What is the difference between a polynomial and a function or can they be used interchangebly?

Question

I have been wondering over this basic question (seems rather trivial at first sight) for a long time:

What is the difference between a polynomial and function?

My confusion arises form the following thoughts:

1. We use same notions to represent both – $f(x)$ is a polynomial or function
2. The operations are quite similar. When we write $f(a)$ in a polynomial or function $f(x)$ we replace all the $x$s with $a$ and find value.

So, polynomials and functions are quite similar. So, can they be used interchangeably?

Answer 1

The simple answer, in the spirit of the comments, is that all polynomials are functions but not all functions are polynomials. A function is simply a rule that assigns a value in the codomain to every value in the domain. A couple simple examples of functions that are not polynomials are $\sin x$ and $|x|$. A less simple one is the Dirichlet function which is $$f(x)=\begin {cases} 1&x \text { rational}\\0&x \text { irrational} \end {cases}$$ Sometimes we view a polynomial as an expression, not as a function. Even if we do not specify the values $x$ can take, we can say $x^2-3x+2=(x-2)(x-1)$

Answer 2

A polynomial maps $x$ to $p(x) = \text{Sum} (a_{i} \;\; x^{i} )$
A function maps $x$ to $f(x)$ where $f(x)$ can have whatever form you want (including possibly a form too complicated to even DEFINE!)

Answer 3

In addition to the other excellent answers, I would add that in some contexts (especially the contexts that high school students find themselves in), the difference between polynomials and functions can be as simple as the difference between expressions and equations. For example, a student might be asked to factor a polynomial like $x^2+10x+25$. It is easy to add "$f(x)=$" to the polynomial to obtain a function. However, one cannot solve an expression like $x^2+10x+25$, even though it can be factored. It can only be solved when it is an equation, i.e., when it is set to a value, e.g. $x^2+10x+25=0$. Similarly, one can manipulate polynomials but they don't really become functions until they are set equal to something, in this case, a function designation like $f(x)$.

Answer 4

It depends who you ask.

If you were to ask an applied mathematician, the answer would probably be something like this: a polynomial is a function of the form $$x\mapsto a_{0}+a_{1}x+a_{2}x^{2}+\cdots+a_{n}x^{n}.$$ Note that this is fairly informal: I haven't specified what the input is. Here, $x$ could be a real number, or any complex number, or a matrix, or a function (even another polynomial!). Similarly, I haven't specified what kind of objects the coefficients $a_{1},\ldots,a_{n}$ need to be. Most people would be happy to at least play along with this definition and figure out the rest from context when it matters.

If you're happy with that, feel free to stop reading here. In particular, what lies below the line, though interesting, probably won't be of much relevance to you unless/until you study abstract algebra, and requires you to know the basics of sets, e.g., how we work with sets and their elements. It would also help greatly to know what matrices are and how to add and multiply them.

I'm going to give the rest of this answer in pretty much as much generality as I can. If you find my explanation too confusing, then that's probably my fault. A good reference for this stuff is Birkhoff and Maclane, A Survey of Modern Algebra - that's where I learned it.

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Some people (myself included) are more pedantic than others. For the more pedantic people, the word 'polynomial' needs a certain qualification: what are the coefficients meant to be like? A polynomial with integer coefficients is ostensibly different to a polynomial with general complex coefficients, which certainly must be different to a polynomial with matrix coefficients.

There is also a deeper issue that some pedantic people (including myself) have, namely the question: what is a variable? That is, when you say 'the variable $x$ in the polynomial $x^{2}+1$', for example, what do you mean by the word 'variable'? What kinds of objects are variables? What do they do? The point of being pedantic is to clear up these kinds of questions.

