Can You Solve Ramanujan's Sum?

We'll solve a problem from Ramanujan on the sum of three cube roots of cosines.

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00:00 Introduction
03:37 The Three Cosines are Roots of One Cubic
04:53 Vieta's Formulas Relating the Coefficients and Roots of a Polynomial
07:25 Cubing the Roots of a Polynomial
12:04 The Three Cube Roots of Cosines are Roots of One Cubic
13:35 Completing the Cube
14:49 The Final Answer

To square the roots of a general polynomial P, see
https://brilliant.org/wiki/transformi....
The key idea is to consider the product P(x) P(-x).

To cube the roots of a general polynomial P, you can use a similar idea. Consider the product P(x) P(wx) P(w^2 x), where w is a cube root of unity.

For more general transformations, see
https://en.wikipedia.org/wiki/Polynom....

The Ramanujan problems in the Journal of the Indian Mathematical Society:
https://www.imsc.res.in/%7Erao/ramanu...

Rhetorical algebra versus symbolic algebra: example from
https://mathenchant.wordpress.com/202...

Thumbnail by Rostislav Demchuk.

Thanks to my daughter, Meena Boppana, for reviewing the video. She is a math tutor:
http://meenaboppana.com