TNPSC GROUP-1/2/4 | Simplification

X + \frac{1}{x} = A x^{2} + \frac{1}{x^{2}} = A^{2} - 2 x^{3} + \frac{1}{x^{3}} = A^{3} - 3A x^{4} + \frac{1}{x^{4}} = A^{4} - 4A^{2} + 2 x^{5} + \frac{1}{x^{5}} = A^{5} - 5A^{3}+ 5A X - \frac{1}{x} = A x^{2} + \frac{1}{x^{2}} = A^{2} + 2 x^{3} - \frac{1}{x^{3}} = A^{3} + 3A x^{4} + \frac{1}{x^{4}} = A^{4} + 4A^{2} + 2 x^{5} - \frac{1}{x^{5}} = A^{5} + 5A^{3}+ 5A