Defining a complex polynomial with certain roots

Question

How can i define a complex polynomial, with real coefficents $a_{0}$ and the form $p(z)=a_{2}z^2 + a_{1}z + a_{0}$ with the roots being at the same time $z_{1}=1-i$ and $z_{2}=-1+i$,

Answer 1

In a quadratic polynomial $p(z)=a_2z^2+a_1z+a_0$ with roots $z_1,z_2$, we have that $\frac{a_0}{a_2}=z_1 \cdot z_2$ and $-\frac{a_1}{a_2}=z_1+z_2$. So we have the following equations $$ \frac{a_0}{a_2}=z_1 \cdot z_2=(1-i)(-1+i)=2i \\ -\frac{a_1}{a_2}=z_1+z_2=(1-i)+(-1+i)=0 $$ Hence $a_1=0$ and we are left with $\frac{a_0}{a_2}=2i$. Since we want $a_0$ real, we can choose $a_0=2, a_2=-i$ and thus $p(z)=-iz^2+2$.