A Polynomial Added to a Polynomial is a Polynomial

A Polynomial Added to a Polynomial is a Polynomial.

Type of mathematical expressions

In mathematics, a
polynomial
is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of improver, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate

x

is

ten
2
− 410
+ seven
. An example with three indeterminates is

x
3
+ 2xyz
ii

yz
+ 1
.

Polynomials appear in many areas of mathematics and science. For example, they are used to class polynomial equations, which encode a wide range of problems, from elementary discussion problems to complicated scientific bug; they are used to define
polynomial functions, which announced in settings ranging from bones chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In avant-garde mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry.

Etymology

[edit]

The discussion
polynomial
joins ii diverse roots: the Greek
poly, meaning “many”, and the Latin
nomen, or “name”. It was derived from the term
binomial
past replacing the Latin root
bi-
with the Greek
poly-. That is, it means a sum of many terms (many monomials). The word
polynomial
was first used in the 17th century.[1]

Annotation and terminology

[edit]

The graph of a polynomial office of caste 3

The
x
occurring in a polynomial is ordinarily called a
variable
or an
indeterminate. When the polynomial is considered as an expression,
10
is a stock-still symbol which does not have whatever value (its value is “indeterminate”). However, when one considers the function defined by the polynomial, then
x
represents the statement of the function, and is therefore called a “variable”. Many authors use these two words interchangeably.

A polynomial
P
in the indeterminate
10
is commonly denoted either as
P
or as
P(x). Formally, the proper name of the polynomial is
P, not
P(10), just the apply of the functional notation
P(x) dates from a time when the distinction between a polynomial and the associated function was unclear. Moreover, the functional annotation is frequently useful for specifying, in a unmarried phrase, a polynomial and its indeterminate. For example, “allow
P(x) exist a polynomial” is a shorthand for “permit
P
be a polynomial in the indeterminate
x“. On the other manus, when it is not necessary to emphasize the proper noun of the indeterminate, many formulas are much simpler and easier to read if the proper noun(s) of the indeterminate(s) exercise non appear at each occurrence of the polynomial.

The ambiguity of having two notations for a single mathematical object may be formally resolved by considering the full general meaning of the functional notation for polynomials. If
a
denotes a number, a variable, some other polynomial, or, more mostly, whatever expression, then
P(a) denotes, past convention, the consequence of substituting
a
for
ten
in
P. Thus, the polynomial
P
defines the role





a



P
(
a
)
,


{\displaystyle a\mapsto P(a),}



which is the
polynomial function
associated to
P. Oftentimes, when using this notation, i supposes that
a
is a number. However, one may use it over any domain where addition and multiplication are defined (that is, any ring). In particular, if
a
is a polynomial then
P(a) is also a polynomial.

More than specifically, when
a
is the indeterminate
x, so the image of
ten
past this function is the polynomial
P
itself (substituting
x
for
x
does not change anything). In other words,





P
(
x
)
=
P
,


{\displaystyle P(x)=P,}



which justifies formally the existence of two notations for the same polynomial.

Definition

[edit]

A
polynomial expression
is an expression that can exist built from constants and symbols chosen
variables
or
indeterminates
by means of addition, multiplication and exponentiation to a non-negative integer ability. The constants are generally numbers, simply may exist any expression that do non involve the indeterminates, and correspond mathematical objects that can exist added and multiplied. Two polynomial expressions are considered as defining the same
polynomial
if they may be transformed, one to the other, by applying the usual properties of commutativity, associativity and distributivity of addition and multiplication. For instance




(
x



ane
)
(
x



ii
)


{\displaystyle (ten-1)(ten-two)}




and





x

ii





three
ten
+
2


{\displaystyle x^{2}-3x+two}




are 2 polynomial expressions that represent the same polynomial; and so, one has the equality




(
x



1
)
(
x



2
)
=

x

2





3
10
+
2


{\displaystyle (x-one)(10-2)=x^{two}-3x+2}



.

