Find a polynomial with integral coefficients whose zeros include $\sqrt{2} + \sqrt{5}$.
By Emily Wilson •
Find a polynomial with integral coefficients whose zeros include $\sqrt{2} + \sqrt{5}$.
I think I can use $-3= (\sqrt{2} + \sqrt{5})(\sqrt{2} - \sqrt{5})$ and a certain telescopic factorisation. The problem is that I don't know how to continue this problem. Is anyone is able to give me a hint?
$\endgroup$ 21 Answer
$\begingroup$You may observe that $$ (\sqrt{2} + \sqrt{5})^2=7+2\sqrt{10} $$ giving $$\left((\sqrt{2} + \sqrt{5})^2-7\right)^2= (2\sqrt{10})^2=40.$$ Thus $\sqrt{2} + \sqrt{5}$ is a root of $$P(X)= (X^2-7)^2-40=X^4-14X^2+9.$$
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