How to tell whether the roots are the only rational roots for a given polynomial Find all the rational roots of the polynomial $p(x)=2x^4-5x^3+7x^2-25x-15$. I only found $x=3, -\frac{1}{2}$. I am not sure whether there is any other rational roots. 

Is there a way to tell whether these are the ONLY rational roots other than doing the long polynomial division?

Thanks.
You can use Vieta's relations: the sum of the roots (in $\mathbf C$) is $\frac 52$. Now the sum of the found roots is already $\frac 52$. So if  there were other rational roots, they would be opposite. Now the equation can be written as 
$$2x^4+7x^2+15=5x(x^2+5),$$
and having opposite roots would imply they're roots of $x^2+5$, which has no real root.
Develop $(x-3)(x+\frac{1}{2})(ax^2+bx+c)$. Find $a,b,c$ by identification. Then, you can calculate the other roots.