Intermediate Value Theorem

If a continuous function changes sign between two points, it has a zero between them.

The Intermediate Value Theorem, applied to polynomials, says that if $f(a)$ and $f(b)$ have opposite signs, then $f$ has at least one real zero between $a$ and $b$ . It traps irrational zeros that no rational-candidate search can land on exactly.

Examples

$f(1) = -3$ and $f(2) = 3$ have opposite signs, so $f$ has a zero between $1$ and $2$ .