Can you find the fourth degree polynomial with integer coefficients that has given roots for 2i and 4-i

Answer 1

if R is a root (zero), then x-R is a factor

complex roots (with i's in them) occur in conjugate pairs

so the roots are ... 2i , -2i , 4-i , 4+i

the factors are ... (x - 2i) (x + 2i) (x + 4 - i) (x + 4 + i)

multiply the factors together to find the polynomial

Answer 2

if a+bi is a root, then so is a-bi
(x-(a+bi))(x-(a-bi)) = (x-a)^2 + b^2
So, your polynomial will be
(x^2+4)((x-4)^2+1) = (x^2+4)(x^2-8x+17) = ...

Answer 3

oops...got the signs wrong in the last two factors
... can't follow my own instructions

should be ... (x - 2i) (x + 2i) (x - 4 - i) (x - 4 + i)