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

Answer:

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.