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Popular Álgebra >

expandir (3a-2b)^5

Solução

expandir

(3 a - 2 b)^{5}

Solução

243 a^{5} - 810 a^{4} b + 1080 a^{3} b^{2} - 720 a^{2} b^{3} + 240 a b^{4} - 32 b^{5}

Passos da solução

(3 a - 2 b)^{5}

Aplicar o teorema do binômio:

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

a = 3 a, b = - 2 b

= i = 0 \sum 5 (i 5) (3 a)^{(5 - i)} (- 2 b)^{i}

Expandir a soma

= \frac{5 !}{0 ! ( 5 - 0 ) !} (3 a)^{5} (- 2 b)^{0} + \frac{5 !}{1 ! ( 5 - 1 ) !} (3 a)^{4} (- 2 b)^{1} + \frac{5 !}{2 ! ( 5 - 2 ) !} (3 a)^{3} (- 2 b)^{2} + \frac{5 !}{3 ! ( 5 - 3 ) !} (3 a)^{2} (- 2 b)^{3} + \frac{5 !}{4 ! ( 5 - 4 ) !} (3 a)^{1} (- 2 b)^{4} + \frac{5 !}{5 ! ( 5 - 5 ) !} (3 a)^{0} (- 2 b)^{5}

Simplificar

\frac{5 !}{0 ! ( 5 - 0 ) !} (3 a)^{5} (- 2 b)^{0} : 243 a^{5}

Simplificar

\frac{5 !}{1 ! ( 5 - 1 ) !} (3 a)^{4} (- 2 b)^{1} : - 810 a^{4} b

Simplificar

\frac{5 !}{2 ! ( 5 - 2 ) !} (3 a)^{3} (- 2 b)^{2} : 1080 a^{3} b^{2}

Simplificar

\frac{5 !}{3 ! ( 5 - 3 ) !} (3 a)^{2} (- 2 b)^{3} : - 720 a^{2} b^{3}

Simplificar

\frac{5 !}{4 ! ( 5 - 4 ) !} (3 a)^{1} (- 2 b)^{4} : 240 a b^{4}

Simplificar

\frac{5 !}{5 ! ( 5 - 5 ) !} (3 a)^{0} (- 2 b)^{5} : - 32 b^{5}

= 243 a^{5} - 810 a^{4} b + 1080 a^{3} b^{2} - 720 a^{2} b^{3} + 240 a b^{4} - 32 b^{5}

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