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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

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