<s>[INST] What is a good technique for solving polynomials? Say for example:
$6x^{3}-17x^{2}-4x+3=0$
I sort of look at it and don't know where to start, other than just guessing what the first one would be and trying to do from there. Is there a good technique for approaching such polynomials?
 [/INST] $f(x)= 6x^3 -17x^2 -4x +3$
Always observe the polynomials behavior for different values of x
$f(-1)=-16$
$f(0)=3$
$f(1)=-12$
Here we see that there are two roots of $f(x)=0$ between $-1$ and $1$
Now the constant part of the term $6x^3$ is $6=(1)(2)(3)$
Check for $x=\frac{-1}{3}, \frac{1}{3},\frac{-1}{2}, \frac{1}{2}$
$f(\frac{1}{3})=0$
$f(x)$ is not an even function so discard $x=\frac{-1}{3}$
$f(\frac{-1}{2})=0$
$f(x)$ is not an even function so discard $x=\frac{1}{2}$
we see that two of the roots are $x=\frac{-1}{2}$ and $x=\frac{1}{3}$
Now $f(x)= (x +\frac{1}{2})(6x^2 -20x^2 +6)$
i.e. $f(x)= (x +\frac{1}{2})(x -\frac{1}{3})(6x -18)$
i.e. $f(x)= (6)(x +\frac{1}{2})(x -\frac{1}{3})(x -3)$
i.e. $f(x)= (1)(2)(3)(x +\frac{1}{2})(x -\frac{1}{3})(x -3)$
the solution set for $f(x)=0$ is $(\frac{-1}{2}, \frac{1}{3}, 3)$
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