A polynomial in a single indeterminate

x

tin ever be written (or rewritten) in the grade






a

n



x

n


+

a

n



ane



x

north



i


+



+

a

two



x

ii


+

a

1


ten
+

a

0


,


{\displaystyle a_{n}x^{due north}+a_{due north-ane}x^{due north-one}+\dotsb +a_{two}ten^{2}+a_{i}x+a_{0},}



where





a

0


,



,

a

n




{\displaystyle a_{0},\ldots ,a_{n}}




are constants that are called the
coefficients
of the polynomial, and




ten


{\displaystyle x}




is the indeterminate.[2]
The word “indeterminate” ways that




ten


{\displaystyle x}




represents no particular value, although any value may exist substituted for it. The mapping that associates the result of this substitution to the substituted value is a function, called a
polynomial office.

This can be expressed more than concisely by using summation notation:










one thousand
=
0


n



a

thou



x

chiliad




{\displaystyle \sum _{k=0}^{northward}a_{k}10^{k}}



That is, a polynomial tin either be zero or can exist written as the sum of a finite number of non-null terms. Each term consists of the production of a number – called the coefficient of the term[a] – and a finite number of indeterminates, raised to non-negative integer powers.

Classification

[edit]

The exponent on an indeterminate in a term is called the degree of that indeterminate in that term; the caste of the term is the sum of the degrees of the indeterminates in that term, and the degree of a polynomial is the largest caste of any term with nonzero coefficient.[three]
Because

x
=
x
one
, the degree of an indeterminate without a written exponent is one.


A term with no indeterminates and a polynomial with no indeterminates are chosen, respectively, a constant term and a
abiding polynomial.[b]
The degree of a abiding term and of a nonzero constant polynomial is 0. The caste of the cypher polynomial 0 (which has no terms at all) is generally treated every bit not divers (merely see beneath).[four]

For example:








5

x

2


y


{\displaystyle -5x^{2}y}



is a term. The coefficient is
−five, the indeterminates are

x

and

y
, the caste of

x

is two, while the degree of

y

is one. The degree of the unabridged term is the sum of the degrees of each indeterminate in it, so in this case the caste is
2 + 1 = 3.

Forming a sum of several terms produces a polynomial. For case, the following is a polynomial:















3

x

2














t
e
r
m






ane



















5
x












t
east
r
thousand






2












+




4












t
e
r
yard






3







.


{\displaystyle \underbrace {_{\,}3x^{2}} _{\begin{smallmatrix}\mathrm {term} \\\mathrm {1} \finish{smallmatrix}}\underbrace {-_{\,}5x} _{\begin{smallmatrix}\mathrm {term} \\\mathrm {two} \stop{smallmatrix}}\underbrace {+_{\,}4} _{\begin{smallmatrix}\mathrm {term} \\\mathrm {3} \cease{smallmatrix}}.}



It consists of three terms: the get-go is degree two, the second is degree ane, and the third is degree nothing.

Polynomials of pocket-sized caste have been given specific names. A polynomial of degree goose egg is a
constant polynomial, or simply a
constant. Polynomials of degree 1, 2 or three are respectively
linear polynomials,
quadratic polynomials
and
cubic polynomials.[3]
For higher degrees, the specific names are not commonly used, although
quartic polynomial
(for degree four) and
quintic polynomial
(for caste five) are sometimes used. The names for the degrees may be practical to the polynomial or to its terms. For example, the term
2x

in

10
2
+ 2x
+ 1

is a linear term in a quadratic polynomial.

The polynomial 0, which may be considered to have no terms at all, is called the
zero polynomial. Unlike other abiding polynomials, its caste is non nil. Rather, the degree of the nothing polynomial is either left explicitly undefined, or divers as negative (either −one or −∞).[5]
The zero polynomial is also unique in that information technology is the merely polynomial in i indeterminate that has an infinite number of roots. The graph of the cipher polynomial,

f(ten) = 0
, is the
ten-axis.

In the case of polynomials in more than than ane indeterminate, a polynomial is called
homogeneous
of
degree

n


if
all
of its not-zero terms take
caste

n

. The zero polynomial is homogeneous, and, as a homogeneous polynomial, its degree is undefined.[c]
For example,

ten
3
y
ii
+ 7ten
ii
y
three
− iiix
five

is homogeneous of degree 5. For more details, run into Homogeneous polynomial.

