Question

Write a polynimial function of least degree with integral coefficeints havung zeros that include -1 and 1+2i

Answer 1

complex roots always occur in pairs ,so
zeros are \[-1,1+2 \iota ,1-2 \iota\]
eq.of polynomial is
\[\left( x+1 \right)\left( x-1+2 \iota \right)\left( x-1- 2 \iota \right)=0\]
\[\left( x+1 \right)\left\{ \left( x-1 \right)+2 \iota \right\}\left\{ \left( x-1 \right) -2 \iota\right\}=0\]
\[\left( x+1 \right)\left\{ \left( x-1 \right)^2-4 \iota^2 \right\}=0,\iota^2=-1\]
simplify and form the equation.