Say I have the polynomial $6x^4+3x^3-x^2-12$
They teach in math class to list the factors of your leading coefficient and the constant, then use synthetic division (or a process of elimination plugging in the function of those factors) until you can find at least one factor that results in $0$. From there you can depress the equation and find the roots. That's great and all, but what does one do to find the roots by hand when that doesn't work? It did not work for me on this equation, despite our instructor stating this method will always work.
The only solution I've found, other than the obvious use of a graphing utility, is the following- use the function of a whole number that gets me closest to $0$. For the example here, I used the function of $1$. From there, I applied the Newton Raphson method. The only issue of that is it only provides an approximation to one root of the function. It is a complete crap-shoot for me to find the other $X-$intercept, or to approximate a good starting point for apply Newton Raphson. It also is tricky for a function that could cross the $X-$axis many times.
What is the actual best approach when a graphing utility is not available?
Unfortunately, if there are no rational roots to the polynomial $p(x)$, then you will have to use the tedious quartic formula.
HOWEVER:
There are special cases where there are no real solutions, yet you can find the $x$-intercepts. Consider:
$$x^4+6x^2+9=0$$
Where the roots are not rational, but you can find the factors.