nAalniaOMliO wrote: A polynomial $p(x)$ with integer coefficients satisfies the equality $$p(\sqrt{2}+\sqrt{3})=\sqrt{2}-\sqrt{3}$$a) Find all possible values of $p(\sqrt{2}-\sqrt{3})$ b) Find an example of any polynomial $p(x)$ which satisfies the condition. Write : $P(x)=Q(x^2)+xR(x^2)$, $Q(x+5)=A(x^2)+xB(x^2)$ $R(x+5)=C(x^2)+xD(x^2)$ Then : $P(\sqrt 2+\sqrt 3)=Q(5+2\sqrt 6)+(\sqrt 2+\sqrt 3)R(5+2\sqrt 6)$ $=A(24)+2\sqrt 6B(24)+(\sqrt 2+\sqrt 3)(C(24)+2\sqrt 6D(24))$ $=A(24)+\sqrt 2\left(C(24)+6D(24)\right)+\sqrt 3\left(C(24)+4D(24)\right)+2\sqrt 6B24)$ Same : $P(\sqrt 2-\sqrt 3)=Q(5-2\sqrt 6)+(\sqrt 2-\sqrt 3)R(5-2\sqrt 6)$ $=A(24)-2\sqrt 6B(24)+(\sqrt 2-\sqrt 3)(C(24)-2\sqrt 6D(24))$ $=A(24)+\sqrt 2\left(C(24)+6D(24)\right)-\sqrt 3\left(C(24)+4D(24)\right)-2\sqrt 6B(24)$ $P(\sqrt 2+\sqrt 3)=\sqrt 2-\sqrt 3$ implies $A(24)=B(24)=0$, $C(24)+6D(24)=1$ and $C(24)+4D(24)=-1$ And so $A(24)=B(24)=0$, $C(24)=-5$ and $D(24)=1$ And $\boxed{P(\sqrt 2-\sqrt 3)=\sqrt 2+\sqrt 3}$ And an example $A(x)=B(x)=0$, $C(x)=-5$, $D(x)=1$ So $Q(x)=0$ $R(x+5)=-5+x$ and so $R(x)=x-10$ And $\boxed{P(x)=x(x^2-10)=x^3-10x}$

nAalniaOMliO wrote: A polynomial $p(x)$ with integer coefficients satisfies the equality $$p(\sqrt{2}+\sqrt{3})=\sqrt{2}-\sqrt{3}$$a) Find all possible values of $p(\sqrt{2}-\sqrt{3})$ b) Find an example of any polynomial $p(x)$ which satisfies the condition. Let $a=\sqrt{2}+\sqrt{3}$ and $b= \sqrt{2}-\sqrt{3}$. $q=Xp+1$ is a non constant polynomial with integer coefficients satisfying $q(a)=0$, hence it is divisible by the minimal polynomial of $a$. So $q=(X^4-10X^2+1)r$ where $r$ is some integer polynomial. So $q(b)=0$ therefore $p(b)=-\frac1{b}=a$ Take $r=1$ to get $p= X^3-10X$ as an example