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Popular Álgebra >

expandir (X+Y)^7

Solução

expandir

(X + Y)^{7}

Solução

X^{7} + 7 X^{6} Y + 21 X^{5} Y^{2} + 35 X^{4} Y^{3} + 35 X^{3} Y^{4} + 21 X^{2} Y^{5} + 7 X Y^{6} + Y^{7}

Passos da solução

(X + Y)^{7}

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 7 (i 7) X^{(7 - i)} Y^{i}

Expandir a soma

= \frac{7 !}{0 ! ( 7 - 0 ) !} X^{7} Y^{0} + \frac{7 !}{1 ! ( 7 - 1 ) !} X^{6} Y^{1} + \frac{7 !}{2 ! ( 7 - 2 ) !} X^{5} Y^{2} + \frac{7 !}{3 ! ( 7 - 3 ) !} X^{4} Y^{3} + \frac{7 !}{4 ! ( 7 - 4 ) !} X^{3} Y^{4} + \frac{7 !}{5 ! ( 7 - 5 ) !} X^{2} Y^{5} + \frac{7 !}{6 ! ( 7 - 6 ) !} X^{1} Y^{6} + \frac{7 !}{7 ! ( 7 - 7 ) !} X^{0} Y^{7}

Simplificar

\frac{7 !}{0 ! ( 7 - 0 ) !} X^{7} Y^{0} : X^{7}

Simplificar

\frac{7 !}{1 ! ( 7 - 1 ) !} X^{6} Y^{1} : 7 X^{6} Y

Simplificar

\frac{7 !}{2 ! ( 7 - 2 ) !} X^{5} Y^{2} : 21 X^{5} Y^{2}

Simplificar

\frac{7 !}{3 ! ( 7 - 3 ) !} X^{4} Y^{3} : 35 X^{4} Y^{3}

Simplificar

\frac{7 !}{4 ! ( 7 - 4 ) !} X^{3} Y^{4} : 35 X^{3} Y^{4}

Simplificar

\frac{7 !}{5 ! ( 7 - 5 ) !} X^{2} Y^{5} : 21 X^{2} Y^{5}

Simplificar

\frac{7 !}{6 ! ( 7 - 6 ) !} X^{1} Y^{6} : 7 X Y^{6}

Simplificar

\frac{7 !}{7 ! ( 7 - 7 ) !} X^{0} Y^{7} : Y^{7}

= X^{7} + 7 X^{6} Y + 21 X^{5} Y^{2} + 35 X^{4} Y^{3} + 35 X^{3} Y^{4} + 21 X^{2} Y^{5} + 7 X Y^{6} + Y^{7}

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