First, since we want to add, subtract and multiply polynomials (otherwise, what good are they?), we specify exactly this of their coefficients: the coefficients of a polynomial should all belong to a set of objects which we can add, subtract and multiply together (but not necessarily divide). Such a set of objects is called a ring (Wikipedia link). In particular, there should be an element of the set which behaves like the number $0$: if you add it to something, it doesn't change the value. We call this the zero element, and often just denote it by $0$ when there is no fear of confusion.

Examples of rings include:

• the set of integers, with the usual number $0$ as the zero element,
• the set of real numbers, with the usual number $0$ as the zero element,
• the set of $n\times n$ real-entry matrices, with the zero matrix (all entries $0$) as the zero element,
• the set of functions $\mathbb{R}\to\mathbb{R}$, with the zero map $x\mapsto0$ as the zero element.

So, suppose we have such a set of objects. We'll call our ring $R$. We want to define what we mean by 'a polynomial with coefficients in $R$'. We do this by considering sequences of elements of $R,$ that is, things of the form $a_{0},a_{1},a_{2},a_{3},\ldots,$ where all the terms of the sequence are elements of $R.$ Here's our definition:

A polynomial with coefficients in $R$ is a sequence of elements of $R,$ $$a_{0},a_{1},a_{2},a_{3},\ldots,$$ such that, past a certain point, all of the remaining terms in the sequence are the zero element. Formally, there exists $n\in\mathbb{N}$ such that $a_{i}=0$ whenever $i> n.$ The least such $n$ is called the degree of the polynomial.

Given such a sequence, when we want to refer to it as a polynomial we avoid the usual notation for sequences as above and instead write something like $$a_{0}+a_{1}x+a_{2}x^{2}+\cdots+a_{n}x^{n}.$$ The use of the letter $x$ here is arbitrary; we could have used pretty much any symbol.

We also define addition and multiplication of polynomials 'in the usual way', that is, we define addition and multiplication of polynomials by way of formulae which mimic the behaviour we expect: for example, the sum of the polynomial $a_{0}+a_{1}x+\cdots+a_{n}x_{n}$ with the polynomial $b_{0}+b_{1}x+\cdots+b_{m}x^{m},$ where $m>n,$ is defined to be the polynomial $$(a_{0}+b_{0})+(a_{1}+b_{1})x+\cdots+(a_{n}+b_{n})x_{n}+b_{n+1}x^{n+1}+\cdots+b_{m}x^{m}.$$ The formula defining multiplication doesn't bear writing down: suffice to say we just 'multiply out the brackets' in the usual manner.

With this definition, a polynomial is a bit like a matrix. It doesn't 'take input' by itself - that is, to highly pedantic people a polynomial is not a function, but it can represent a function. If you know some linear algebra, this is a bit like how a matrix is not itself a function, but instead represents a function (in particular, a linear transformation with certain choices of bases). The point is that polynomials themselves are 'just' formal expressions which we can manipulate freely; we don't need to be tied down to the somewhat vague notion of 'a variable'.

So how does this relate to the usual definition? Given a ring $R,$ we denote by $R[x]$ the set of polynomials with coefficients in $R,$ written as above. Now, if $\alpha\in R$ then we can define a function $\mathrm{ev}_{\alpha}\colon R[x]\to R$ given by $$\mathrm{ev}_{\alpha}(a_{0}+a_{1}x+\cdots+a_{n}x^{n}) = a_{0}+a_{1}\alpha+\cdots+a_{n}\alpha^{n}.$$ That is, the function $\mathrm{ev}_{\alpha}$ takes polynomials as input and "evaluates" them at $\alpha.$

The usual definition of 'polynomial' given at the top of this answer, fits in as follows: if $f:R\to R$ is a function such that, for some fixed polynomial $p(x)\in R[x],$ $f$ is given for all $\alpha\in R$ by $$f(\alpha) = \mathrm{ev}_{\alpha}(p(x)),$$ then $f$ is a polynomial function - or, for short, a 'polynomial'.

A final question to think about: how do we deal with multivariate polynomials, like $1+x+xy+y^{2}$?