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Popular Álgebra >

expandir (x-3)^7

Solução

expandir

(x - 3)^{7}

Solução

x^{7} - 21 x^{6} + 189 x^{5} - 945 x^{4} + 2835 x^{3} - 5103 x^{2} + 5103 x - 2187

Passos da solução

(x - 3)^{7}

Aplicar o teorema do binômio:

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

a = x, b = - 3

= i = 0 \sum 7 (i 7) x^{(7 - i)} (- 3)^{i}

Expandir a soma

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

Simplificar

\frac{7 !}{0 ! ( 7 - 0 ) !} x^{7} (- 3)^{0} : x^{7}

Simplificar

\frac{7 !}{1 ! ( 7 - 1 ) !} x^{6} (- 3)^{1} : - 21 x^{6}

Simplificar

\frac{7 !}{2 ! ( 7 - 2 ) !} x^{5} (- 3)^{2} : 189 x^{5}

Simplificar

\frac{7 !}{3 ! ( 7 - 3 ) !} x^{4} (- 3)^{3} : - 945 x^{4}

Simplificar

\frac{7 !}{4 ! ( 7 - 4 ) !} x^{3} (- 3)^{4} : 2835 x^{3}

Simplificar

\frac{7 !}{5 ! ( 7 - 5 ) !} x^{2} (- 3)^{5} : - 5103 x^{2}

Simplificar

\frac{7 !}{6 ! ( 7 - 6 ) !} x^{1} (- 3)^{6} : 5103 x

Simplificar

\frac{7 !}{7 ! ( 7 - 7 ) !} x^{0} (- 3)^{7} : - 2187

= x^{7} - 21 x^{6} + 189 x^{5} - 945 x^{4} + 2835 x^{3} - 5103 x^{2} + 5103 x - 2187

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