# How do you find a fourth degree polynomial given roots 2i and 4-i?

Apr 30, 2017

Assuming that the 4-th degree polynomial is of real coefficients, then the conjugates $- 2 i$ and $4 + i$ are also roots

#### Explanation:

So we know the four roots of the polynomial, and then one of the possible polynomials is:

(x-2i) * (x+2i) * ((x-(4-i)) * ((x-(4+i)) = (x^2-4) * (x^2-17) = x^4-4x^2-17x^2+68= x^4-21x^2+68

So a possible answer is ${x}^{4} - 21 {x}^{2} + 68$, but of course for any constant $k$ also any polynomial

$k \cdot \left({x}^{4} - 21 {x}^{2} + 68\right)$ is also a solution