1. \( x^{5}=25 x \)

To solve the equation \( x^{5} = 25x \), you can first rearrange it to form \( x^{5} - 25x = 0 \). This can be factored as \( x(x^{4} - 25) = 0 \). From here, you can set \( x = 0 \) or solve \( x^{4} - 25 = 0 \) by recognizing it as a difference of squares, yielding \( (x^2 - 5)(x^2 + 5) = 0 \). The real solutions emerge from \( x^2 - 5 = 0 \), giving \( x = \pm\sqrt{5} \). Upon closer inspection, it's important to note common mistakes. A frequent pitfall is neglecting the simpler solutions, like \( x = 0 \), when handling higher degree polynomials. A systematic approach to factoring or using the rational root theorem is helpful in ensuring no solutions are overlooked!