Suppose that all four of the numbers \(2 - \sqrt{5}, \;4+\sqrt{10}, \;14 - 2\sqrt{7}, \;-\sqrt{2}\)are roots of the same nonzero polynomial with rational coefficients. What is the smallest possible degree of the polynomial?
Since we have rational coefficients....
If 2 - √5 is a root then so is 2 + √5
If 4+ √10 is a root then so is 4 - √10
If 14 - 2√7 is root then so is 14 + 2√7
Lastly....if - √2 is a root then so is √2
So.....the smallest possible degree is 8
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