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# Polynomial

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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