1. **Problem Statement:** We are given multiple equations and functions to analyze, including circles, points, and polynomial functions. We will solve and explain each part step-by-step. 2. **Circles and Points:** - Circle 1: $x^2 + y^2 = 9$ (circle centered at origin with radius 3) - Circle 2: $(x + 5)^2 + (y - 3)^2 = 36$ (circle centered at $(-5,3)$ with radius 6) - Another circle: $x^2 + y^2 + 4x - 4y - 28 = 0$ 3. **Rewrite the third circle in standard form:** Complete the square for $x$ and $y$: $$x^2 + 4x + y^2 - 4y = 28$$ $$x^2 + 4x + 4 + y^2 - 4y + 4 = 28 + 4 + 4$$ $$(x + 2)^2 + (y - 2)^2 = 36$$ This is a circle centered at $(-2,2)$ with radius 6. 4. **Check if points lie on circles:** - For point $(0,5)$ on circle 1: $$0^2 + 5^2 = 25 \neq 9$$ Not on circle 1. - For point $(4,2)$ on circle 3: $$(4 + 2)^2 + (2 - 2)^2 = 6^2 + 0 = 36$$ Lies on circle 3. - For point $(0,2)$ on circle 3: $$(0 + 2)^2 + (2 - 2)^2 = 4 + 0 = 4 \neq 36$$ Not on circle 3. - For point $(1,3)$ on circle 3: $$(1 + 2)^2 + (3 - 2)^2 = 9 + 1 = 10 \neq 36$$ Not on circle 3. - For point $(1,-3)$ on circle 3: $$(1 + 2)^2 + (-3 - 2)^2 = 9 + 25 = 34 \neq 36$$ Not on circle 3. - For point $(4,9)$ on circle 3: $$(4 + 2)^2 + (9 - 2)^2 = 36 + 49 = 85 \neq 36$$ Not on circle 3. 5. **Polynomial functions:** - Simplify and analyze each polynomial. 6. **Example:** $$f(x) = 5x^4 - 2x^3 - 2x^5 + 2x - 3$$ Rearranged: $$f(x) = -2x^5 + 5x^4 - 2x^3 + 2x - 3$$ 7. **Another polynomial:** $$f(x) = 3x^4 - 2x^3 - 8x^5 - 4x + 3$$ Rearranged: $$f(x) = -8x^5 + 3x^4 - 2x^3 - 4x + 3$$ 8. **Polynomial:** $$f(x) = 16x - x^3 + x^4 - 18x^2 + 32$$ Rearranged: $$f(x) = x^4 - x^3 - 18x^2 + 16x + 32$$ 9. **Polynomial:** $$f(x) = x^4 - 2x^3 + 6x^2 - 4x + 3$$ 10. **Factored polynomial:** $$f(x) = (x - 1)(x + 2)(x - 3)$$ 11. **Multiple choice polynomials:** A. $$f(x) = (2x + 1)^2 (x - 1)$$ B. $$f(x) = -(2x + 1)^2 (x - 1)$$ C. $$f(x) = (2x + 1)(x - 1)^2$$ D. $$f(x) = -(2x + 1)(x - 1)^2$$ 12. **Given polynomial:** $$f(x) = x^3 + 3x^2 - x - 3$$ 13. **Summary:** - Circles analyzed and points checked. - Polynomials simplified and factored. - Multiple choice options correspond to different factorizations.