Given that we know the roots of the first equation is there anyway to use them or the relationship between these two quintic polynomials to discern the roots of the second equation. is there any relationship between the roots of the two equations: (e is a variable)
1)$x^5-3 \sqrt{\frac{e}{5}}x^3+ex+f$
2)$x^5+ex+f$
Help appreciated.
COMMENT.-If $x_1,x_2,x_3,x_4,x_5$ are the roots of $(1)$ and $t_1,t_2,t_3,t_4,t_5$ are those in $(2)$ you only have for the five values of $t_i$ the two equations $$s_4=\sum t_{i_1}t_{i_2}t_{i_3}t_{i_4}=\sum x_{i_1}x_{i_2}x_{i_3}x_{i_4}=e\\s_5=t_1t_2t_3t_4t_5=x_1x_2x_3x_4x_5=f$$ only two equation for five unknowns, so it is impossible to determine the values of $t_i$.
In general a basic fact is that each irreducible polynomial of $n$ degree has "its" $n$ roots any of them cannot be root of another irreducible polynomial.
Example: The real root of the irreducible $x^3+3x+3=0$ is $$x_0=\left(\frac12(\sqrt{13}-3\right)^{1/3}-\left(\frac{2}{\sqrt{13}-3}\right)^{1/3}$$ This $x_0$ cannot be root of any irreducible distinct of $x^3+3x+3=0$.