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# polynomials

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Let $$f(x)$$ be a polynomial with integer coefficients such that $$f(5)=f(7)=20$$. What is the smallest possible value of $$|f(0)|$$?

Dec 21, 2018

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$$\text{consider }g(x) = f(x) - 20\\ g(x) = a(x-5)(x-7) = a(x^2 - 12x+35)\\ \text{and for integer coefficients it's clear that }a \in \mathbb{Z}\\ g(0) = 35a\\ f(0) = g(0)+20 = 35a+20\\ |f(0)| = |35a+20|\\ \text{It should be fairly clear that }|f(0)| \text{ is minimized by }a=-1\\ \text{and thus }|f(0)|=15 \text{ is the smallest possible value}$$

There's holes in this. f isn't necessarily degree 2.  I'll need to think about it a bit to make it more rigorous.

Dec 21, 2018