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Hi can someone help me with this problem?

 

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.

 

Thanks!

 Feb 25, 2021
 #1
avatar+939 
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If a polynomial has a root of the form \(a + \sqrt{b}\) then it must also have another root of the form \(a - \sqrt{b}.\)

 

So, our quartic is:

 

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

 Feb 25, 2021
 #2
avatar+282 
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Thank you so much!

Caffeine  Feb 26, 2021

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