Why can one equate polynomials by "inside power"s, when solving through expansion?

Question

Why can one equate $x^3-x+\epsilon=0$ by "inside power"s, when solving through expansion?

https://www.maths.nottingham.ac.uk/plp/pmzjb1/G13AMD/book2.pdf, p.1

The author expands $x^3-x+\epsilon=0$ with $x=x_0+\epsilon x_1 + \epsilon^2 x_2 + O(\epsilon^3)$, like

$$(x_0+\color{red}{\epsilon x_1} + \epsilon^2 x_2)^3-(x_0+\color{red}{\epsilon x_1} + \epsilon^2 x_2)+\color{red}{\epsilon}=0 \space (eq.1)$$ (and further)

Then groups terms by $\epsilon$ powers.

Then equates terms grouped by $\epsilon$ powers with what's "inside powers" in $(eq.1)$.

So that e.g. for $O(\epsilon)$ level:

$$(3x_0-1)x_1+1=\color{red}{2x_1}+\color{red}{1}=0$$

My question:

Why is one allowed to equate things that are "inside powers", like the first redded term?

Answer 1

he is trying to find an approximate solution for small value of $\epsilon$.SInce right hand side is zero, the author equates all the coefficients of $\epsilon$ to Zero.

Answer 2

Since when $\epsilon=0$ the given equation has simple roots at $x=-1,0,1$ labeled by $x_1,x_2,x_3$, respectively, we can apply inverse function theorem near each $x_i$ to conclude that parametrized root $x_i(\epsilon)$ is in fact an analytic function of $\epsilon\in \mathbb{C}$. (with the domain being $|\epsilon|<\eta$ for sufficiently small $\eta>0$.)