Final Answer:
The solution to the equation f(x) = x⁵ - 3x⁴ - 5x³ + 5x² - 6x + 8 = 0 is x = 1.
Explanation:
To solve the given equation f(x) = x⁵ - 3x⁴ - 5x³ + 5x² - 6x + 8 = 0, we can use the rational root theorem and synthetic division. By applying the rational root theorem, we find that the potential rational roots are ±1, ±2, ±4, and ±8. Upon testing these potential roots using synthetic division, we find that x = 1 is a root of the given equation. Therefore, the solution to the equation f(x) = x⁵ - 3x⁴ - 5x³ + 5x² - 6x + 8 = 0 is x = 1.
Using synthetic division: 1 | 1 -3 -5 5 -6 8
| 1 -2 -7 -2 -8
-------------------------
| 1 -2 -7 -2 0
Therefore, by synthetic division, we have (x-1)(x⁴-2x³-7x²-2x) = 0. Since x=1 is a root of the equation x⁴-2x³-7x²-2x=0, we can factorize it as (x-1)(x³-x²-6x)=0. Solving for x in this factorized form gives us x=1 as the solution to the original equation f(x)=0.