What actually is a polynomial?

Question

I can perform operations on polynomials. I can add, multiply, and find their roots. Despite this, I cannot define a polynomial.

I wasn't in the advanced mathematics class in 8th grade, then in 9th grade I skipped the class and joined the more advanced class. This question isn't about something I don't understand; it's something I missed.

My classes have not covered what really a polynomial is. I can generate one, but not define one. The internet has yielded incomplete definitions: "Consisting of multiple terms" or "A mathematical expression containing 2 or more terms and variables."

Take the following expressions for example:

$2x^2-x+12-2x^2+x-12$. Consists of multiple terms, but can also be expressed as $0$. Is zero a polynomial?

What about $x^{-1}$? $x^{-1}$ have -1 zeroes?">I've been told this one isn't a polynomial, but I don't understand why.

Is $x^2+x+1-x^{-1}-x^{-2}-x^{-3}$? a polynomial? It contains both positive and negative exponents?

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tl;dr: What actually is the mathematical definition of a polynomial? Is $0$ a polynomial, and why isn't $x^{-1}$ a polynomial under this definition?

Answer 1

A polynomial in the indeterminate $x$ is an expression that can be obtained from numbers and the symbol $x$ by the operations of multiplication and addition.

$0$ is a polynomial, because it is a number.

Any positive integer power of $x$ is a polynomial, because you can get it by multiplying the appropriate number of $x$'s together (e.g. $x^3 = x \cdot x \cdot x$). But negative and non-integer powers of $x$ are not polynomials (e.g. $x^{-1}$ is not a polynomial), because those operations only give you positive integer powers of $x$.

Answer 2

This is a non-simple question, unfortunately. Polynomials can be defined in very abstract and potentially incomprehensible terms.

Formally, a polynomial in one variable--say $x$--with real coefficients can be defined as an expression that can be equivalently expressed as a real linear combination of finitely-many terms of the form $x^n$ (where $n$ is a non-negative integer, and $x^0:=1$).

$0$ is a polynomial, since it can be written (for example) as $0x^0.$ However, $x^2+x+1-x^{-1}-x^{-2}-x^{-3}$ is not a polynomial, since it has negative exponents. Neither is $\sqrt{x}$ a polynomial, since it has non-integer exponents. Neither is $1+x+x^2+x^3+\cdots$ a polynomial, since it cannot be expressed in finitely-many non-$0$ terms. On the other hand, the following is a polynomial: $-1+(x-x)+(x^2-x^2)+(x^3-x^3)+\cdots.$ In particular, it is equivalent to the (constant) polynomial $-1x^0.$

Answer 3

A polynomial is a mathematical expression (as opposed to an equation) where all terms are either added or subtracted from each other (if there is more than one term), each term contains some real number constant, and each term contains a variable with a non-negative power. You cannot have infinitely many terms. The number one is a polynomial. Likewise, zero is a polynomial. Any term with a negative powered variable invalidates the entire expression from being a polynomial.

Edit: In regards to the expression that simplifies to zero, both the original expression and zero are polynomials. The expression with negative powers is not a polynomial. If you had an expression with negative powers that simplified to zero, my understanding is that the unsimplified expression is not a polynomial, but the simplified expression, 0, is a polynomial.

Edit 2: No you cannot have infinitely many terms.

Answer 4

Normally we define a polynomial such that it can be written as $\sum_{i=0}^n a_ix^i$ for some $a_i\in \mathbb R$ where $i,n\in \mathbb N$. This is the reason why $x^{-i}$ isn't a polynomial. though it can be treated as a composition between a function and a polynomial.

The other reason is that when you start dealing with $\sum^n_{i=0}\frac{a_i}{x^i}$ you start to lose properties that all polynomials share. Like for instance $P(x)$ doesn't exist for $x=0$.

Answer 5

A polynomial (in one variable) is an expression of the form $$ p(x) = a_0+a_1x+a_2x^2+\ldots+a_nx^n$$ where the coefficients $a_i$ are some kind of number (or more generally they're elements of a Ring). The exponents $1,2,\ldots n$ must all be integers.

Unless we've been silly and $a_n=0,$ $n$ is called the degree of the polynomial. We can formalize this by defining the largest $n$ such that $a_n\ne0$ as the degree.

