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

Did you know that polynomial equations like this one can model a variety of real-world phenomena, from economics to engineering? By adjusting the coefficients, you can simulate diverse systems, making them powerful tools in various fields. Imagine using them to predict population growth or even the stress on materials under different conditions! When solving this polynomial equation, a common mistake is overlooking synthetic division or factoring techniques which can simplify the problem. Always check for possible rational roots using the Rational Root Theorem, and don't forget to verify each potential solution by plugging it back into the original equation! This ensures no sneaky errors sneak by unnoticed.