1. Find factors of the constant term (-1): ±1
2. Find factors of the leading coefficient (5): ±1, ±5
3. Possible rational roots: ±1/1, ±1/5
4. Use synthetic division with each possible root.
5. Since f(1) = 0, x = 1 is a root.
6. Factor polynomial using synthetic division with x = 1 as a root: (x - 1)(5x^2 - 6x + 1)
7. Solve 5x^2 - 6x + 1 = 0 to find remaining roots: (5x - 1)(x - 1)
8. Remaining roots: x = 1/5, x = 1
9. Roots of f(x) = 5x^3 - 11x^2 + 7x - 1 are x = 1, x = 1/5, and x = 1.
