Solution 3.4:7b
From Förberedande kurs i matematik 2
According to the factor theorem, a polynomial that has the zeros \displaystyle -\text{1}+i and \displaystyle -\text{1}-i must contain the factors \displaystyle z-\left( -\text{1}+i \right) and \displaystyle z-\left( -\text{1}-i \right). An example of such a polynomial is
\displaystyle \left( z-\left( -\text{1}+i \right) \right)\left( z-\left( -\text{1}-i \right) \right)=z^{2}+2z+2
NOTE: If one wants to have all the polynomials which have only these zeros, the answer is
\displaystyle C\left( z+1-i \right)^{m}\left( z+1+i \right)^{n}
where
\displaystyle C
is a non-zero constant and
\displaystyle m
and
\displaystyle n
are positive integers.
