If You Are Happy And You Don't Know It You'll Get This One Right

Algebra Level 3

Consider the optimal polynomial of degree n to be ( x 1 ) n ({x}-1)^n .The product of the roots of an optimal polynomial of degree n is denoted S n S_n . Find n = 2 S n \prod_{n=2}^{\infty}S_n .

And yes I decided to use latex. And yes(for those looking at me in the face with discontent), there are idiots who can't solve this problem.


The answer is 1.

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2 solutions

Bento Cheese
Sep 15, 2019

the root of any polynomial ( x 1 ) n (x-1)^n when n n is greater than 2 is 1, because of that, the product of is just multiplying 1 on to infinity, which gives back, 1 (btw I'm surprised that people don`t like latex, it's very useful)

Aaryan Vaishya
Sep 15, 2019

I had the inspiration for a harder problem on Vieta's formulas but instead I came up with this.Anyway, all the roots are one so the product 1x1x1x1x1x1x1x1x1x1x1 . . . ... is 1.

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