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There is a unique polynomial P(x) of degree 4 with rational coefficients and leading coefficient 1 which has $$\sqrt{2}+\sqrt{5}$$ as a root. What is P(1)?

May 30, 2019

$$\text{An earlier similar problem established that the roots are }\\ r = \sqrt{2}+\sqrt{5},~\sqrt{2}-\sqrt{5},~-\sqrt{2}+\sqrt{5},~-\sqrt{2}-\sqrt{5}\\~\\ \text{which results in }\\~\\ p(x) = x^4-14 x^2+9\\ p(1) = 1-14+9 = -4$$