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This was given in the text:

 

And this is the question I'm curious about.

 

 

Now I managed to prove this result to my satisfaction, but not in the way I think they want me to. I set out by trying to find the derivative of a polynomial Q(x). As I've been watching some 3b1b I tried to think about the derivative visually as a change in volume and what happens when you get to an n-dimensional volume with sides (x-ai). From this I worked out that the derivative was the sum of all possible combinations of the linear factors of Q(x) where one is left out (because \(\frac{d}{dx} (x-{a}_{i}) = 1\) ). Now I can't work out how to do product notation in LaTeX but regardless, when x = ak all the parts with (x - ak) as a factor go away and you're left with the one product which is a constant times all the factors multiplied together except for (x - ak). And that's the same as appears on the last line of the proof in the text.

 

But that's not very rigorous, and I'd be interested in where they're going with the hint. Thoughts?  

Mynameiszac  Sep 6, 2018
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I don't understand how your n-dimensional volume thingy works, but differentiating Q(x) using the product rule will produce the correct expression. 

Guest Sep 7, 2018
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I guess I just haven't learned a product rule for n factors yet. Here's where I got the volume thingy from: https://www.youtube.com/watch?v=YG15m2VwSjA

Mynameiszac  Sep 7, 2018

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