This question has completely floored me.

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Let $\displaystyle f(x) = x^7 -2x^6-2x^5-3x^4$. Evaluate the Legendre symbol

$\displaystyle \displaystyle \left( \frac{f(x)}{19} \right) $

for all integers x.

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I have factorised f(x) and have the following, by multiplicativity:

$\displaystyle \displaystyle \left( \frac{f(x)}{19} \right) = \left( \frac{x}{19} \right)^4 \left( \frac{x-3}{19} \right) \left( \frac{x^2 + x +1}{19} \right)$

and the first term must be +1 (as it is raised to an even power). But I've been manipulating the other two for the last two hours and have gotten pretty much nowhere. Any ideas?