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Popular Álgebra >

expandir (x+y)^{11}

Solução

expandir

(x + y)^{11}

Solução

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

Passos da solução

(x + y)^{11}

Aplicar o teorema do binômio:

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

a = x, b = y

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

Expandir a soma

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

Simplificar

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

Simplificar

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

Simplificar

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

Simplificar

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

Simplificar

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

Simplificar

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

Simplificar

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

Simplificar

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

Simplificar

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

Simplificar

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

Simplificar

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

Simplificar

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

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

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