Question

Cho \(\left\{{}\begin{matrix}x+y=4\\x.y=1\end{matrix}\right.\)

và x > y .

a) Tính \(x^2+y^2,x^4+y^4,x^3+y^3,x^5+y^5\)

b) Tính \(x^2-y^2,x^4-y^4,x^3-y^3,x^6-y^6\)

Answer 1

\(\left(x+y\right)^2-2xy=x^2+y^2=4^2-2.1=14\)

\(x^4+y^4=\left(x^2+y^2\right)^2-2x^2y^2=14^2-2=196-2=194\)

\(x^3+y^3=\left(x+y\right)\left(x^2+y^2-xy\right)=4\left(14-1\right)=52\)

\(\left(x^4+y^4\right)\left(x+y\right)=194.4=776\Leftrightarrow x^5+y^5+x^4y+y^4x=\left(x^5+y^5\right)+xy\left(x^3+y^3\right)=\left(x^5+y^5\right)+1.52=\left(x^5+y^5\right)+52=776\Rightarrow x^5+y^5=724\)

Answer 2

\(\left\{{}\begin{matrix}x+y=4\\xy=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x^2+2xy+y^2=16\\4xy=4\end{matrix}\right.\Rightarrow x^2+2xy-4xy+y^2=\left(x-y\right)^2=12mà:x>y\Leftrightarrow x-y>0\Rightarrow x-y=\sqrt{12}=2\sqrt{3};x+y=2.2\Rightarrow\left\{{}\begin{matrix}x=\sqrt{3}+2\\y=2-\sqrt{3}\end{matrix}\right.\)

\(x^2-y^2=\left(x-y\right)\left(x+y\right)=4.2\sqrt{3}=8\sqrt{3}\)

\(\left(x^2+y^2\right)\left(x^2-y^2\right)=8\sqrt{3}.14=112\sqrt{3}\Rightarrow x^4-y^4=112\sqrt{3}\)

\(\left(x^3-y^3\right)=\left(x-y\right)\left(x^2+xy+y^2\right);x^6-y^6=\left(x^3+y^3\right)\left(x^3-y^3\right)tựlm\)