Polynomial

A polynomial sums the multiples of powers of a variable or indeterminate:

$P(x)=a_{0}+a_{1}x+a_{2}x^{2}+\dotsb +a_{n}x^{n},\quad n\geq 0$

or briefly with the sum character:

$P(x)=\sum _{i=0}^{n}a_{i}x^{i},\quad n\geq 0$

Here is $\textstyle \sum$ the sum sign, the numbers $a_{i}$ are the coefficients (these can be, for example, real numbers or, more generally, elements from any ring), and is the indefinite.

Exponents of the powers are natural numbers. Furthermore, the sum is always finite. Infinite sums of multiples of powers with natural number exponents of an indefinite are called formal power series.

For mathematics and physics there are some important special polynomials.

In elementary algebra, one identifies this expression with a function in (a polynomial function). In abstract algebra, a strict distinction is made between a polynomial function and a polynomial as an element of a polynomial ring. In school mathematics, a polynomial function is often also called an integer function.

This article also explains the mathematical terms: leading coefficient, normalizing a polynomial and absolute member.

Etymology

The word polynomial means something like "multi-named". It comes from the Greek πολύ polý "much" and όνομα onoma "name". This name goes back to Euclid's Elements. In Book X he calls a two-membered sum a+b δύο ὀνομάτων (ek dýo onomátōn): "consisting (of) two names". The term polynomial goes back to Viëta: in his Isagoge (1591) he uses the expression polynomia magnitudo for a multimember quantity.

Polynomials in abstract algebra

→ Main article: Polynomial ring

Definition

In abstract algebra one defines a polynomial as an element of a polynomial ring R[X] . This in turn is the extension of the coefficient ring by an indefinite, (algebraically) free element . Thus contains R[X] the powers $X^{n}$ , $n\in \mathbb {N}$ and their linear combinations $\textstyle a_{0}+\sum _{k=1}^{n}a_{k}X^{k}$ with $a_k\in R$ . These are also already all elements, i.e., each polynomial is uniquely defined by the sequence

$(a_0,a_1,\dots,a_n,0,0,\dots)\in R\times R\times R\times\dots$

of its coefficients.

Construction

Conversely, a model of the polynomial ring can be $R\times R\times R\times\dots$ constructed R[X] by the set of finite sequences in For this purpose, on R[X] an addition " " is defined as a limb-wise sum of the sequences and a multiplication " $\cdot$ " is defined by convolution of the sequences. Thus, if $a=(a_n)_{n\in \N_0}$ and $b=(b_n)_{n\in \N_0}$ , then

$a+b:=(a_n+b_n)_{n\in \N_0}$

and

$a\cdot b:=\left(\sum_{i=0}^{n} a_ib_{n-i}\right)_{n\in \N_0}=\left(\sum_{i+j=n} a_ib_j\right)_{n\in \N_0},$

Now R[X] with these links is itself a commutative ring, the polynomial ring (in an indefinite) over .

Identifying the indefinite as a sequence , so that $X^2=X\cdot X=(0,0,1,0,0,\dotsc)$ , $X^3=X^2\cdot X=(0,0,0,1,0,0,\dotsc)$ etc., so any sequence can $(a_0,a_1,a_2,\dotsc)\in R[X]$ again be represented in the intuitive sense as a polynomial as

$(a_0,a_1,a_2,\dotsc) = a_0 + a_1\cdot X + a_2\cdot X^2 + \dotsb = a_0+\sum_{n\in\N_{>0}} a_n\cdot X^n.$

Connection with the analytical definition

Now consider that, according to the premise, there exists a natural number $n\in \N_0$ exists such that $a_{i}=0$ for all i>n , then according to the above considerations, any polynomial $f\in R[X]$ over a commutative unitary ring can be written uniquely as $f=a_0 + a_1\cdot X + \dotsb + a_n\cdot X^n$ . However, is not a function as in analysis or elementary algebra, but an infinite sequence (an element of the ring R[X] ) and is not an "unknown", but the sequence $(0,1,0,0,\dotsc)$ . However, one can use as a "pattern" to form a polynomial function (i.e., a polynomial in the ordinary analytic sense) afterwards. For this one uses the so-called substitution homorphism.

