Vieta: An Opera in Two Acts by Andrei Golovanov
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 is . Let me introduce performer Gustav von !"
Loud round of applause. Gustav von 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 . 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!)
The answer is 0.11111111.
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1 solution
Through all the empty words and unneeded phrasing we have:
A is the sum of the roots of 4 x 1 2 − 3 6 x 1 1 + 3 x 7 + 7 x 2 + 1 = 0 . By Vieta's formula , we know that the sum of the roots is − a 1 1 a 1 2 = − 4 − 3 6 = 9 . Therefore we know that A is 9 .
Next, in part two we need to find the product of the roots of 3 A x 3 − 3 = 0 . As we know A ( 9 ) , we can substitute it in there to get the equation 2 7 x 3 − 3 = 0 . Now we simply need to find the product of the roots of that equation. By Vieta (again) we know that it is − a 3 a 0 = − 2 7 − 3 = 0 . 1 1 1 (to the first few decimal places).