Here's an excerpt from abstract algebra book that I'm reading and my question is given later:

The difference between a polynomial and a polynomial function is mainly a difference of viewpoint. Given $a(x)$ with coefficients in $F$: if $x$ is regarded merely as a placeholder, then $a(x)$ is a polynomial; if $x$ is allowed to assume values in $F$, then $a(x)$ is a polynomial function.

Then it goes:

Remember that two polynomials $a(x)$ and $b(x)$ are equal iff corresponding coefficients are equal, whereas two functions $a(x)$ and $b(x)$ are equal $a(x) = b(x)$ for every $x$ in their domain. These two notions of equality do not always coincide!

For example, consider the following two polynomials in $\mathbb{Z}_5[x]$: $$ a(x) = x^5 + 1 $$

$$ b(x) = x - 4 $$

You may check that $a(0) = b(0), a(1) = b(1), \ldots, a(4) = b(4)$, hence $a(x)$ and $b(x)$ are equal functions from $\mathbb{Z}_5$ to $\mathbb{Z}_5$.

My question: can anyone tell me why and how $a(0)=b(0)$ for the above two functions?

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    $\begingroup$ Which book is it? $\endgroup$ – Edoardo Lanari Jun 23 '13 at 14:43
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    $\begingroup$ @Lano I'm really having hard times understanding this book. The explanations are really vague. It's Pinter's "A book of Abstract Algebra". I thought it's the easiest book in the market. $\endgroup$ – mvr950 Jun 23 '13 at 15:51
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    $\begingroup$ I asked you because that "..mainly a difference of viewpoint" is misleading, it contradicts the fact that those two objects are different, so not the same stuff under different lights. $\endgroup$ – Edoardo Lanari Jun 23 '13 at 15:53

$a(0) = (0)^5 + 1 \equiv 1$ mod $5$.

$b(0) = (0) - 4 = -4 \equiv1$ mod $5$

Remember numbers in $\mathbb{Z}_5$ are the same if they differ by a multiple of 5.

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    $\begingroup$ However, the original text may be confusing - The main point is not that the constant term $+1$ of the first is the same as the constant term $-4$ of the second polynomial. In fact $x^5-4=x^5+1$ as polynomial in $\mathbb Z_5$ because $-4=1$ here! Rather the essential point is that $x^5$ and $x$ are different polynoials (different degrees!) that take the same values for each input in $\mathbb Z_5$. $\endgroup$ – Hagen von Eitzen Jun 23 '13 at 13:12

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