Example 2: Solve: \( x^{5}-x^{4}-3 x^{3}+3 x^{2}-4 x+4=0 \)

The polynomial equation \( x^{5}-x^{4}-3 x^{3}+3 x^{2}-4 x+4=0 \) is a high-degree equation and can be tackled using various methods. One approach to find its roots is to use the Rational Root Theorem, which suggests that potential rational roots are factors of the constant term (in this case, ±1, ±2, ±4). Testing these values could lead you to potential integer roots. Once a root is found, synthetic division can be employed to simplify the polynomial into a lower-degree equation, making it easier to analyze. You might also consider numerical methods or graphing tools if analytical roots remain elusive. Just remember, every polynomial of degree \( n \) has \( n \) roots (counting multiplicities), so don't lose hope if some methods seem messy!