If the coefficients of $$p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$$ are rational, the Conjugate Radical Roots theorem states that if the equation $p(x)=0$ has a root of the form $s+t\sqrt{u}$ where $\sqrt{u}$ is irrational, then the equation must also have the conjugate radical, $s-t\sqrt{u}$, as a root.
How to prove that statement? Can anyone show it please?
Best Answer
Suppose that $s+t\sqrt u$ is a root of $p(x)$ and that $\sqrt u\notin\mathbb Q$. Then $s+t\sqrt u$ is also a root of the polynomial$$q(x)=x^2-2sx+s^2-t^2u,$$ since$$q(x)=\bigl(x-s-t\sqrt u)\bigl(x-s+t\sqrt u\bigr).$$By polynomial long division, you can express $p(x)$ as $q(x)q^\star(x)+r(x)$, where $q^\star(x),r(x)\in\mathbb{Q}[x]$ and the degree of $r(x)$ is smaller that $2$ (which is the degree of $q(x)$). But, since $s+t\sqrt u$ is a root of both $p(x)$ and $q(x)$, it must be a root of $r(x)$ too, and the only polynomial in $\mathbb{Q}[x]$ with degree smaller than $2$ with an irrational root is the null polynomial. Therefore, $q(x)\mid p(x)$, and, since $s-t\sqrt u$ is root of $q(x)$, this implies that $s-t\sqrt u$ is also a root of $p(x)$.