The commutative constabulary of improver can be used to rearrange terms into any preferred order. In polynomials with one indeterminate, the terms are usually ordered according to degree, either in “descending powers of

x
“, with the term of largest caste first, or in “ascending powers of

x
“. The polynomial
3x
2
– vten
+ iv

is written in descending powers of

ten
. The first term has coefficient
3, indeterminate

10
, and exponent
2. In the 2d term, the coefficient
is
−5
. The third term is a abiding. Because the
degree
of a non-zippo polynomial is the largest degree of any one term, this polynomial has degree two.[half dozen]

Two terms with the same indeterminates raised to the aforementioned powers are chosen “similar terms” or “like terms”, and they can be combined, using the distributive police force, into a unmarried term whose coefficient is the sum of the coefficients of the terms that were combined. It may happen that this makes the coefficient 0.[seven]
Polynomials tin can exist classified by the number of terms with nonzero coefficients, then that a i-term polynomial is called a monomial,[d]
a two-term polynomial is called a binomial, and a 3-term polynomial is called a
trinomial. The term “quadrinomial” is occasionally used for a four-term polynomial.



A
existent polynomial
is a polynomial with existent coefficients. When information technology is used to define a office, the domain is non so restricted. However, a
real polynomial function
is a function from the reals to the reals that is defined past a real polynomial. Similarly, an
integer polynomial
is a polynomial with integer coefficients, and a
complex polynomial
is a polynomial with complex coefficients.




A polynomial in one indeterminate is called a
univariate polynomial, a polynomial in more than ane indeterminate is called a
multivariate polynomial. A polynomial with two indeterminates is chosen a
bivariate polynomial.[two]
These notions refer more to the kind of polynomials one is generally working with than to individual polynomials; for instance, when working with univariate polynomials, ane does not exclude constant polynomials (which may issue from the subtraction of non-constant polynomials), although strictly speaking, abiding polynomials do not incorporate any indeterminates at all. It is possible to further classify multivariate polynomials as
bivariate,
trivariate, and and then on, according to the maximum number of indeterminates allowed. Again, and then that the set of objects nether consideration exist closed nether subtraction, a report of trivariate polynomials usually allows bivariate polynomials, and so on. It is also mutual to say simply “polynomials in

x,
y
, and

z
“, listing the indeterminates allowed.

The evaluation of a polynomial consists of substituting a numerical value to each indeterminate and conveying out the indicated multiplications and additions. For polynomials in one indeterminate, the evaluation is usually more efficient (lower number of arithmetic operations to perform) using Horner’south method:





(
(
(
(
(

a

n


ten
+

a

n



i


)
x
+

a

northward



2


)
x
+



+

a

3


)
x
+

a

2


)
x
+

a

1


)
x
+

a

0


.


{\displaystyle (((((a_{north}x+a_{n-one})x+a_{north-2})ten+\dotsb +a_{iii})x+a_{2})x+a_{1})x+a_{0}.}



Arithmetic

[edit]

Addition and subtraction

[edit]

Polynomials can be added using the associative law of add-on (grouping all their terms together into a single sum), possibly followed by reordering (using the commutative law) and combining of similar terms.[7]
[8]
For case, if





P
=
iii

10

2





2
x
+
5
ten
y



2


{\displaystyle P=3x^{2}-2x+5xy-ii}




and




Q
=



iii

ten

2


+
three
x
+
iv

y

two


+
eight


{\displaystyle Q=-3x^{2}+3x+4y^{2}+viii}



then the sum





P
+
Q
=
3

10

2





2
x
+
5
x
y



2



3

ten

2


+
3
x
+
4

y

ii


+
viii


{\displaystyle P+Q=3x^{2}-2x+5xy-two-3x^{2}+3x+4y^{ii}+8}



can be reordered and regrouped every bit





P
+
Q
=
(
3

x

two





3

x

2


)
+
(



ii
x
+
iii
10
)
+
5
x
y
+
iv

y

ii


+
(
8



2
)


{\displaystyle P+Q=(3x^{2}-3x^{two})+(-2x+3x)+5xy+4y^{two}+(8-two)}



and so simplified to





P
+
Q
=
x
+
5
ten
y
+
iv

y

2


+
half dozen.


{\displaystyle P+Q=x+5xy+4y^{2}+six.}



When polynomials are added together, the outcome is some other polynomial.[9]

Subtraction of polynomials is similar.

