<s>[INST] Pre Calculus Synthetic Division two of the roots of the equation $2x^3-3x^2+px+q=0$ are $3$ and $-2$. Find the third root of the equation.
 [/INST] The cubic must factor as $2(x-r_1)(x-r_2)(x-r_3)$ and the coefficient of $x^2$ is $2 \cdot (-3/2).$ So the sum of the roots is $3/2$ and you know two roots are $3,-2$ with sum $1$, forcing the third root to be $1/2$.
Note that we do not have to find $p,q$ to use this method, since the sum of the roots of a monic cubic gives the negative of the coefficient of $x^2$, and the given equation may be divided by $2$ to be
$$x^3-(3/2)x^2+(p/2)x+(q/2)=0.$$
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