Irrational Root Theorem

An irrational zero of the form $a+\sqrt b$ in a rational-coefficient polynomial always comes paired with its conjugate $a-\sqrt b$ .

The Irrational Root Theorem: if $f$ has rational coefficients and $a+\sqrt b$ is a zero (with $a,b$ rational, $\sqrt b$ irrational), then $a-\sqrt b$ is also a zero. Requires rational coefficients — strictly stronger than the Conjugate Pair Theorem's real-coefficient hypothesis.

Examples

If $1+\sqrt3$ is a zero of a rational-coefficient polynomial, $1-\sqrt3$ must also be a zero.