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




Writing the polynomial as factors...


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

Can anyone help me out here?

  • $\begingroup$ “the real coefficients of a polynomial” or “a polynomial with only real coefficients”? $\endgroup$ – Martin R Oct 9 '18 at 20:22
  • $\begingroup$ Only real coefficients. $\endgroup$ – Boris Grunwald Oct 9 '18 at 20:24

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:

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

  • $\begingroup$ Thanks, I think this makes sense. $\endgroup$ – Boris Grunwald Oct 9 '18 at 20:34
  • $\begingroup$ And sorry, it was the smallest possible degree $\endgroup$ – Boris Grunwald Oct 9 '18 at 20:39

Polynomial with only real coefficients is $$( z-10) \, ( z-2i+8)\,(z+2 i+8) \, ( z-i-3) \, ( z+i-3)\\= {{z}^{5}}-118 {{z}^{3}}-68 {{z}^{2}}+3160 z-6800 $$


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