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QuestionFind a polynomial equation of smallest degree with a zero at $x=-1+i$.
AnswerComplex zeroes of polynomials come in pairs. If $x=-1+i$ is a zero of the polynomial, then the conjugate $x=-1-i$ is a zero of the polynomial too.
Thus $(x-(-1+i))$ and $(x-(-1-i))$ are factors of the polynomial.
$(x-(-1+i))(x-(-1-i))$
$=((x+1)-(i))((x+1)+(i))$
$=x^{2}+2x+1+1$
$=x^{2}+2x+2$
$x^{2}+2x+2=0$ is one such polynomial equation