IME / ITA ⇒ (IME - 1989) Binômio de Newton Tópico resolvido

[brain_tnt]

Determine o coeficiente de [tex3]x^{-9}[/tex3] no desenvolvimento de [tex3]\left(x^2+\frac{1}{x^5}\right)^2\cdot \left(x^3+\frac{1}{x^4}\right)^5.[/tex3]

[Thadeu]

[tex3]\(\frac{x^7+1}{x^5}\)^2\,.\,\(\frac{x^7+1}{x^4}\)^5\\\frac{(x^7+1)^2}{x^{10}}\,.\,\frac{(x^7+1)^5}{x^{20}}\\\frac{\(x^7+1\)^7}{x^{30}}[/tex3]

No binômio do numerador, [tex3](x^7+1)^7[/tex3], vamos encontrar o termo [tex3]x^{21}[/tex3]

[tex3]\binom{7}{n}\(x^7\)^{7-n}\,(1)^{n}\\\binom{7}{n}\,x^{49-7n}\,\Rightarrow\,49-7n=21\,\Rightarrow\,n=4\\\binom{7}{4}\,x^{21}=\(\frac{7!}{4!\cdot 3!}\)x^{21}\,\Rightarrow\,\(\frac{7\,\cdot \,6\,\cdot \,5\,\cdot \,4!}{4!\,6}\)x^{21}=35x^{21}[/tex3]
Logo:
[tex3]\frac{35x^{21}}{x^{30}}=35x^{-9}[/tex3]