Reference 1: https://en.wikipedia.org/wiki/Vieta%27s_formulas
Reference 2: Selected Problems of the Vietnamese Mathematical Olympiad (1962-2009), Le Hai Chau, and Le Hai Khoi, published by World Scientific;
Amazon India link:
Without solving the cubic equation, , compute the sum of the eighth powers of all roots of the equation.
Approach: we want to be able to express the sum of the eighth powers of the three roots in terms of the three Viete’s relations here.
If , , are roots of the given cubic equation then, by Viete’s relations between roots and coefficients, we can say the following:
Furthermore, from , it follows that
But, and so .