Multiplication

[edit]

Polynomials can also be multiplied. To expand the product of two polynomials into a sum of terms, the distributive constabulary is repeatedly applied, which results in each term of one polynomial being multiplied by every term of the other.[7]
For example, if










P





=
2
x
+
3
y
+
5







Q





=
2
x
+
5
y
+
x
y
+
1








{\displaystyle {\begin{aligned}\color {Cherry}P&\color {Crimson}{=2x+3y+v}\\\color {Blueish}Q&\color {Blue}{=2x+5y+xy+one}\end{aligned}}}



so












P






Q






=




(



2
x









2
x



)


+


(



ii
x









5
y



)


+


(



2
x









x
y



)


+


(



2
10









ane



)






+


(



three
y









2
x



)


+


(



iii
y









five
y



)


+


(



3
y









10
y



)


+


(



3
y









one



)






+


(



5









ii
x



)


+


(



5









five
y



)


+


(



5









x
y



)


+


(



five









ane



)






{\displaystyle {\begin{array}{rccrcrcrcr}{\color {Cherry}{P}}{\colour {Blue}{Q}}&{=}&&({\color {Red}{2x}}\cdot {\color {Blue}{2x}})&+&({\color {Red}{2x}}\cdot {\color {Blue}{5y}})&+&({\colour {Red}{2x}}\cdot {\color {Bluish}{xy}})&+&({\colour {Red}{2x}}\cdot {\color {Blueish}{1}})\\&&+&({\color {Cherry}{3y}}\cdot {\color {Bluish}{2x}})&+&({\color {Ruddy}{3y}}\cdot {\color {Blue}{5y}})&+&({\color {Cherry}{3y}}\cdot {\colour {Blue}{xy}})&+&({\color {Red}{3y}}\cdot {\color {Blue}{1}})\\&&+&({\colour {Reddish}{5}}\cdot {\color {Blue}{2x}})&+&({\color {Carmine}{5}}\cdot {\color {Bluish}{5y}})&+&({\color {Crimson}{5}}\cdot {\color {Bluish}{xy}})&+&({\colour {Red}{5}}\cdot {\color {Blueish}{1}})\finish{array}}}



Carrying out the multiplication in each term produces









P
Q


=



four

x

2




+


10
x
y


+


two

x

2


y


+


2
ten






+


half dozen
x
y


+


15

y

2




+


3
x

y

2




+


3
y






+


10
x


+


25
y


+


v
x
y


+


five.






{\displaystyle {\begin{array}{rccrcrcrcr}PQ&=&&4x^{2}&+&10xy&+&2x^{2}y&+&2x\\&&+&6xy&+&15y^{2}&+&3xy^{2}&+&3y\\&&+&10x&+&25y&+&5xy&+&5.\stop{array}}}



Combining like terms yields









P
Q


=



4

ten

2




+


(
10
x
y
+
6
x
y
+
5
ten
y
)


+


ii

x

two


y


+


(
2
x
+
10
x
)






+


15

y

ii




+


3
x

y

ii




+


(
3
y
+
25
y
)


+


5






{\displaystyle {\begin{array}{rcccrcrcrcr}PQ&=&&4x^{2}&+&(10xy+6xy+5xy)&+&2x^{2}y&+&(2x+10x)\\&&+&15y^{2}&+&3xy^{2}&+&(3y+25y)&+&5\end{array}}}



which tin be simplified to





P
Q
=
4

ten

2


+
21
x
y
+
2

x

2


y
+
12
x
+
fifteen

y

2


+
iii
10

y

2


+
28
y
+
5.


{\displaystyle PQ=4x^{2}+21xy+2x^{2}y+12x+15y^{ii}+3xy^{two}+28y+5.}



As in the instance, the product of polynomials is always a polynomial.[ix]
[4]

Composition

[edit]

Given a polynomial




f


{\displaystyle f}




of a single variable and some other polynomial
g
of whatever number of variables, the limerick




f



g


{\displaystyle f\circ g}




is obtained by substituting each copy of the variable of the first polynomial past the second polynomial.[4]
For example, if




f
(
ten
)
=

10

ii


+
2
x


{\displaystyle f(ten)=ten^{2}+2x}




and




g
(
x
)
=
3
10
+
two


{\displaystyle g(x)=3x+2}




then




(
f



thousand
)
(
ten
)
=
f
(
g
(
x
)
)
=
(
iii
x
+
2

)

2


+
ii
(
3
x
+
2
)
.


