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Question

Write the simplest polynomial with zeros of i and square root of 2. 

Answer 1

Hi Joanna.

 

For a polynomial to have zeros of sqrt(2) and i, it must also have a zero of -i, since complex zeros always travel as conjugates.

 

You would start by writing this:

 

P(x) = (x - √(2))*(x - i)*(x + i)

 

In this case, all of the above values of x result in P(x) = 0

 

This polynomial then becomes:

 

P(x) = (x - √(2))*(x^2 + 1)

 

Which then becomes

 

P(x) = x^3  - √(2)*x^2 + x - √(2)

 

This is the simplest polynomial with the given zeros.

 

Hope this helps.