All polynomials are continuous, which is essentially saying that their graph could be traced without lifting your pencil from the paper. So then, if $\alpha$ is a root of a nonconstant polynomial, $p$,* the polynomial should pass $0$ at $\alpha$ and take a value near $0$ for $x$ near $\alpha$. But we made the assumption that $p$ is nonconstant so $p$ must take small positive/negative values. But then since $p$ is continuous, for a given side of $\alpha$, in the neighbourhood of $\alpha$, it should have the same sign for all values it takes. It should be strictly negative on one side and strictly positive on the other. This is essentially the intermediate value theorem.
By testing values near $\alpha$, we can check their sign and determine whether $p$ is above or below $0$. The root, $\alpha$ must always lie between an $a$ and $b$, such that $p(a)$ and $p(b)$ have opposite signs. When we test a value, $c$, between $a$ and $b$, we can check its sign and discern whether $\alpha$ is in $(a,c)$ or $(c,b)$. This is the method of bisection. We are iteratively improving the bounds by testing values in the interval. I believe there is no general closed formula for the $n$'th bound without a recurrence relation, since it relies on you actively choosing which side of $c$ contains $\alpha$.
* Ignoring roots of even multiplicity; if that is the case, the function would not have a change of sign and the method of bisection would fail.