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Despite the fact that $\forall n, n^3 + 2n \equiv 0 \pmod 3$, I understand that $n^3 + 2n$ (considered as a polynomial with coefficients in $\mathbb Z/3\mathbb Z$) is not equal to the zero polynomial.

What is the value of defining polynomials in this (strange) way? What situations does it make things simpler?

I ask this because it seemed natural to me to define polynomials as a subset of functions, so I was surprised by this.

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    You can think of polynomials as functions, i.e. as pairings of inputs and outputs. However, they usually arise when you are looking at multiplying variables, etc. so it is hard to tell what the function looks like but easy to see its polynomial form.2012-01-12
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    That's not really the usual meaning of the word "formal". Polynomials are just elements of some ring, and you shouldn't confuse a polynomial with its image under *evaluation* at a point. The usual meaning of "formal" comes in when talking about (infinite) power series. Those too form a ring, but unless they converge they don't define an evaluable *function*.2012-01-12

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