How to get the force constant? Suppose we have a spring with a difference . When it is streched by x , the restoring force is not proportional to x instead ,

F = $x^3$ + $x^2$ + $x$

Now , for normal springs F = kx 
where  k : Spring constant
If we want to find out the spring constant for the  given spring then how will we proceed ? (For this I think we need the definition of spring constant for such cases)
 A: Yes indeed: we need the definition of spring constant for such cases. For small enough $x$ you can neglect the $x^2$ and $x^3$ so that
$$
F\approx x
$$
which means that $k=1$. For a more general force, we can always define
$$
k\equiv  \lim_{x\to x_\mathrm{eq}}\frac{\mathrm dF}{\mathrm dx}
$$
where
$$
F(x_\mathrm{eq})\equiv 0
$$
defines $x_\mathrm{eq}$.
A: $F = kx$ is obviously a linear equation.  Accordingly, since two points determine a line, a minimum of two spring positions at two displacements can be used to calculate the value of $k$ (the boundary condition of zero displacement with zero force may be one of those points).  For the equation $F = k3x^3 + k2x^2 + k1x$, there are three constants rather than one (which happen to all equate to "1" in the OP's question), so the question of calculating one spring constant probably doesn't make sense from a purely mathematical perspective.  Nevertheless, in order to calculate the three constants, a minimum of four different spring displacements would have to be measured by using four different forces (the boundary condition of zero displacement with zero force may be one of those points), and a cubic polynomial curve fit would then be used to calculate the associated constants for the cubic equation.
