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Find a monic quartic polynomial f(x) with rational coefficients whose roots include x=1-sqrt2 and x=2+sqrt7. Give your answer in expanded form.

 Nov 14, 2021
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rational coefficients

 

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\(f(x)=(x-1+\sqrt{2} )\times (x-2-\sqrt{7})\\ f(x)=(x^2-2x-x\sqrt{7}-x+2+\sqrt{7}+x\sqrt{2}-2\sqrt{2} -\sqrt{2\cdot 7})\\ f(x)=x^2+(-2-\sqrt{7}-1+\sqrt{2})x+\sqrt{7}-2\sqrt{2} -\sqrt{2\cdot 7} \)

\(f(x)=x^2-4.2315x-5.1559\)

Does the function with rational coefficients exist? I do not think so.

laugh  !

 Nov 14, 2021
edited by asinus  Nov 15, 2021

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