If 2 + √3 is a polynomial root, name another root of the polynomial, and explain how you know it must also be a root.
If $2 + \sqrt{3}$ is a polynomial root, then its conjugate $2 - \sqrt{3}$ must also be a root.
We can prove this by considering a polynomial $P(x)$ with coefficients in a field $F$, where $2 + \sqrt{3}$ is a root. By the conjugate root theorem, if $2 + \sqrt{3}$ is a root of $P(x)$, then its conjugate $2 - \sqrt{3}$ must also be a root.
The conjugate of $2 + \sqrt{3}$ is obtained by changing the sign of the term involving the square root. Therefore, $2 - \sqrt{3}$ is the conjugate of $2 + \sqrt{3}$. Since $2 + \sqrt{3}$ is a root of $P(x)$, substituting $x = 2 + \sqrt{3}$ into $P(x)$ gives us $P(2 + \sqrt{3}) = 0$. Consequently, substituting $x = 2 - \sqrt{3}$ into $P(x)$ yields $P(2 - \sqrt{3}) = 0$.
Thus, $2 - \sqrt{3}$ is another root of the polynomial $P(x)$, and we know this through the conjugate root theorem.