<s>[INST] Zeros of a Fourth Degree Polynomial Given that $\left(\sqrt3+\sqrt5\right)$ is one zero of a fourth-degree polynomial with integer coefficients and leading coefficient 1, how can the constant term of this polynomial be found?
I know that $\left(\sqrt3-\sqrt5\right)$ must also be a root because it is the conjugate. How can I determine the other two roots (and ultimately, the constant term) beyond what I have right now?
$(x^2-2\sqrt3x-2)(x-r_1)(x-r_2)$
Thanks!
 [/INST] We know that $\sqrt 3+\sqrt 5$ is a root of the polynomial $\sqrt 3+\sqrt 5=x$, so we begin from there and eliminate radicals to obtain a polynomial over $\Bbb Z$:
\begin{align}
\sqrt 3+\sqrt 5&=x\\
\sqrt 3&=x-\sqrt 5\\
3&=x^2-2\sqrt 5 x+5\\
2\sqrt 5x&=x^2+2\\
20x^2&=x^4+4x^2+4\\
0&=x^4-16x^2+4.
\end{align}
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