Question

Como posso ter os desenvolvimentos dos seguintes binomios: a) (X-1)^6 b) (X+Y/X)^4

Answer 1

Olá, Benedito.

Binômio de Newton: ${\left(x+y\right)}^n=\sum_{k=0}^n{n \choose k}x^{n-k}y^k$


$a)\,\,{\left(x-1\right)}^6=\sum_{k=0}^6{6 \choose k}x^{6-k}(-1)^k=\\\\={6 \choose 0}x^{6}(-1)^0+{6 \choose 1}x^{5}(-1)^1+{6 \choose 2}x^{4}(-1)^2+{6 \choose 3}x^{3}(-1)^3+\\\\+{6 \choose 4}x^{2}(-1)^4+{6 \choose 5}x(-1)^5+{6 \choose 6}x^{0}(-1)^6$


$b)\,\,(x+\frac y x)^4=\sum_{k=0}^4{4 \choose k}x^{4-k}(\frac y x)^k=\\\\={4 \choose 0}x^{4}(\frac y x)^0+{4 \choose 1}x^{3}(\frac y x)^1+{4 \choose 2}x^{2}(\frac y x)^2+{4 \choose 3}x(\frac y x)^3+{4 \choose 4}x^{0}(\frac y x)^4$


Desenvolva agora os coeficientes binomiais, lembrando que:

$\binom n k = \frac{n!}{(n-k)!k!}$