Note, however, that different polynomials can induce the same polynomial function. For example, if the residue class ring $\mathbb Z/3\mathbb Z=\{\bar0,\bar1,\bar2\}$ , then the polynomials induce {\displaystyle $f,g \in (\mathbb Z/3\mathbb Z)[X]$

$f=X(X-\bar1)(X-\bar2)=X^3-\bar3X^2+\bar2X=X^3-X$

and

the zero polynomial g=0

both the zero mapping $0\in \operatorname{Abb}\left(\mathbb Z/3\mathbb Z,\mathbb Z/3\mathbb Z\right)$ , that is: $f(x)=g(x)=\bar 0=0(x)$ for all $x\in\mathbb Z/3\mathbb Z.$

However, for polynomials over the real or integers, or in general any infinite integrity ring, a polynomial is determined by the induced polynomial function.

The set of polynomial functions with values in also forms a ring (subring of the function ring), but this is rarely considered. There is a natural ring homomorphism from R[X] into the ring of polynomial functions whose kernel is the set of polynomials inducing the zero function.

Generalizations

Polynomials in several indefinites

general, any sum of monomials of the form $a_{i_1,\dotsc,i_n}X_1^{i_1}\dotsm X_n^{i_n}$ is understood to be a polynomial (in several indefinites):

$P(X_1, \dotsc, X_n) = \sum_{i_1,\dotsc,i_n}a_{i_1,\dotsc,i_n}X_1^{i_1}\dotsm X_n^{i_n}$

Read: "Large-p from large-x-1 to large-x-n (is) equal to the sum over all i-1 to i-n from a-i-1-to-l-n times large-x-1 to the power of i-1 to large-x-n to the power of i-n."

By a monomial ordering it is possible to arrange the monomials in such a polynomial and thus generalize terms like leading coefficient.

The quantity $i_1+\dotsb+i_n$ is called the total degree of a monomial $X_1^{i_1}\dotsm X_n^{i_n}$ . If all (nonvanishing) monomials in a polynomial have the same total degree, it is called homogeneous. The maximum total degree of all non-vanishing monomials is the degree of the polynomial.

The maximum number of possible monomials of a given degree is

$\binom{n+k-1}{k},$

Read: "n+k-1 over k" or "k from n+k-1".

where is the number of occurring indeterminates and the degree. Illustratively, here we consider a problem of combinations with repetition (reclining).

Summing the number of possible monomials of degree $0$ to we get the number of possible monomials in a polynomial of a certain degree:

$\binom{n+k}{k}$

Read: "n+k over k" or "k from n+k".

If all indeterminates are in a certain way "equal", the polynomial is called symmetric. What is meant is: if the polynomial does not change when the indeterminates are interchanged.

Also, the polynomials in the indefinites $X_1, \dotsc, X_n$ over the ring form a polynomial ring written as $R[X_1, \dotsc, X_n]$ .

Formal power series

Going to infinite series of the form

$f = \sum_{i=0}^\infty a_i X^i$

Read: "f (is) equal to the sum of i equals zero to infinity of a-i (times) (upper case) x to the power of i."

over, one obtains formal power series.

Laurent polynomials and Laurent series

If one also allows negative exponents in a polynomial, one obtains a Laurent polynomial. Corresponding to the formal power series, formal Laurent series can also be considered. These are objects of the form

$f=\sum _{i=-N}^{\infty }a_{i}X^{i}.$
Read: "f (is) equal to the sum of i equals minus (upper case) n to infinity of a-i (times) (upper case) x to the power of i."

Posynomial functions

If one allows several variables and arbitrary real powers, one obtains the notion of a posynomial function.

Questions and Answers

Q: What is a polynomial?

A: A polynomial is a kind of mathematical expression that is a sum of several mathematical terms called monomials, which are numbers, variables, or products of numbers and several variables.

Q: How do mathematicians, scientists and engineers use polynomials?

A: Mathematicians, scientists and engineers all use polynomials to solve problems.

Q: What operations can be used in an algebraic expression to make it a polynomial?

A: In order for an algebraic expression to be considered a polynomial, the only arithmetic operations that can be used are addition, subtraction, multiplication and whole number exponentiation. If harder operations such as division or square roots are used then the algebraic expression is not considered a polynomial.

Q: What type of equations can be formed using polynomials?

A: Polynomials are often used to form both polynomial equations (such as 7x^4(-3)x^3+19x^2(-8)x+197=0) and also polynomial functions (such as f(x)=7x^4(-3)x^3+19x^2(-8)x+197).

Q: What subject does one need to understand in order to work with polynomials?

A: In order to work with polynomials one needs to understand algebra which is a gateway course into all technical subjects.