I have a quintic polynomial where the coefficients depends on a parameter $c$, i.e. $$ a_0(c)+a_1(c)x+a_2(c)x^2+a_3(c)x^3+a_4(c)x^4+x^5 $$ I know that the roots of the polynomial are real and non-negative, and that $a_0(c)=\alpha c^2$, so that for $c=0$ one has $a_0(c)=0$ and therefore in this case one root is $x=0$. I want to calculate how the root $x=0$ (the smallest root) is perturbed for $c\neq0$ at the second order in $c$. As far I understand, a typical approach is to consider a solution proportional to $c$ $$ x=c z $$ and make the substitution in the polynomial. $$ a_0(c)+a_1(c)cz+a_2(c)c^2 z^2+a_3(c)c^3 z^3+a_4(c)c^4 z^4+c^5 z^5, $$ where $a_0(c)=\alpha c^2$. Since I am interested in the second order in $c$, I throw away the last three terms in the polynomial, and I write the first three coefficients as $a_0(c)=\alpha c^2$ (this is exact), $a_1(c)=\beta+\gamma c$, and $a_2(c)=\delta$, that is $$ \alpha c^2+(\beta+\gamma c)cz+\delta c^2 z^2+O(c^3) $$
Now my question is: is it possible to neglect the term in $z^2$ and therefore obtain the approximate root $$ x=cz=\frac{\alpha c^2}{\beta+\gamma c}? $$
If my reasoning is wrong, can you provide some guidelines?