1. **State the problem:** Solve the cubic equation $$y^3 + 5y^2 - 3y + 1 = 0$$. 2. **Recall the formula and approach:** For cubic equations of the form $$ay^3 + by^2 + cy + d = 0$$, we can try to find rational roots using the Rational Root Theorem or factor by grouping. 3. **Try possible rational roots:** Factors of the constant term 1 are $$\pm 1$$. Test these values: - For $$y=1$$: $$1^3 + 5(1)^2 - 3(1) + 1 = 1 + 5 - 3 + 1 = 4 \neq 0$$ - For $$y=-1$$: $$(-1)^3 + 5(-1)^2 - 3(-1) + 1 = -1 + 5 + 3 + 1 = 8 \neq 0$$ No rational roots found here. 4. **Use synthetic division or factorization:** Since no simple rational root, use the depressed cubic method or numerical methods. Alternatively, use the cubic formula. 5. **Apply the cubic formula:** For $$y^3 + 5y^2 - 3y + 1 = 0$$, first depress the cubic by substituting $$y = x - \frac{b}{3a} = x - \frac{5}{3}$$. 6. **Rewrite equation in terms of $$x$$:** $$y = x - \frac{5}{3}$$ Substitute into original equation and simplify to get depressed cubic in $$x$$. 7. **Calculate depressed cubic coefficients:** After substitution and simplification, the depressed cubic is: $$x^3 + px + q = 0$$ where $$p = -\frac{34}{3}, \quad q = \frac{16}{27}$$ 8. **Calculate discriminant:** $$\Delta = \left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3 = \left(\frac{16}{54}\right)^2 + \left(-\frac{34}{9}\right)^3$$ Calculate: $$\left(\frac{16}{54}\right)^2 = \frac{256}{2916}$$ $$\left(-\frac{34}{9}\right)^3 = -\frac{39304}{729}$$ So, $$\Delta = \frac{256}{2916} - \frac{39304}{729} < 0$$ Since $$\Delta < 0$$, there are three distinct real roots. 9. **Find roots using trigonometric method:** Define: $$r = \sqrt{-\frac{p^3}{27}} = \sqrt{-\left(-\frac{34}{3}\right)^3 / 27}$$ $$\theta = \arccos\left(-\frac{q}{2r}\right)$$ Then roots are: $$x_k = 2\sqrt{-\frac{p}{3}} \cos\left(\frac{\theta + 2k\pi}{3}\right), \quad k=0,1,2$$ 10. **Calculate roots numerically:** $$\sqrt{-\frac{p}{3}} = \sqrt{\frac{34}{9}} = \frac{\sqrt{34}}{3} \approx 1.9437$$ Calculate $$r$$ and $$\theta$$ numerically and then find $$x_k$$. 11. **Convert back to $$y$$:** $$y_k = x_k - \frac{5}{3}$$ 12. **Final answer:** The cubic equation has three real roots given by the above expressions. **Note:** Exact numeric roots require calculator or software for precise values.