{\displaystyle (f\circ grand)(x)=f(g(ten))=(3x+2)^{2}+2(3x+2).}

A composition may be expanded to a sum of terms using the rules for multiplication and division of polynomials. The composition of two polynomials is another polynomial.[10]

Division

[edit]

The division of one polynomial by another is not typically a polynomial. Instead, such ratios are a more full general family of objects, called
rational fractions,
rational expressions, or
rational functions, depending on context.[11]
This is analogous to the fact that the ratio of ii integers is a rational number, not necessarily an integer.[12]
[13]
For case, the fraction
1/(x
2
+ 1)

is not a polynomial, and information technology cannot be written as a finite sum of powers of the variable
ten.

For polynomials in one variable, there is a notion of Euclidean segmentation of polynomials, generalizing the Euclidean sectionalisation of integers.[due east]
This notion of the segmentation

a(x)/b(x)

results in two polynomials, a
quotient

q(x)

and a
remainder

r(ten)
, such that

a
=
b
q
+
r

and
degree(r) < caste(b). The quotient and remainder may be computed by whatsoever of several algorithms, including polynomial long division and constructed partition.[xiv]

When the denominator

b(x)

is monic and linear, that is,

b(x) =
x

c

for some constant
c, so the polynomial residual theorem asserts that the remainder of the division of

a(x)

by

b(ten)

is the evaluation

a(c)
.[13]
In this case, the quotient may be computed by Ruffini’s rule, a special example of synthetic division.[15]

Factoring

[edit]

All polynomials with coefficients in a unique factorization domain (for example, the integers or a field) as well have a factored class in which the polynomial is written equally a product of irreducible polynomials and a constant. This factored form is unique up to the order of the factors and their multiplication by an invertible abiding. In the case of the field of complex numbers, the irreducible factors are linear. Over the existent numbers, they have the caste either i or two. Over the integers and the rational numbers the irreducible factors may have any degree.[16]
For example, the factored form of





v

x

3





5


{\displaystyle 5x^{three}-5}



is





5
(
10



1
)

(


x

2


+
10
+
1

)



{\displaystyle 5(x-1)\left(ten^{2}+x+one\right)}



over the integers and the reals, and





5
(
x



i
)

(

x
+



1
+
i


3



ii



)


(

x
+



one



i


3



2



)



{\displaystyle 5(10-1)\left(x+{\frac {1+i{\sqrt {3}}}{ii}}\right)\left(x+{\frac {1-i{\sqrt {3}}}{2}}\right)}



over the complex numbers.

The computation of the factored form, called
factorization
is, in general, too difficult to be done by hand-written computation. However, efficient polynomial factorization algorithms are bachelor in most computer algebra systems.

Calculus

[edit]

Calculating derivatives and integrals of polynomials is particularly simple, compared to other kinds of functions. The derivative of the polynomial




P
=

a

northward



10

due north


+

a

due north



1



x

due north



1


+



+

a

two



ten

2


+

a

ane


ten
+

a

0


=





i
=
0


n



a

i



x

i




{\displaystyle P=a_{n}x^{northward}+a_{due north-i}x^{due north-1}+\dots +a_{ii}x^{ii}+a_{one}x+a_{0}=\sum _{i=0}^{northward}a_{i}x^{i}}

with respect to
ten
is the polynomial




due north

a

northward



x

north



1


+
(
north



1
)

a

n



1



x

n



2


+



+
ii

a

2


10
+

a

one


=





i
=
ane


n


i

a

i



x

i



1


.


{\displaystyle na_{n}x^{north-1}+(n-1)a_{due north-1}x^{n-two}+\dots +2a_{ii}x+a_{i}=\sum _{i=one}^{n}ia_{i}10^{i-1}.}

Similarly, the general antiderivative (or indefinite integral) of




P


{\displaystyle P}




is








a

n



x

n
+
1




n
+
one



+




a

n



1



10

north



n


+



+




a

2



x

iii



three


+




a

1



x

ii



2


+

a

0


x
+
c
=
c
+





i
=
0


n






a

i



x

i
+
1




i
+
i





{\displaystyle {\frac {a_{n}ten^{north+1}}{n+1}}+{\frac {a_{north-1}ten^{north}}{n}}+\dots +{\frac {a_{2}x^{three}}{three}}+{\frac {a_{1}ten^{2}}{ii}}+a_{0}x+c=c+\sum _{i=0}^{due north}{\frac {a_{i}x^{i+one}}{i+1}}}

where
c
is an capricious abiding. For example, antiderivatives of

10
ii
+ 1

have the class


1
/
3

x
3
+
x
+
c
.

