given the real part of 1 root of a polynomial with degree 4, find all the other roots.

Question

ive tried - long divsion and equating the remainder to zero, and substituting 2 + ib in the equation,

but both are too complex and didnt give me the right answer, which is

$$z = \frac{3 \pm \sqrt{5} }{2}$$ and 2 + i and 2 - i

)

Answer 1

If you substitute $2+bi$ into the equation you get $$11+7ib-12b^2-7ib^3+b^4=0.$$ So $7ib(1-b^2)=0$ and since $b=0$ is no solution you have $b=\pm 1$. If you divide your original polynominal by $(z-(2+i))(z-(2-i)) = z^2 - 4z +5$ you have to solve $$z^2+3z+1 =0.$$

Can you continue?