1. We are given the function $$y=3x^7+2x^6-8x^5+6x^4-3x^3+3x^2+4x-5$$ and asked to analyze or work with it. 2. This is a polynomial function of degree 7. Polynomial functions are sums of terms of the form $$ax^n$$ where $$a$$ is a coefficient and $$n$$ is a non-negative integer. 3. Important rules: - The degree of the polynomial is the highest power of $$x$$, here it is 7. - The leading term is $$3x^7$$ which dominates the behavior for large $$|x|$$. - Polynomial functions are continuous and smooth everywhere. 4. To analyze or graph this function, we can look for intercepts, extrema, and end behavior. 5. Intercepts: - To find the y-intercept, set $$x=0$$: $$y=3(0)^7+2(0)^6-8(0)^5+6(0)^4-3(0)^3+3(0)^2+4(0)-5 = -5$$ So the y-intercept is at $$(0,-5)$$. - To find x-intercepts, solve $$3x^7+2x^6-8x^5+6x^4-3x^3+3x^2+4x-5=0$$ which may require numerical methods or graphing. 6. Extrema: - To find local maxima or minima, take the derivative: $$y' = 21x^6 + 12x^5 - 40x^4 + 24x^3 - 9x^2 + 6x + 4$$ - Set $$y' = 0$$ and solve for $$x$$ to find critical points. 7. The function is complex, so graphing tools or numerical solvers are recommended for detailed analysis. Final answer: The polynomial is $$y=3x^7+2x^6-8x^5+6x^4-3x^3+3x^2+4x-5$$ with y-intercept at $$(0,-5)$$ and derivative $$y' = 21x^6 + 12x^5 - 40x^4 + 24x^3 - 9x^2 + 6x + 4$$ for finding extrema.