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Use dominant balance for algebraic roots to separate the three scales. On writing $x=ϵX$, the equation becomes $X^3-X^2+2ϵX+2{ϵ}^3=0$. Its nonzero simple limiting root is $X=1$, so one root has $x_1\sim ϵ$.

For the two remaining roots put $x={ϵ}^{2}Y$. Division by ${ϵ}^5$ gives $ϵY^3-Y^2+2Y+2ϵ=0$. The nonzero limiting root is $Y=2$, giving $x_2\sim 2{ϵ}^2$. Finally $x={ϵ}^{3}Z$ gives, after division by ${ϵ}^6$,Its simple limiting root is $Z=-1$. These three distinct balances account for all roots of the cubic, andFor the root of smallest magnitude, set $Z=-1+bϵ+O({ϵ}^2)$. The order-$ϵ$ coefficient is $-1+2b$, so $b=1/2$. HenceEach rescaled root is simple at $ϵ=0$, justifying the regular perturbation of a simple algebraic root rather than treating the original triple root as an ordinary regular perturbation.

A regular perturbation admits an asymptotic expansion about the limiting problem on the relevant scale without losing its solutions or conditions. A singular perturbation changes the limiting structure, so some solutions require a different scale: a differential equation may lose an order, or a polynomial may lose its degree.

The two finite roots approach $s=\pm \sqrt{2}$. Set $x=s+ϵa+O({ϵ}^2)$. Substitution into the polynomial gives $s^2-2=0$ at leading order and $s^3+2sa=0$ at the next order. Since $s^2=2$, $a=-1$ for either sign. Each branch is a regular perturbation of a simple algebraic root, withThe limiting quadratic has only two roots, whereas the original equation has three. The missing root is a divergent algebraic root in a singular perturbation. The dominant balance for algebraic roots sets $x={ϵ}^{-1}X$, giving $X^3+X^2=2{ϵ}^2$. Its nonzero leading root is $X=-1$. Put $X=-1+a{ϵ}^2+O({ϵ}^4)$; the left side is $a{ϵ}^2+O({ϵ}^4)$, so $a=2$. Hence the second nonzero term occurs at order $ϵ$, not order one:For small nonzero real $ϵ$ these three branches are real and distinct; the divergent branch changes sign with $ϵ$. The two bounded branches are regular even though the full root problem is singular.

A regular perturbation admits an asymptotic expansion about the limiting problem without changing its relevant scale or losing solutions or conditions. A regular perturbation of a simple algebraic root follows from the implicit function theorem. A full problem can still be a singular perturbation even when some individual solution branches are regular.