SOLUTION: Can a function with the complex roots 5,sqrt(2), and 3i be a fourth-degree polynomial with rational coefficients? Explain.

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Question 715740: Can a function with the complex roots 5,sqrt(2), and 3i be a fourth-degree polynomial with rational coefficients? Explain.
Answer by josgarithmetic(39621) About Me  (Show Source):
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No. Must be fifth degree at least. The sqrt(2) would require also -sqrt(2) so that you have a way to get an expression of degree 2 with only rational coefficients. Similar with the 3i root. You need to also have a -3i root. You also were given the root 5. If you then count all the needed roots, there are five of them.