New to this concept don't understand
Find all the complex roots of the polynomial
z
z^6 -z^5+2z^4-5z^3+3z^2
Compare the number of roots you obtain with the number predicted by the Fundamental
Theorem of Algebra and discuss.
Well it's a sixth order polynomial. How many solutions do you expect from the fundamental theorem of algebra.
Then you're trying to solve $\displaystyle \begin{align*} z^6 - z^5 + 2z^4 - 5z^3 + 3z^2 = 0 \end{align*}$, what methods do you know to solve polynomials? Can you at least pull out a factor?
It is fairly easy to find 4 (non-distinct) integer solutions.
(The "rational root theorem" is useful here: If the polynomial equation $\displaystyle a_nx^n+ a_{n-1}x^{n-1}+ \cdot\cdot\cdot+ a_1x+ a_0= 0$ has rational root $\displaystyle x= \frac{p}{q}$ then q must evenly divide the leading coefficient, $\displaystyle a_n$, and p must evenly divide the constant term, $\displaystyle a_0$. That gives you numbers to try.)