Notice that constants are allowed. $p(x) = 3$ is a zero-th degree polynomial.

You asked about zero. Yes, $p(x) =0$ is considered to be a polynomial. However, you'll notice that there is a problem with the definition of degree here since there is no coefficient that is nonzero. The degree of the zero polynomial is thus undefined.

This allows us to say that if we multiply two polynomials $w(x)=p(x)q(x)$ with $p$ of degree $n$ and $q$ of degree $m,$ then $w$ has degree $n+m.$ (Notice how the zero polynomial would mess this up if its degree were defined to be zero like the other constants.)

You're right that simplification is important. The $x$ is just a symbol and we can always "combine like terms" $$ a_lx^l+b_lx^l= (a_l+b_l)x^l.$$ We always combine all the terms together and simplify in order to get an expression into the form above with only one term for each power before we do things like consider the degree.

Notice we can add two polynomials according to the simplification rule and get a polynomial as a result. This is a good reason to consider zero to be a polynomial... it allows the sum of two polynomials to always be a polynomial. Likewise we can multiply two polynomials according to the the distributive property, the rule $$ (a_mx^m)(a_lx^l) = a_ma_l x^{m+l},$$ and the additive simplification rule. The result will be another polynomial.

Yes, the exponents all need to be positive. Of course other expressions are possible but they aren't called polynomials. Terms like $x^{-3}$ are considered part of the family of rational functions (or as a commenter noted, the Laurent polynomials, not to be confused with the (unqualified) polynomials). This is just a definition and thus somewhat arbitrary (though good definitions are important for organization). It's just like saying $-4$ is an integer but not a natural number. It's true by definition, and yes, a bit arbitrary, but nonetheless useful and a nearly universal convention.

EDIT As Paul Sinclair pointed out in the comments, there are also polynomials in multiple variables. For instance $$p(x,y) = A + Bx + Cy +Dx^2+Exy+Fy^2$$ is the general degree two polynomial in two variables. The degree of a term is just the sum of the degrees with respect to the individual variables. So a term like $3xy$ has degree two and a term like $3x^4y^5z$ would have degree $4+5+1=10.$ The degree of a polynomial is the degree of its highest-degree term with nonzero coefficient.

Answer 6

I will give you a rigorous definition.

Definition 1. A quadratic polynomial in the variable $x$ is an expression of the form $$ a x^2 + bx + c, $$ where $a$, $b$ and $c$ are real numbers and $a \not = 0$.

Example 1. Take $a=1$, $b=2$ and $c=0$. You can then see that $$ x^2 + 2x $$ is a quadratic polynomial.

More generally, we have the following definition of a polynomial (not necessarily quadratic).

Definition 2. A polynomial in the variable $x$ is either $0$ or an expression of the form $$ a_n x^n + a_{n-1}x^{n-1} + \dots + a_1 x + a_0, $$ where $n$ is a non-negative integer, $a_n, a_{n-1}, \dots, a_1, a_0$ are real numbers and $a_n \not = 0$. The non-negative integer $n$ is said to be the degree of the polynomial.

Example 2. The expression $x^{-1}$ is not a polynomial. While it is indeed an expression of the form $a_n x^{n}$, where $n = -1$ and $a_n = 1$, the integer $n$ is not positive, contradicting our definition.

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Further remarks.

You can define the addition and the multiplication of polynomials in the way you are used to. This implies that $$ x+ 2x + 3x^2 = 3x^2 + 3x + 0 $$ and $$ (x-2)(x+2) = x^2 + 0x - 4 $$ are also polynomials, by definition.

Answer 7

Note: in this answer I will try to motivate the definition which is used in more advanced contexts such as "abstract algebra". This may go beyond what is in a typical pre-algebra book, but I hope it will show how the mathematics community has found a way to come up with a workable definition, even it is less obvious at first.

It is hard to define polynomials because there is a tension between several of their key properties, which don't quite agree:

1. A polynomial can be written as an expression in the form $a_0 + a_1x + a_2x^2 + \cdots + a_nx^n$ for some $n \geq 0$ and some choice of coefficients $a_0, \ldots, a_n$.

