Polynomial Function 7Aaa2E

1. **State the problem:** We are given the polynomial function $$f(x) = 2x^5 - 3x^3 + 2x^2 + x - 3$$ and we want to understand or analyze it. 2. **Identify the function:** This is a polynomial function of degree 5, with terms involving powers of $$x$$ from 5 down to 0 (constant term). 3. **Important rules:** Polynomial functions are continuous and differentiable everywhere. The degree (5) tells us the general shape and number of roots (up to 5 real roots). 4. **Evaluate or simplify:** We can evaluate $$f(x)$$ at specific values or find its derivative for slope analysis. 5. **Derivative:** Using the power rule $$\frac{d}{dx} x^n = n x^{n-1}$$, the derivative is: $$f'(x) = \frac{d}{dx} (2x^5) - \frac{d}{dx} (3x^3) + \frac{d}{dx} (2x^2) + \frac{d}{dx} (x) - \frac{d}{dx} (3)$$ $$= 10x^4 - 9x^2 + 4x + 1$$ 6. **Summary:** The function is $$f(x) = 2x^5 - 3x^3 + 2x^2 + x - 3$$ and its derivative is $$f'(x) = 10x^4 - 9x^2 + 4x + 1$$ which can be used to find critical points and analyze the graph. This completes the analysis of the given polynomial function.