For e.g:Let the question be:
Q)Locate the roots of the following polynomial:$$f(x)=x^3-6x+2$$
For this,by putting various values and by bruteforcing,we see that the polynomial changes sign at -2,1,3 and thereby making the roots lie in $(-\infty,-2),(-2,1),(1,3)$.However,in the book the answer is more precise and given as $(-3,-1),(0,1),(1,3)$.How did this come as -3,-1 and 0 are not leading to sign change specifically as there is a sign change for (-3,1),(-4,1) and (-5,1),(-3,-2) too ?
Also,how can this be done for the general equation$$f(x)=a_1x^n+a_2x^{n-1}+.......+a_{n-1}x+a_n$$without actually solving it?
$$ \begin{array}{c|c|c|c|c|c|c|c} x & -3 & -2 & -1 & 0 & 1 & 2 & 3\\ \hline f(x) & -7& 6 & 7 & 2 & -3 & -2 & 11 \end{array} $$
So the sign changes between $-3$ and $-2$, between $0$ and $1$ and between $2$ and $3$.