1. **State the problem**: Given the cubic function $$F(x) = -\frac{1}{9}x^3 + 7x^2 - 7x - 15,$$ find its roots, critical points, and analyze sign changes. 2. **Find roots of the function**: The roots (zeros) are given as $$x = -1, 3, 5$$ and confirmed by factorization or synthetic division. 3. **Evaluate function at key points**: - $$F(0) = -15$$ 4. **Confirm factorization via synthetic division**: - Dividing by \((x+1)\) gives zero remainder. - Using quadratic formula on derived quadratic factors to verify roots at 3 and 5. 5. **Solve quadratic part with formula**: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(-15)}}{2(1)} = \frac{8 \pm \sqrt{64 + 60}}{2} = \frac{8 \pm \sqrt{124}}{2} = \frac{8 \pm 2\sqrt{31}}{2} = 4 \pm \sqrt{31}$$ which approximates roots near 3 and 5. 6. **Sign chart**: - On \(x < -1\), function is negative. - Between \(-1 < x < 3\), function is positive. - Between \(3 < x < 5\), function is negative. - For \(x > 5\), function is positive. 7. **Graph information**: Roots at $$x = -1, 3, 5$$ and point $$F(0) = -15$$ Final answer summary: - Roots: $$\boxed{-1, 3, 5}$$ - Function value at 0: $$F(0) = -15$$ - Sign changes around roots as detailed above.