2. The sum of two polynomials is a polynomial. The product of two polynomials is a polynomial. Overall, the collection of polynomials is the smallest collection that includes all the numbers, $x$, and is closed under addition and multiplication.

3. The expressions $(x+1)(x-1)$ and $x^2-1$ determine the same polynomial.

If we want to use something like (1) as a definition, we end up with the issue that $x$ and $2x$ are defined to be polynomials, but $x + 2x$ is a polynomial according to (2) but is not literally in the form shown in (1). So we have to define a "simplification" operation.

If we want to use something like (2) as a definition, then we still have the issue of defining when two polynomials are equal, as (3) points out.

In general, although it is tempting to define polynomials in terms of "expressions", this causes more trouble than it is worth. So it is common in more advanced texts to define polynomials as follows:

A polynomial (over the real numbers) is a sequence of real numbers $(a_i : i \in \mathbb{N})$ in which at most finitely many of the terms are nonzero. Two polynomials are equal when they are the same sequence.

So $(2,1,0,0,\ldots)$ and $(0,1,3,0,0,\ldots)$ are polynomials according to this definition. Of course, the "polynomial" $(2,1,0,0,\ldots)$ is meant to stand for $2 +x$, and $(0,1,3,0,0,\ldots)$ stands for $x + 3x^2$. But in these definitions we do not define the polynomials in terms of the expressions. Rather, we view the expressions as nothing more than notation - shorthand - for the sequences which are actually polynomials.

We continue the definition by defining addition of polynomials using the formula $(a_n) + (b_n) = (a_n + b_n)$.

Multiplication is defined in a way analogous to the Cauchy Product: $(a_n)(b_n)$ is defined to be the sequence $(c_n)$ where $$ c_k = \sum_{i=0}^k a_i b_{k-i}. $$ This is exactly the formula you would discover if you multiply polynomials in the usual, pre-algebra style.

In this way, the collection of polynomials in the variable $x$ is identified with the ring $\mathbb{R}[x]$, which is also defined as the set of finitely-supported sequences of reals with the operations shown above. These definitions of the operations take care of simplification automatically, so we do not need to worry about "unsimplified" polynomials in the formal definition.

Answer 8

There are lots of good answers here and they are all essentially correct, even though they are different! I will try to contribute another, which is somewhat more abstract than the others. I normally wouldn't try this for a high school student, but your very good question deserves different kinds of answers. Maybe this one will help.

It's the "what actually is" in your question that I want to address. In mathematics at a more advanced level you don't think as much about what something "is" as you do about how it "behaves". (The same is true in object oriented programming languages = you say you're studying computer science. If you're learning Java you know about this.)

To manipulate polynomials (which you know how to do) all you really need to know is the sequence of coefficients. We'll assume for the moment that those coefficients are ordinary numbers. It's useful to start those coefficients with the constant term. since the degree (which is the place that holds the last nonzero coefficient) isn't fixed. So the polynomial $$ 8x^3 + 5x + 7 $$ is "really just" the sequence $$ (7, 5, 0, 8) $$ or, if you like $$ (7, 5, 0, 8, 0, 0, \ldots) $$ where the zeroes go on forever.

What "really just" means there is that if you know the sequences of coefficients for two polynomials you can calculate out the sequence for their sum. Just add the sequences element by element. You can also calculate their product. It's a little harder to write down the algorithm, but you can figure it out if you understand how writing a polynomial the high-school way with powers of $x$ makes the multiplication automatic.

You can even divide one polynomial by another as long as you're willing to allow yourself a remainder (and allow fractions for the coefficients). You may in fact have learned how to do that and called it "synthetic division".

You can also "evaluate" a polynomial at a number $n$ when you know its coefficients.

What all this means in practice is that you don't need "$x$" or its powers to think about polynomials. The "variable" just helps to keep the polynomial arithmetic straight. And that's so useful that we almost always write polynomials with an $x$ rather than as a sequence of coefficients.

