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# Find a monic quartic polynomial

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Find a monic quartic polynomial f(x) with rational coefficients whose roots include x = 1 - sqrt(2) and x = 2 + sqrt(7).

Apr 17, 2021

$$(x-(1-\sqrt{2}))(x-(1+\sqrt{2}))(x-(2+\sqrt{7}))(x-(2-\sqrt{7}))=\\ (x^2-x(1+\sqrt{2})-x(1-\sqrt{2})+(1-\sqrt{2})(1+\sqrt{2}))(x^2-x(2+\sqrt{7})-x(2-\sqrt{7})+(2+\sqrt{7})(2-\sqrt{7}))=\\ (x^2-2x-1)(x^2-2x-3)=\\ \boxed{x^4-6x^3+4x^2+10x+3}$$