As the question states, I am looking to solve the following problem:
Find a $p \neq 0$ quadratic $q(X)$ with integral coefficients such that $q(2+\sqrt{3}) = 0$
I know how to find the answer if the roots are given, but they are not here. I was also given a hint saying "try using the fact that $(x-a)(x+a)=x-a^2$". However, I'm still not sure what to do.
The second part of the question is very similar and asks for a non-zero polynomial, but hopefully with some help on this one I can do that by myself. Thanks you for any help.
Best Answer
You are given one root so your polynomial is $(q-2-\sqrt 3)(q-r)=0$, where $r$ is the other root. If you write it in terms of $x=q-2$ you get $(x-\sqrt 3)(x-r)=0$ Now can you find $r$ so the polynomial has rational coefficients?