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Find a monic quartic polynomial f(x) with rational coefficients whose roots include x = 1 - sqrt(2) and x = 2 + sqrt(7).

 May 15, 2022
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Since the coefficients are rational and the roots are irrational, the roots are \(x = 1\pm \sqrt 2\) and \(x = 2\pm \sqrt 7\).

 

The quartic polynomial required is 

\(\quad(x - (1 + \sqrt 2))(x - (1 - \sqrt 2))(x - (2 + \sqrt 7))(x - (2 - \sqrt 7))\\ = (x^2 - 2x - 1)(x^2 - 4x - 3)\)

 

Can you expand that on your own?

 May 15, 2022

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