Suppose I have a cubic equation as $$15x^3-4x^2-25x+14=0$$
By Hit and Trial method I know that one of the roots is $x=1$.
And hence I can solve the cubic equation wit ease as it will take the form of $$(x-1)(ax^2+bx+c)=0$$
But what if the cubic equation is $$15 x^3-64 x^2-69 x+70 = 0$$
One of the roots is $x=5$
How do I guess that? Like...I have to try for $x=0,1,-1,2,-2,3,4,5$. It takes time and huge calculations...
I am only talking about simple roots like $x \in \mathbb{Z}$ (integers) and maybe $x \in [-5,5]$ or maybe $x=\frac{1}{2}$ or $x=\frac{-1}{2}$
The methods I am aware of is drawing an approx graph of the cubic equation using maxima/minima or using Newton's method with maybe 2 iterations and check for integers near that.
So are there any easy yet reliable methods of guessing one root.