# 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!

• The other two roots are $-\sqrt3+\sqrt5$ and $-\sqrt3-\sqrt5$ – Brian Cheung Jan 8 '17 at 6:27

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 $\Bbb Z$: