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Popular Álgebra >

expandir (1+x)^{11}

Solução

expandir

(1 + x)^{11}

Solução

1 + 11 x + 55 x^{2} + 165 x^{3} + 330 x^{4} + 462 x^{5} + 462 x^{6} + 330 x^{7} + 165 x^{8} + 55 x^{9} + 11 x^{10} + x^{11}

Passos da solução

(1 + x)^{11}

Aplicar o teorema do binômio:

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

a = 1, b = x

= i = 0 \sum 11 (i 11) \cdot 1^{(11 - i)} x^{i}

Expandir a soma

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

Simplificar

\frac{11 !}{0 ! ( 11 - 0 ) !} \cdot 1^{11} x^{0} : 1

Simplificar

\frac{11 !}{1 ! ( 11 - 1 ) !} \cdot 1^{10} x^{1} : 11 x

Simplificar

\frac{11 !}{2 ! ( 11 - 2 ) !} \cdot 1^{9} x^{2} : 55 x^{2}

Simplificar

\frac{11 !}{3 ! ( 11 - 3 ) !} \cdot 1^{8} x^{3} : 165 x^{3}

Simplificar

\frac{11 !}{4 ! ( 11 - 4 ) !} \cdot 1^{7} x^{4} : 330 x^{4}

Simplificar

\frac{11 !}{5 ! ( 11 - 5 ) !} \cdot 1^{6} x^{5} : 462 x^{5}

Simplificar

\frac{11 !}{6 ! ( 11 - 6 ) !} \cdot 1^{5} x^{6} : 462 x^{6}

Simplificar

\frac{11 !}{7 ! ( 11 - 7 ) !} \cdot 1^{4} x^{7} : 330 x^{7}

Simplificar

\frac{11 !}{8 ! ( 11 - 8 ) !} \cdot 1^{3} x^{8} : 165 x^{8}

Simplificar

\frac{11 !}{9 ! ( 11 - 9 ) !} \cdot 1^{2} x^{9} : 55 x^{9}

Simplificar

\frac{11 !}{10 ! ( 11 - 10 ) !} \cdot 1^{1} x^{10} : 11 x^{10}

Simplificar

\frac{11 !}{11 ! ( 11 - 11 ) !} \cdot 1^{0} x^{11} : x^{11}

= 1 + 11 x + 55 x^{2} + 165 x^{3} + 330 x^{4} + 462 x^{5} + 462 x^{6} + 330 x^{7} + 165 x^{8} + 55 x^{9} + 11 x^{10} + x^{11}

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