Polynomial with complex zeroes

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Find a polynomial equation of smallest degree with a zero at $x=6+7i$.

**Answer**

Complex zeroes of polynomials come in pairs. If $x=6+7i$ is a zero of the polynomial, then the conjugate $x=6-7i$ is a zero of the polynomial too.

Thus $(x-(6+7i))$ and $(x-(6-7i))$ are factors of the polynomial.

$(x-(6+7i))(x-(6-7i))$

$=((x-6)-(7i))((x-6)+(7i))$

$=x^{2}-12x+36+49$

$=x^{2}-12x+85$

$x^{2}-12x+85=0$ is one such polynomial equation