How do I choose the correct range of numbers to find the zeros...
Trial and error, We know from Descartes rule of signs that there is exactly one positive root, and 0 or 2 negative roots. Also we know that for large positive x f(x) is positive, and "large" negative x f(x) is negative from the sign of the x^3 term.
So evaluate f(0) it should be -ve, take positive steps untill f changes sign (in this case between 1 and 2). Then as the sum of the roots is equal to -the coefficient of x^2, we conclude that the negative roots if they exist, are >=-5.
There are almost certainly other methods of establishing a suitable range.
RonL
A rather simple method (that computer programs use) is the "bisection method".
First find an interval that has only one zero and work in that interval.
Example,
We can to find the zero of this. That is to solve,
You can draw a graph and see where it intersects only once and work in that interval. Look below.
So work on interval,
Now, chose the midpoint does it work?
too large.
Meaning the solution must be in the interval.
.
Now chose the midpoint there does it work?
too little.
Meaning the solution must be in the interval.
Choose the midpoint there does it work?
too large.
Thus, the solution is on the interval
And so on.