Here"s one excerpt from summary algebra book that I"m reading and also my concern is offered later:

The difference between a polynomial and a polynomial role is mostly a distinction of viewpoint. Given$a(x)$ v coefficients in $F$: if $x$ is regarded merely as a placeholder, then $a(x)$ is a polynomial;if $x$ is enabled to assume values in $F$, then $a(x)$ is a polynomial function.

You are watching: What is the difference of the polynomials?

Then the goes:

Remember that two polynomials $a(x)$ and $b(x)$ are equal iff corresponding coefficients are equal, whereastwo features $a(x)$ and also $b(x)$ room equal $a(x) = b(x)$ for every $x$ in your domain. These two notions that equality perform not always coincide!

For example, think about the complying with two polynomials in $\stclairdrake.netbbZ_5$:$$a(x) = x^5 + 1$$

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

You may check that $a(0) = b(0), a(1) = b(1), \ldots, a(4) = b(4)$, therefore $a(x)$ and also $b(x)$ are equal functionsfrom $\stclairdrake.netbbZ_5$ come $\stclairdrake.netbbZ_5$.

My question: deserve to anyone tell me why and also how $a(0)=b(0)$ because that the over two functions?


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request Jun 23 "13 at 12:46
user35885user35885
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Lano I'm really having tough times expertise this book. The explanations are really vague. It's Pinter's "A publication of abstract Algebra". I believed it's the easiest book in the market. $\endgroup$
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Jun 23 "13 at 15:51
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$a(0) = (0)^5 + 1 \equiv 1$ mod $5$.

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

Remember number in $\stclairdrake.netbbZ_5$ room the very same if they differ by a multiple of 5.


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answered Jun 23 "13 at 12:58
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Eric AuldEric Auld
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