Question

Cho biểu thức:

\(P=\left(\frac{x^3+y^3}{x+y}-xy\right):\left(x\sqrt{x}-y\sqrt{x}-x\sqrt{y}+y\sqrt{y}\right)\)

a) Rút gọn P

b) Tính P biết x,y là hai nghiệm của pt \(t^2-2015t+2016=0\)

Answer 1

a/

\(P=\left(\frac{x^3+y^3}{x+y}-xy\right):\left(x\sqrt{x}-y\sqrt{x}-x\sqrt{y}+y\sqrt{y}\right)\)

\(=\left(\frac{x^3+y^3-xy\left(x+y\right)}{x+y}\right):\left(\sqrt{x}\left(x-y\right)-\sqrt{y}\left(x-y\right)\right)\)

\(=\left(x-y\right)^2.\frac{1}{\left(\sqrt{x}-\sqrt{y}\right)\left(x-y\right)}=\sqrt{x}+\sqrt{y}\)

b/ Áp dụng vi-et ta có: \(\left\{\begin{matrix}x+y=2015\\xy=2016\end{matrix}\right.\)

\(P=\sqrt{x}+\sqrt{y}\)

\(\Rightarrow P^2=x+y+2\sqrt{xy}\)

\(=2015+2\sqrt{2016}\)

\(\Rightarrow P=\sqrt{2015+2\sqrt{2016}}\)