For polynomials whose coefficients come from more than abstract settings (for instance, if the coefficients are integers modulo some prime

p
, or elements of an arbitrary ring), the formula for the derivative can nonetheless exist interpreted formally, with the coefficient

ka

k


understood to mean the sum of
k
copies of

a

thousand

. For example, over the integers modulo

p
, the derivative of the polynomial

x

p

+
10

is the polynomial
1.[17]

Polynomial functions

[edit]

A
polynomial function
is a office that can be defined by evaluating a polynomial. More precisely, a function

f

of one argument from a given domain is a polynomial function if there exists a polynomial






a

n



x

north


+

a

north



ane



10

n



ane


+



+

a

ii



x

2


+

a

one


x
+

a

0




{\displaystyle a_{n}x^{northward}+a_{n-1}x^{n-1}+\cdots +a_{2}x^{2}+a_{ane}x+a_{0}}



that evaluates to




f
(
x
)


{\displaystyle f(x)}




for all
x
in the domain of
f
(hither,

n

is a non-negative integer and

a
0,
a
1,
a
two, …,
anorth


are constant coefficients). By and large, unless otherwise specified, polynomial functions have circuitous coefficients, arguments, and values. In item, a polynomial, restricted to take real coefficients, defines a role from the complex numbers to the complex numbers. If the domain of this function is too restricted to the reals, the resulting function is a real role that maps reals to reals.

For example, the office

f
, divers by





f
(
x
)
=

x

3





x
,


{\displaystyle f(x)=10^{3}-x,}



is a polynomial function of one variable. Polynomial functions of several variables are similarly defined, using polynomials in more than 1 indeterminate, as in





f
(
x
,
y
)
=
2

10

iii


+
4

ten

two


y
+
x

y

5


+

y

ii





vii.


{\displaystyle f(x,y)=2x^{three}+4x^{2}y+xy^{v}+y^{2}-7.}



According to the definition of polynomial functions, at that place may be expressions that apparently are not polynomials only even so define polynomial functions. An case is the expression






(


1




ten

2




)


2


,


{\displaystyle \left({\sqrt {one-x^{2}}}\correct)^{2},}




which takes the same values as the polynomial




1




x

two




{\displaystyle 1-x^{2}}




on the interval




[



ane
,
1
]


{\displaystyle [-ane,i]}



, and thus both expressions define the aforementioned polynomial function on this interval.

Every polynomial function is continuous, smooth, and entire.

Graphs

[edit]

A polynomial office in one real variable can be represented by a graph.

  • The graph of the zero polynomial


    f(x) = 0

    is the

    ten
    -centrality.

  • The graph of a caste 0 polynomial


    f(ten) =
    a
    0
    , where

    a
    0
    ≠ 0
    ,

    is a horizontal line with


    y
    -intercept

    a
    0

  • The graph of a degree one polynomial (or linear part)


    f(x) =
    a
    0
    +
    a
    ane
    ten
    , where

    a
    one
    ≠ 0
    ,

    is an oblique line with


    y
    -intercept

    a
    0


    and slope

    a
    1
    .

  • The graph of a degree 2 polynomial


    f(x) =
    a
    0
    +
    a
    one
    x
    +
    a
    2
    10
    2
    , where

    a
    2
    ≠ 0

    is a parabola.

  • The graph of a degree 3 polynomial


    f(x) =
    a
    0
    +
    a
    1
    x
    +
    a
    2
    x
    two
    +
    a
    3
    x
    three
    , where

    a
    3
    ≠ 0

    is a cubic curve.

  • The graph of whatsoever polynomial with caste two or greater


    f(x) =
    a
    0
    +
    a
    1
    x
    +
    a
    2
    x
    2
    + ⋯ +
    a

    n

    ten

    n

    , where

    a

    due north

    ≠ 0 and
    north
    ≥ 2

    is a continuous non-linear curve.

A non-abiding polynomial part tends to infinity when the variable increases indefinitely (in absolute value). If the caste is college than one, the graph does not have whatever asymptote. It has two parabolic branches with vertical direction (one co-operative for positive
x
and one for negative
x).

Polynomial graphs are analyzed in calculus using intercepts, slopes, concavity, and end beliefs.

