Finding Polynomials with Integer Coefficients & \sqrt(2) + i Zero

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To find a polynomial with integer coefficients that has \(\sqrt{2} + i\) as a zero, one must consider the conjugate \(\sqrt{2} - i\) as well, since polynomials with real coefficients require complex roots to occur in conjugate pairs. Expanding the product of the factors \((x - (\sqrt{2} + i))(x - (\sqrt{2} - i))\) leads to a polynomial in the form \(x^2 - 2\sqrt{2}x + 3\). The resulting polynomial has integer coefficients when simplified correctly. The discussion emphasizes the importance of including both roots to ensure the polynomial meets the criteria.
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Homework Statement



Find a polynomial with integer coefficient for which \sqrt(2) + i is a zero.

Homework Equations



The Attempt at a Solution



I'm not sure where to really start with this one. It is on my review sheet, and I can't remember how to do it. Could someone give me a hand?
 
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duki said:

Homework Statement



Find a polynomial with integer coefficient for which \sqrt(2) + i is a zero.

Homework Equations



The Attempt at a Solution



I'm not sure where to really start with this one. It is on my review sheet, and I can't remember how to do it. Could someone give me a hand?

consider (sqrt(2)+i)2...what are you left with after expanding and simplifying?
how about (sqrt(2)+i)4?
 
Ok, as an answer I got:

x^2 - 2 \sqrt{2}x + 2 - i
Does that look right?
 
Latex isn't working, so I got

x^2 - 2sqrt(2)x + 2 - i
 

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