How do I find and plot the roots of a polynomial with complex roots on an Argand diagram? e.g. f(z) =z^3 -3z^2 + z + 5 where one of the roots is known to be 2+i

For a polynomial with real coefficients, use that roots come in complex conjugate pairs. Therefore, another root is 2-i (and we know for this example that the final root must be real). Write the factorised function from what we know so far. For this polynomial, the factorised equation must therefore look like (z-z₁)(z-z₂)(z-z₃) where z₁ and z₂ are the 2 already known roots, and z₃ is real.Expand the known part. Expanding (z-(2+i))(z-(2-i)) gives (z^2 -4z +5) - be careful with the algebra when expanding. Comparing coefficients of (z^2 -4z +5)(z-z₃)=z^3 -3z^2 + z + 5 shows that z₃ is -1.An Argand diagram plot is simply a plot of imaginary against real components. The roots are (2+i1), (2-i1), and (-1 +i0), so the points to plot will be (2,-1), (2,+1), and (-1, 0).