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

The difference in between a polynomial and also a polynomial function is mainly a distinction of viewpoint. Given$a(x)$ through coefficients in $F$: if $x$ is regarded simply as a placeholder, climate $a(x)$ is a polynomial;if $x$ is enabled to assume worths in $F$, then $a(x)$ is a polynomial function.

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

Then it goes:

Remember that two polynomials $a(x)$ and also $b(x)$ room equal iff corresponding coefficients space equal, whereastwo features $a(x)$ and also $b(x)$ room equal $a(x) = b(x)$ for every $x$ in their domain. These two notions the equality do not constantly coincide!

For example, think about the following two polynomials in $\wgc2010.orgbbZ_5$:$$a(x) = x^5 + 1$$

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

You may examine that $a(0) = b(0), a(1) = b(1), \ldots, a(4) = b(4)$, hence $a(x)$ and $b(x)$ space equal functionsfrom $\wgc2010.orgbbZ_5$ come $\wgc2010.orgbbZ_5$.

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

abstract-algebra functions polynomials
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inquiry Jun 23 "13 in ~ 12:46
Lano I'm yes, really having hard times knowledge this book. The explanations room really vague. It's Pinter's "A publication of summary Algebra". I assumed it's the easiest publication in the market. $\endgroup$
Jun 23 "13 in ~ 15:51
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$a(0) = (0)^5 + 1 \equiv 1$ mode $5$.

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

Remember numbers in $\wgc2010.orgbbZ_5$ room the exact same if they different by a lot of of 5.

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answered Jun 23 "13 at 12:58

Eric AuldEric Auld
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