 x^{16}

Semplificare

\frac{10 !}{3 ! ( 10 - 3 ) !} (x^{2})^{7} \cdot 5^{3} : 15000 x^{14}

Semplificare

\frac{10 !}{4 ! ( 10 - 4 ) !} (x^{2})^{6} \cdot 5^{4} : 131250 x^{12}

Semplificare

\frac{10 !}{5 ! ( 10 - 5 ) !} (x^{2})^{5} \cdot 5^{5} : 787500 x^{10}

Semplificare

\frac{10 !}{6 ! ( 10 - 6 ) !} (x^{2})^{4} \cdot 5^{6} : 3281250 x^{8}

Semplificare

\frac{10 !}{7 ! ( 10 - 7 ) !} (x^{2})^{3} \cdot 5^{7} : 9375000 x^{6}

Semplificare

\frac{10 !}{8 ! ( 10 - 8 ) !} (x^{2})^{2} \cdot 5^{8} : 17578125 x^{4}

Semplificare

\frac{10 !}{9 ! ( 10 - 9 ) !} (x^{2})^{1} \cdot 5^{9} : 19531250 x^{2}

Semplificare

\frac{10 !}{10 ! ( 10 - 10 ) !} (x^{2})^{0} \cdot 5^{10} : 9765625

= x^{20} + 50 x^{18} + 1125 x^{16} + 15000 x^{14} + 131250 x^{12} + 787500 x^{10} + 3281250 x^{8} + 9375000 x^{6} + 17578125 x^{4} + 19531250 x^{2} + 9765625

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