Finally, this abstract view lends itself to further abstraction! All you need to know to manipulate polynomials (written as sequences) is how to add and multiply the coefficients. So the coefficients themselves might be polynomials. So, for example, you can think of $$ 4x^2y^3 + 6xy^3 - 2xy^2 $$ as "a polynomial in $x$ whose coefficients are polynomials in $y$": $$ (0, -2y^2 + 6y^3 , 4y^3) = ((0), (0, 0, -2, 6), (0, 0, 0, 4)) $$ or as "a polynomial in $y$ whose coefficients are polynomials in $x$". (You write that one.)

The coefficients can even be matrices, when you learn what matrices are and how to add and multiply them.

Further thoughts:

You can think of the algorithms for addition and multiplication you learned a long time ago as like the arithmetic of polynomials, only more complicated. When you "collect like powers of $x$" in a polynomial, you just add up what you see. When you "collect like powers of $10$" in ordinary arithmetic you have to simplify further by "carrying", so replacing, say, $21 + 7 \times 10$ by $1 + 9 \times 10$.

If you relax the requirement that the coefficients be $0$ from some point on then you are dealing with a (formal) power series, traditionally written $$ a_0 + a_1 x + a_2 x^2 + \cdots = \sum_{n=0}^\infty a_n x^n . $$ You can add these and multiply them with the usual polynomial rules. They are "formal" power series because trying to evaluate them by substituting a value for $x$ is much more subtle than it is for polynomials. You'll study that in calculus. (And formal power series have uses that don't depend on evaluation.)

Then you can decide allow a few terms with negative powers, like $$ 4x^{-3} + 7x^{-1} + \text{ an ordinary formal power series} . $$ These are called "Laurent series"; they come up when you study functions of a complex variable. You have lots of nice mathematics to look forward to.

Answer 9

One way (Off the top of my head) to resolve the problem of of $x^{-1}$ not being a polynomial and $0$ being one is that all polynomials are the result of integrating $0$ a finite number of times.

Answer 10

Added: 15/12/2018

Although I still think the ideas in this answer are great, in retrospect the exposition is lacking. As one commenter says, this answer would be infinitely more useful if it actually explained things rather than just stating stuff. Consequently, I would request that someone edit or totally rewrite it to make the answer more comprehensible. If there's any takers, please comment below. If there are no takers, I might try myself, although I'm not sure where to even start.

the exposition and lack of explanatory

The other answers do a great job of giving a non-technical explanation. For users of the website who are a little further along in their studies, here's a fairly technical answer.

Philosophically speaking, I think that the concept polynomial with coefficients in $R$ somehow "is" the endofunctor $U \circ F : \mathbf{Set} \rightarrow \mathbf{Set}$, where $U$ is the forgetful functor $R\mathbf{Alg} \rightarrow \mathbf{Set}$ and $F$ is its left-adjoint. This ties in with Carl's answer, namely that:

The sum of two polynomials is a polynomial. The product of two polynomials is a polynomial. Overall, the collection of polynomials is the smallest collection that includes all the numbers, x, and is closed under addition and multiplication.

The reason this is a good description of polynomials is because:

• Carl is being vague, and just emphasizing polynomials with integer coefficients
• An object of $\mathbb{Z}\mathbf{Alg}$ is just a ring
• The signature $(+,\times,0,1)$ is a sufficiently large to state the axioms of ring theory, so we just need closure under these operations (and Carl is being vague and not including $0$ and $1$.)

The reason this is an incomplete answer is because

• it doesn't tell how to decide whether or not two polynomials are equal.

So, how do we decide whether or not two polynomials are equal? By applying the axioms of ring theory, of course! Two polynomials with integer coefficients are equal if, and only if, the axioms of ring theory can be used to prove they're equal. Otherwise, they're distinct. Seen from this vantage point, it's not too surprising that the category $\mathbb{Z}\mathbf{Alg}$ of rings has a direct connection to polynomials.

