A Quartic equation tough to solve without computer The equation is $4x^3+\frac{127}{x}=2016, x\neq0$.
By rational roots test, it is easy to see that the equation does not have rational roots. By Descartes rule of signs there exists a negative root, hence it has atleast two real roots. After entering this equation in WolframAlpha, the equation is seen to possess two real and two purely complex roots. 
My question pertains as to whether there is any simple procedure to obtain the rootsi.e. without using Ferrari's solution or Numerical methods or computer algebra systems? Another observation is $2016=\underline{126}\times16$ and there is $127$ on the LHS. Any ideas. Thanks beforehand.
 A: We see that $127$ is a prime number, therefore if the polynomial is reducible over $\mathbb{Z}$(or, equivalently, $\mathbb{Q}$), then it should factor as $$(ax^2+bx+1)(cx^2+dx+127)=4x^4-2016x+127$$ Equating coefficients gives us four equations to solve:$$\begin{cases} ac=4\\ad+bc=0\\127a+c+bd=0\\127b+d=-2016\end{cases}$$ whence $$4x^4-2016x+127=(2x^2-16x+1)(2x^2+16x+127)$$ which can then be factored into complex factors using well known formula. We take note that the key factor for the ease was the primality of $127$. 
A: Why not use the tangent method. If $p(x)=4x^4-2016x+127$ then we know that a solution will lie in the interval $[-1,1]$, since the function is continuous (even though it is not defined for $x=0$) and $p(-1)>0, p(1)<0$. The tangent method states that we can find the solution of an equation in an interval $[a,b]$ by iterating $x_{n+1}=x_n-\frac{f(x)}{f'(x)}$ where $x_0=a$. The derivative of $p(x)$ is $p'(x)=16x^3-2016$; we have that 
$$
x_1=-1-\frac{p(-1)}{p'(-1)}=\frac{115}{2032} \\
x_2=x_1-\frac{p(x_1)}{p'(x_1)}=\frac{541300119662413}{8592602591386304} \\
x_3=x_2-\frac{p(x_2)}{p'(x_2)}=0.0629960629\\
$$
and the accuracy keeps getting higher. The other root is between $7$ and $8$ since $p(7)=-4381$ and $p(8)=383$; we can set $x_1=7$ and do the same process again
$$
x_1=-7-\frac{p(7)}{p'(7)}=8,261808756 \\
x_2=x_1-\frac{p(x_1)}{p'(x_1)}=7,961028858 \\
x_3=x_2-\frac{p(x_2)}{p'(x_2)}=7,937148598 \\
x_4=x_3-\frac{p(x_3)}{p'(x_3)}=7,937003942 \\
$$
So the two roots of the equation are $x_1=0.0629960629$ and $x_2=7,937003942$
Note that these calculations have been made only with a scientific calculator and not with a computer even though these are only the real roots and not the complex one.
