Polynomial with complex zeroes

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Find a polynomial equation of smallest degree with a zero at $x=4-5i$.

**Answer**

Complex zeroes of polynomials come in pairs. If $x=4-5i$ is a zero of the polynomial, then the conjugate $x=4+5i$ is a zero of the polynomial too.

Thus $(x-(4-5i))$ and $(x-(4+5i))$ are factors of the polynomial.

$(x-(4-5i))(x-(4+5i))$

$=((x-4)-(-5i))((x-4)+(-5i))$

$=x^{2}-8x+16+25$

$=x^{2}-8x+41$

$x^{2}-8x+41=0$ is one such polynomial equation