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:

https://www.amazon.in/Selected-Problems-Vietnamese-Olympiad-Mathematical/dp/9814289590/ref=sr_1_1?s=books&ie=UTF8&qid=1510149044&sr=1-1&keywords=selected+problems+of+the+vietnamese+mathematical+olympiad

Question:

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. 

Answer:

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

Then,

But, and so .

More later,

Nalin Pithwa.