Question

Записать разложение бинома: $( \sqrt{x} - \frac{1}{ \sqrt{x}} )^{5}$

Answer 1

$(\sqrt{x}-\frac{1}{\sqrt{x}})^5=\\=C^0_5(\sqrt{x})^5*(\frac{1}{\sqrt{x}})^0-C^1_5(\sqrt{x})^4*(\frac{1}{\sqrt{x}})^1+C^2_5(\sqrt{x})^3*(\frac{1}{\sqrt{x}})^2-\\-C^3_5(\sqrt{x})^2*(\frac{1}{\sqrt{x}})^3+C^4_5(\sqrt{x})^1*(\frac{1}{\sqrt{x}})^4-C^5_5(\sqrt{x})^0(\frac{1}{\sqrt{x}})^5=\\=1*\sqrt{x^5}-5*\frac{x^2}{\sqrt{x}}+10*\frac{\sqrt{x^3}}{x}-10*\frac{x}{\sqrt{x^3}}+5*\frac{\sqrt{x}}{x^2}-1*\frac{1}{\sqrt{x^5}}=$
$=\sqrt{x^5}-5\sqrt{x^3}+10\sqrt{x}-\frac{10}{\sqrt{x}}+\frac{5}{\sqrt{x^3}}-\frac{1}{\sqrt{x^5}}$

$C_n^m=\frac{n!}{k!(n-k)!}$ - биномиальный коэффициент.