Question

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?

Answer 1

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.