Question

what is the degree of the simplest polynomial with integer coefficients that has √2 , 2 , √2 + i , and 2 + 2i as some of its zeros?

a. 6
b. 7
c. 8
d. 9

Answer 1

If the polynomial has integer coefficients, then any complex roots with non-zero imaginary part must occur along with their conjugates. If a + bi is a root, then so is a - bi.

The given polynomial would then have at least 6 roots: √2, 2, √2 + i and √2 - i, and 2 + 2i and 2 - 2i. So the minimum degree is (A) 6.