Polynomial with complex zeroes

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Find a polynomial equation of smallest degree with a zero at $x=5+9i$.

**Answer**

Complex zeroes of polynomials come in pairs. If $x=5+9i$ is a zero of the polynomial, then the conjugate $x=5-9i$ is a zero of the polynomial too.

Thus $(x-(5+9i))$ and $(x-(5-9i))$ are factors of the polynomial.

$(x-(5+9i))(x-(5-9i))$

$=((x-5)-(9i))((x-5)+(9i))$

$=x^{2}-10x+25+81$

$=x^{2}-10x+106$

$x^{2}-10x+106=0$ is one such polynomial equation