By the way, I think it's similarly the case that the concept $R$-linear combination "is" the endofunctor $U \circ F$, with $R\mathbf{Alg}$ is replaced by $R\mathbf{Mod}$. In fact, there's a whole dictionary of such things:

$R \mathbf{Alg} \mapsto \mbox{Polynomial with coefficients in $R$}$

$R \mathbf{Mod} \mapsto \mbox{$R$-linear combination}$

$\mathbf{Mon} \mapsto \mbox{Word}$

$\mathbf{Grp} \mapsto \mbox{Reduced Word}$

$\mathbf{PSet} \mapsto \mbox{Element}$

$\mathbf{SupLat} \mapsto \mbox{Subset}$

$\mathbf{Magma} \mapsto \mbox{Catalan Tree}$

etc. On the left we have concrete categories, and on the right we have the monads that they define, and are defined by. The technical concept that underlies this correspondence is that of a monadic adjunction. This is all well-known, of course, but I like reassuring myself that apparently abstract concepts give meaningful, coherent answers to the kinds of questions Year 9 students might ask their apparently-humble math tutor. This is the kind of thing that got me excited about mathematics in the first place :)

Answer 11

A polynomial is any element of a free extension of a ring (which in this answer is taken to mean "commutative ring with a multiplicative identity"). Thus, a polynomial can only be defined with respect to a given ring, say the ring $R$. The simplest free extension of $R$ is generated by augmenting $R$ with a single free element, say $x$, and is denoted by $R[x]$. Here free means that the elements of $R[x]$ are unconstrained by any condition apart from the ring axioms and any particular condition on the elements of $R$. Every element of $R[x]$ can be written in the form $\sum_{k=0}^n a_kx^k$, where $n\in \Bbb N$ and $a_k\in R$ for $k=0,...,n$, with the usual operations of addition and multiplication for such elements. In this context, the element $x$ is often called a variable.

Generally a ring can be freely extended by any number of variables, even infinitely many; the elements of such extensions are still called polynomials; and the resulting rings are called polynomial rings. As an example, we have the polynomial ring $R[x,y,z]$ in three variables.

Often the base ring is $\Bbb R$. In this case, note that the ordered-field structure of $\Bbb R$ does not extend to $\Bbb R[x]$, although division of elements of $\Bbb R[x]$ by nonzero elements of $\Bbb R$ is still definable. Another common example is $\Bbb C[z]$, where the name of the variable is $z$, rather than $x$, by convention. Other base rings often encountered are $\Bbb Z$ and $\Bbb Q$.

Added note: It may be asked why we need to have such an abstract definition of a polynomial. Indeed, for each of the familiar rings $\Bbb Z$, $\Bbb Q$, $\Bbb R$, and $\Bbb C$, the related polynomial rings are isomorphic to the corresponding rings of polynomial functions; for example, we could identify the element $x^8-2x^6+x^4$ in $\Bbb R[x]$ with the polynomial function $x\mapsto x^8-2x^6+x^4$ on $\Bbb R$. Sadly this doesn't work in general. In the case of the "clock arithmetic" ring $\Bbb Z_{12}$, the polynomial function $x\mapsto x^8-2x^6+x^4$ on $\Bbb Z_{12}$ is indistinguishable from the zero function, although the polynomial $x^8-2x^6+x^4$ is a perfectly good member of $\Bbb Z_{12}[x]$ in its own right.

Answer 12

A small remark to the role of $x$ in the spirit of the answer of @EthanBolker and @CarlMummert.

A representation of $x$:

We already know according to the given answers a polynomial \begin{align*} a_0+a_1x+a_2x^2+\cdots a_nx^n \end{align*} can be represented by the coefficients $a_0,\ldots, a_n$ as tuple with infinite many elements \begin{align*} (a_0,a_1,a_2,\cdots,a_n,0,0,\cdots) \end{align*} whereby all but finitely many elements are zero.

Question: But, what about the role of $x$ and why can we add and multiply $x$ with polynomials in more or less the same way as we can add and multiply the coefficients (i.e. the elements of the ring)?

Let's consider elements of $\mathbb{R}$ as coefficients of a polynomial and let's take e.g. \begin{align*} p(x)=7+5x+8x^3 \end{align*} We can represent this polynomial as \begin{align*} (7,5,0,8,0,0,\ldots) \end{align*}

We now pick out the special element $(0,1,0,0,\ldots)$, denote it with $$x:=(0,1,0,0,\ldots)$$ and using the Cauchy-product $\sum_{k=0}^n a_kb_{n-k}$ in order to multiply these tuples we can write \begin{align*} (7,5,0,8,0,\ldots)&=(7,0,0,0,0,\ldots)+(0,5,0,0,0,\ldots)+(0,0,0,8,0,\ldots)\\ &=(7,0,0,0,0,\ldots)+(5,0,0,0,0,\ldots)\cdot \color{blue}{x}+(8,0,0,0,0,\ldots)\cdot \color{blue}{x^3}\tag{1} \end{align*}

