Question

What is all the roots of the polynomial `f(x)=x^4+x^3-3x^2-17x-30`, with the given zero `-1+2i`?

Answer 1

(x+1-2i)(x+1+2i)(x-3)(x+2)

Explanation:

Rewrite -1+2i as -(1-2i) and this has to be considered a zero of the given polynomial. Then, its conjugate -(1+2i) would also be a zero of the polynomial. This x+1-2i and x+1+2i would be the factors of the given polynomial.

Since x+1-2i and x+1+2i are factors, their product (x+1-2i)(x+1+2i) `-> (x+1)^2 +4 -> x^2 +2x+5` would divide f(x). Long division is given below

The quotient `x^2-x-6` can be factorised as (x-3)(x+2). Thus all the factors of f(x) would be

(x+1-2i)(x+1+2i)(x-3)(x+2)