Vieta: An Opera in Two Acts by Andrei Golovanov

Algebra Level 5

A picture of the scene The lights turn on, the curtain opens and Act One of the world-famous opera, Vieta, commences.

"The sum of the roots of 4x1236x11+3x7+7x2+1=04x^{12}-36x^{11}+3x^7+7x^2+1=0 is AA. Let me introduce performer Gustav von AA!"

Loud round of applause. Gustav von AA sings The Joy of Vieta, by Wolfram Alphaeus Mozart. The curtain drops.

After a short interlude, the curtain reopens to reveal a brightly lit stage. In big letters is written 3Ax33=03Ax^3-3=0. Gustav von Schnieber appears to a roar of applause and sings a solo piece in basso: The Product of the Roots of that 'Nomial. He finishes by revealing that number. What was it? (It is a recurring number so just input the first few decimal places. Gustav von Schnieber, however, is still on stage finishing saying it!)


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