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 find its roots (where $f(x)=0$), its critical points (where $f'(x)=0$), and its behavior at infinity. 5. For example, consider the polynomial $f(x) = x^3 - 3x^2 + 2x$. 6. Find roots by factoring: $$x^3 - 3x^2 + 2x = x(x^2 - 3x + 2) = x(x-1)(x-2)$$ so roots are $x=0,1,2$. 7. Find derivative: $$f'(x) = 3x^2 - 6x + 2$$. 8. Find critical points by solving $f'(x)=0$: $$3x^2 - 6x + 2 = 0$$. 9. Use quadratic formula: $$x = \frac{6 \pm \sqrt{(-6)^2 - 4 \cdot 3 \cdot 2}}{2 \cdot 3} = \frac{6 \pm \sqrt{36 - 24}}{6} = \frac{6 \pm \sqrt{12}}{6} = \frac{6 \pm 2\sqrt{3}}{6} = 1 \pm \frac{\sqrt{3}}{3}$$. 10. These critical points indicate local maxima or minima. 11. The polynomial's end behavior is dominated by $x^3$, so as $x \to \infty$, $f(x) \to \infty$ and as $x \to -\infty$, $f(x) \to -\infty$. 12. This analysis helps understand the shape and key features of the polynomial function.