1. The problem is to understand and analyze a polynomial function. 2. A polynomial function is a function of the form $$f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$$ where $a_n, a_{n-1}, \ldots, a_0$ are constants and $n$ is a non-negative integer. 3. Important rules: - The degree of the polynomial is the highest power of $x$ with a non-zero coefficient. - The polynomial is continuous and smooth everywhere. - The number of roots (real or complex) is equal to the degree. 4. To analyze a polynomial, we find its intercepts, extrema, and end behavior. 5. For example, consider the polynomial $$f(x) = x^3 - 3x^2 + 2x$$. 6. Find the roots by factoring: $$f(x) = x(x^2 - 3x + 2) = x(x-1)(x-2)$$ So the roots are $x=0, 1, 2$. 7. Find the derivative to locate extrema: $$f'(x) = 3x^2 - 6x + 2$$ 8. Solve $f'(x) = 0$: $$3x^2 - 6x + 2 = 0$$ Using quadratic formula: $$x = \frac{6 \pm \sqrt{(-6)^2 - 4 \cdot 3 \cdot 2}}{2 \cdot 3} = \frac{6 \pm \sqrt{36 - 24}}{6} = \frac{6 \pm \sqrt{12}}{6} = \frac{6 \pm 2\sqrt{3}}{6} = 1 \pm \frac{\sqrt{3}}{3}$$ 9. These are the $x$-values of local maxima and minima. 10. Evaluate $f(x)$ at these points to find the extrema values. Final answer: The polynomial $f(x) = x^3 - 3x^2 + 2x$ has roots at $0, 1, 2$ and local extrema at $x = 1 \pm \frac{\sqrt{3}}{3}$.