I was looking for a way to factor quadrinomial with large numbers without using the remainder theorem to check each factor, for example:
$$x^3+300x^2+30000x-953125$$
Is there a way to quickly find the factor?, or I must check each $953125$ factors with the remainder theorem until I get 0 remainder?
$953125=61\cdot 5^6$, so only has $28$ factors (counting positive and negative). On the positive side, since all the terms containing $x$ are positive, the polynomial will continue to grow with $x$. By considering the linear and constant terms, you need $x$ to be around $30$, so I would check $x-25$ and be done. It turns out to work. When $x$ is negative, $300x^2$ is the only positive term, so it needs to equal the sum of all the rest. The linear term requires $x \lt -100$ and the cubic term requires $x \gt -300$. There are no factors in that range. Having found $x=25$ is a root, you could also divide it out to get a quadratic, check the discriminant, and know the roots are complex. Here is the check in Alpha