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# Polynomials

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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).

Oct 21, 2021

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Well

one of the factors will be   $$x-(1-\sqrt2) = x-1+\sqrt2$$

think of this as (x-1)+sqrt2

If you introduce another factor as   $$(x-1)-\sqrt2$$

Then a factor will be

$$[(x-1)-\sqrt2][(x-1)+\sqrt2]\\ =(x-1)^2-(\sqrt2)^2\\ =x^2-2x+1-2\\ =x^2-2x-1$$

now do something similar with the  $$x=2+\sqrt7$$   root

Oct 21, 2021