\(x^3-xy+1=2y-x\)
\(\Leftrightarrow x^3+x+1=xy+2y\)
\(\Leftrightarrow x^3+x+1=y\left(x+2\right)\)
\(\Leftrightarrow y=\dfrac{x^3+x+1}{x+2}\)
-Vì \(x,y\) là các số nguyên nên:
\(\left(x^3+x+1\right)⋮\left(x+2\right)\)
\(\Rightarrow\left(x^3+2x^2-2x^2-4x+5x+10-9\right)⋮\left(x+2\right)\)
\(\Rightarrow\left[x^2\left(x+2\right)-2x\left(x+2\right)+5\left(x+2\right)-9\right]⋮\left(x+2\right)\)
\(\Rightarrow\left[\left(x+2\right)\left(x^2-2x+5\right)-9\right]⋮\left(x+2\right)\)
-Vì \(\left(x+2\right)\left(x^2-2x+5\right)⋮\left(x+2\right)\)
\(\Rightarrow9⋮\left(x+2\right)\)
\(\Rightarrow\left(x+2\right)\in\left\{1;3;9;-1;-3;-9\right\}\)
\(\Rightarrow x\in\left\{-1;1;7;-3;-5;-11\right\}\) (tmđk)
*Với \(x=-1\) thì \(y=\dfrac{\left(-1\right)^3+\left(-1\right)+1}{\left(-1\right)+2}=-1\) (tmđk)
*Với \(x=1\) thì \(y=\dfrac{1^3+1+1}{1+2}=1\)(tmđk)
*Với \(x=7\) thì \(y=\dfrac{7^3+7+1}{7+2}=39\)(tmđk)
*Với \(x=-3\) thì \(y=\dfrac{\left(-3\right)^3+\left(-3\right)+1}{\left(-3\right)+2}=29\)(tmđk)
*Với \(x=-5\) thì \(y=\dfrac{\left(-5\right)^3+\left(-5\right)+1}{\left(-5\right)+2}=43\)(tmđk)
*Với \(x=-11\) thì \(y=\dfrac{\left(-11\right)^3+\left(-11\right)+1}{\left(-11\right)+2}=149\)(tmđk)
