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algebra help

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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 5. Give your answer in expanded form.

May 26, 2020

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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 5. Give your answer in expanded form.

$$(x-(1-\sqrt{2}) )(x + 1-\sqrt{2} )(x-(2+\sqrt{5} ))(x+2+\sqrt{5})$$

$$(x^2 -(1-\sqrt{2})^2)(x^2 - (2+\sqrt{5})^2)$$

$$x^4 -[(1-\sqrt{2})^2+(2+\sqrt{5})^2]x^2+(1-\sqrt{2})^2(2+\sqrt{5})^2$$

Can you finish it off from there?

May 26, 2020