1. The problem is to understand and analyze a polynomial function. 2. A polynomial function is generally expressed as $$f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$$ where $a_n, a_{n-1}, \ldots, a_0$ are constants and $n$ is a non-negative integer. 3. Important rules include: the degree of the polynomial is the highest power of $x$ with a non-zero coefficient, and the function is continuous and smooth. 4. To analyze a polynomial, we can find its roots (where $f(x)=0$), its critical points (where the derivative $f'(x)=0$), and its end behavior (as $x \to \pm \infty$). 5. For example, consider $f(x) = x^3 - 3x^2 + 2x$. 6. Find roots by solving $x^3 - 3x^2 + 2x = 0$. 7. Factor out $x$: $x(x^2 - 3x + 2) = 0$. 8. Factor quadratic: $x(x-1)(x-2) = 0$. 9. Roots are $x=0, 1, 2$. 10. Find derivative: $f'(x) = 3x^2 - 6x + 2$. 11. Set derivative to zero: $3x^2 - 6x + 2 = 0$. 12. Solve quadratic: $x = \frac{6 \pm \sqrt{(-6)^2 - 4 \cdot 3 \cdot 2}}{2 \cdot 3} = \frac{6 \pm \sqrt{36 - 24}}{6} = \frac{6 \pm \sqrt{12}}{6} = \frac{6 \pm 2\sqrt{3}}{6} = 1 \pm \frac{\sqrt{3}}{3}$. 13. These are critical points where the function has local maxima or minima. 14. The end behavior for a cubic with positive leading coefficient is $f(x) \to \infty$ as $x \to \infty$ and $f(x) \to -\infty$ as $x \to -\infty$. 15. Thus, the polynomial $f(x) = x^3 - 3x^2 + 2x$ has roots at 0, 1, and 2, critical points at $1 \pm \frac{\sqrt{3}}{3}$, and behaves like a cubic function at the extremes.