What is the significance of constant polynomials?

Question

$5=5$ is an equation but is it an arithmetic equation or an algebraic equation? I ask because $5$ is a real number as well as a constant polynomial. If it is an algebraic equation then it should have a graph too. Please help.

Answer 1

$(x^2+x+1)(x-1)=x^3-1$ is an equation true for all $x$

$P(x)=P(x)$ is an equation whose solutions are all $x\in \mathbb{F}$, the field we take the $x$ from.

And $5=5$ makes no exception: it is an indeterminate equation just like the previous one: solutions are all $x$.

I don't understand what "graph" are you talking about.

If it is $y=P(x)$ then it exists also for $5=5$ and is the horizontal line $y=5$

Answer 2

It really depends on your context. My calculus professor used to call something he wrote on the board a "pile of chalk dust", if it did not have a proper meaning. In the same way $$5=5$$ are just some black pixels on your screen. You have to add context to express something. For example the expression sin are just three letters. A religious person reading these letters might think of something different than a mathematician.

In the same way, the expression $$5=5$$ can be seen in two (or more) different ways.

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The first one is just an identity of numbers. For example, if you are looking for a solution of the equation $$x-5=0,$$ then $x=5$ is a solution. So here the "="-sign means equality of real numbers. In that context $5=5$ can be seen as such an equation between real numbers as well. Though such a simple equation might look odd, there are more complicated equations as well, like $$\sum_{n=1}^∞\frac{1}{n^2} = \frac{\pi^2}{6}.$$

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In the context of two function $f,g:ℝ→ℝ$ the expression $f=g$ means, that for all $x∈ℝ$ they map to the same function value: $$f(x)=g(x),\qquad \forall x∈ℝ.$$ But still nothing has changed: "$f=g$" still represents the equality of two elements - $f$ and $g$. That kind of equality is just a bit more complicated than the equality between real numbers, which is not surprising as functions are a bit more complicated than real numbers.

If your function is a polynomial function it has the shape: $$p(x)=a_0+a_1x+a_2x^2 +…+a_nx^n.$$ So it is correct, that constant functions, such as $f(x)=a_0$, are polynomial functions. And they map all values $x∈ℝ$ to a constant function value, e.g. $y=a_0$. Their graph is therefore a horizontal line given by the set:

$$G(f)=\{(x,a_0)\,|\, x∈ℝ\}.$$

Small remark: I used the term "polynomial function" on purpose, as there is a distinction between polynomial and polynomial function. Depending on your background this might be interesting for you. If you only know about functions that map from $ℝ$ to $ℝ$, then you don't have to care about that, right now.