The right-hand side of (1) shows that all elements $a\in\mathbb{R}$ can be represented as \begin{align*} (\color{blue}{a},0,0,0,\ldots) \end{align*} while the indeterminate $x$ has a specific representation \begin{align*} (0,\color{blue}{1},0,0,0,\ldots) \end{align*} which is zero at the first coordinate but one at the second contrary to all other elements of the ring. In fact $x$ is an element of an extension ring in which all elements of the ring can be embedded.

This element $x$, called indeterminate or transcendental element has the following three properties

• $x\cdot 1=1\cdot x =x$

• $ax=xa\qquad\qquad\qquad \text{for all } a\in\mathbb{R}$

• $a_0+a_1x+\ldots+a_nx^n=0 \quad(a_i\in\mathbb{R}) \qquad\Longleftrightarrow\qquad a_i=0,i=0,1,\ldots,n$

These properties of $x$ are fundamental and enables customary calculation with polynomials.

Answer 13

For finitely many variables $x_i$ comprising a vector $\mathbb{x}$, define $\mathbb{x}^\boldsymbol{\alpha}:=\prod_i x_i^{\alpha_i}$. Such an expression, multiplied by a constant called the coefficient, is a monomial of degree $\left| \alpha\right| :=\sum_i \alpha_i$.

A polynomial is a sum of finitely many monomials with non-zero coefficients. The zero polynomial is the case where the number of such monomials is zero. The polynomial's degree is the supremum of the monomials' degrees, so the zero polynomial has degree $-\infty$. Any non-zero polynomial has at least one monomial, and among these some monomial has maximum degree, and if there is exactly one of these its coefficient is the leading coefficient. It is customary to write a polynomial as a sum over monomials of degree at most its degree, so for non-zero polynomials in one variable a unique non-zero leading coefficient exists.

Answer 14

This isn't a definition suitable for pre-calculus, but I would say that a polynomial in a variable $x$ is anything whose $n^\text{th}$ derivative with respect to $x$ vanishes everywhere (i.e. is equal to zero everywhere), for some integer $n \geq 0$.

The nice thing about this definition is that it talks about how the polynomial behaves, rather than how you write it (so $\cos(2 \cos^{-1} x)$ is also a polynomial in $x$). It also generalizes appropriately to more abstract objects such as rings, functions, etc. as long as you define derivatives appropriately.

Answer 15

You ask if $x^{-1}$ is a polynomial and other answers are saying it is not. That's okay, but... you should look up the term "Laurent polynomial."

Answer 16

The simplest answer: a polynomial is a linear combination of a finite number of monomials.
See Wikipedia for monomial; also binomial and trinomial.

As the Wikipedia Monomial article says in the lead, in some contexts monomial may have negative integer exponents (for example in Laurent polynomials).

For ordinary polynomials (with positive exponents) a degree of a polynomial is the highest exponent among all monomial terms (those actually present in a polynomial, i.e. with non-zero coefficients) in case of one-variable polynomials, or a highest sum of exponents in case of multi-variable polynomials.
Examples:

• $2x^7+5x+2$ is of degree $7$ (which is the highest one among $7$, $1$ and $0$)
• $3qt^3+5q^2 + t$ is of degree $4$ (which is the highest among $1+3$ from $q^\color{red}1t^\color{red}3$, $2$ and $1$)

Answer 17

This definition presented in ncatlab.org is pretty helpful:

Let $R$ be a commutative ring. A polynomial with coefficients in $R$ is an element of a polynomial ring over $R$ . A polynomial ring over $R$ consists of a set $X$ whose elements are called “variables” or “indeterminates”, and a function $X\to R[X]$ to (the underlying set of) a commutative $R$ -algebra that is universal among such functions, so that $R[X]$ is the free commutative $R$ -algebra generated by $X$ ; a polynomial is then an element of the underlying set of $R[X]$ .

The link to the original article is here.