Finding real coefficients of polynomial with complex roots

Question

I have been tasked with finding the just real coefficients of a polynomial with the roots:

$z_{0}=10$

$z_{1}=3-i$

$z_{2}=-8+2i$

Writing the polynomial as factors…

$(z-10)(z-3+i)(z+8-2i)$

..obviously does not work since this results in complex coefficients.

Can anyone help me out here?

Answer 1

There is not enough information, say the degree of a given polynomial.

Anyway, since it has (only) real coeficients if $a+bi$ is one root, then $a-bi$ is also a root of the polynomial.

So this polynomial is divisible with:

$$q(z)=(z-10)\color{red}{(z-3+i)(z-3-i)}\color{blue}{(z+8-2i)(z+8+2i)} $$ $$= (z-10)(z^2-6z+10)(z^2+16z+68)$$