given 3i is a zero of P(x)=x^4-3x^3+19x^2-27x+90 find all other zeros
Complex roots always come in pairs, as complex conjugates.
One root is 3i, i.e. x = 0 + 3i.
The complex conugate of this root is x = 0 - 3i.
So the two roots are x = 3i, x = -3i.
Then two factors of the polynonial are (x - 3i) and (x + 3i). Their product is also a factor, i.e (x - 3i)(x + 3i).
(x - 3i)(x + 3i) = (x^2 + 9)
The polynomial is: x^4-3x^3+19x^2-27x+90.
Using (x^2 + 9) as a facrtor gives us,
(x^2 + 9)(x^2 + 10 - 3x)
The quadratic x^2 - 3x + 10 also has complex roots.
x = {3 +/- sqrt(9 - 4*1*10)}/2 = 3/2 +/- (i/2)*sqrt(31)
The four roots are: x = 3i, x = -3i, x = 3/2 + (i/2)*sqrt(31), x = 3/2 - (i/2)*sqrt(31)