By Rational Root Theorem, all rational roots of a polynomial are in the form $\frac{p}{q}$, where $p$ divides the constant term $-180$ and $q$ divides the leading coefficient $1$. One such root is $6$. Factor the polynomial by dividing it by $z-6$.
Consider $z^{4}+6z^{3}+6z^{2}+11z+30$. By Rational Root Theorem, all rational roots of a polynomial are in the form $\frac{p}{q}$, where $p$ divides the constant term $30$ and $q$ divides the leading coefficient $1$. One such root is $-5$. Factor the polynomial by dividing it by $z+5$.
$$\left(z+5\right)\left(z^{3}+z^{2}+z+6\right)$$
Consider $z^{3}+z^{2}+z+6$. By Rational Root Theorem, all rational roots of a polynomial are in the form $\frac{p}{q}$, where $p$ divides the constant term $6$ and $q$ divides the leading coefficient $1$. One such root is $-2$. Factor the polynomial by dividing it by $z+2$.
$$\left(z+2\right)\left(z^{2}-z+3\right)$$
Rewrite the complete factored expression. Polynomial $z^{2}-z+3$ is not factored since it does not have any rational roots.
