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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?

 May 27, 2019
 #1
avatar+102434 
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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

 

 

 

cool cool cool

 May 28, 2019

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