How do you find a polynomial with complex coefficients of the smallest possible degree for which i and 1+i are zeros and in which the coefficient of the highest power is 1?

1 Answer
KillerBunny ยท George C.
Oct 25, 2015

The polynomial is x^2 -x(2i+1) - 1 + i.

Explanation:

Since using complex numbers a polynomial of degree n has always exactly n solutions x_1,...,x_n, and can be written as (x-x_1)(x-x_2)...(x-x_n), if we want two numbers to be solution, the smallest possible degree is two, and the polynomial can be written as

(x-i)(x-(i+1))=(x-i)(x-i-1)

We can expand it into x^2-ix-x-ix+i^2+i, and since i^2=-1, it becomes

x^2 -x(2i+1) - 1 + i.

I haven't fully understood the second question, but if you want the polynomial to have two solution, you can't go for a polynomial of degree one, because it will have only one solution.

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