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# Polynomial Help, thanks :D

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Find a monic polynomial of degree $$4$$ in $$x$$ with rational coefficients such that $$\sqrt{2}+\sqrt{3}$$ is a root of the polynomial.

Feb 17, 2020

#1
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$$\text{Rational coefficients means that if \sqrt{2}+\sqrt{3} is a zero of the polynomial then}\\ \text{-\sqrt{2}+\sqrt{3},~\sqrt{2}-\sqrt{3}, and -\sqrt{2}-\sqrt{3} must be as well}\\ p(x) = (x-(\sqrt{2}+\sqrt{3})) (x-(-\sqrt{2}+\sqrt{3})) (x-(\sqrt{2}-\sqrt{3})) (x-(-\sqrt{2}-\sqrt{3})) = \\x^4-10 x^2+1$$

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Feb 18, 2020
#2
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Ahh, very nice strategy. So concise. Thank you!

Feb 18, 2020