Equations

[edit]

A
polynomial equation, also chosen an
algebraic equation, is an equation of the class[eighteen]






a

n



x

n


+

a

n



i



x

n



1


+



+

a

ii



x

2


+

a

1


x
+

a

0


=
0.


{\displaystyle a_{n}10^{n}+a_{n-1}x^{n-1}+\dotsb +a_{2}x^{2}+a_{1}x+a_{0}=0.}



For example,





3

x

2


+
4
x



5
=
0


{\displaystyle 3x^{2}+4x-5=0}



is a polynomial equation.

When considering equations, the indeterminates (variables) of polynomials are also called unknowns, and the
solutions
are the possible values of the unknowns for which the equality is true (in general more than one solution may exist). A polynomial equation stands in dissimilarity to a
polynomial identity
similar
(10
+
y)(x

y) =
x
2

y
2
, where both expressions stand for the aforementioned polynomial in different forms, and every bit a effect any evaluation of both members gives a valid equality.

In elementary algebra, methods such as the quadratic formula are taught for solving all first degree and second degree polynomial equations in one variable. There are also formulas for the cubic and quartic equations. For college degrees, the Abel–Ruffini theorem asserts that in that location tin can not exist a full general formula in radicals. Nevertheless, root-finding algorithms may be used to find numerical approximations of the roots of a polynomial expression of whatever degree.

The number of solutions of a polynomial equation with real coefficients may not exceed the degree, and equals the degree when the complex solutions are counted with their multiplicity. This fact is called the key theorem of algebra.

Solving equations


[edit]

A
root
of a nonzero univariate polynomial

P

is a value
a
of
x
such that

P(a) = 0
. In other words, a root of
P
is a solution of the polynomial equation

P(10) = 0

or a goose egg of the polynomial function defined by

P
. In the instance of the cypher polynomial, every number is a zilch of the respective role, and the concept of root is rarely considered.

A number

a

is a root of a polynomial

P

if and only if the linear polynomial

x

a

divides

P
, that is if in that location is some other polynomial

Q

such that

P
= (10

a) Q
. It may happen that a power (greater than
1) of

x

a

divides

P
; in this example,

a

is a
multiple root
of

P
, and otherwise

a

is a
uncomplicated root
of

P
. If

P

is a nonzero polynomial, there is a highest power

m

such that
(x

a)
m


divides

P
, which is called the
multiplicity
of

a

as a root of

P
. The number of roots of a nonzero polynomial

P
, counted with their respective multiplicities, cannot exceed the degree of

P
,[19]
and equals this degree if all complex roots are considered (this is a consequence of the fundamental theorem of algebra). The coefficients of a polynomial and its roots are related by Vieta’southward formulas.

Some polynomials, such as

x
2
+ i
, do not take any roots among the existent numbers. If, withal, the set of accepted solutions is expanded to the complex numbers, every non-constant polynomial has at least i root; this is the fundamental theorem of algebra. By successively dividing out factors

x

a
, one sees that any polynomial with complex coefficients tin be written as a constant (its leading coefficient) times a product of such polynomial factors of degree ane; as a consequence, the number of (complex) roots counted with their multiplicities is exactly equal to the degree of the polynomial.

At that place may be several meanings of “solving an equation”. Ane may want to limited the solutions equally explicit numbers; for case, the unique solution of
2x
− 1 = 0

is
one/2. Unfortunately, this is, in general, impossible for equations of degree greater than one, and, since the ancient times, mathematicians have searched to express the solutions as algebraic expressions; for case, the golden ratio




(
one
+


5


)

/

2


{\displaystyle (1+{\sqrt {5}})/ii}




is the unique positive solution of





x

ii





x



1
=
0.


{\displaystyle x^{ii}-x-i=0.}




In the ancient times, they succeeded but for degrees one and two. For quadratic equations, the quadratic formula provides such expressions of the solutions. Since the 16th century, similar formulas (using cube roots in add-on to square roots), although much more complicated, are known for equations of caste three and four (see cubic equation and quartic equation). Simply formulas for degree 5 and college eluded researchers for several centuries. In 1824, Niels Henrik Abel proved the striking outcome that there are equations of degree 5 whose solutions cannot exist expressed by a (finite) formula, involving simply arithmetic operations and radicals (meet Abel–Ruffini theorem). In 1830, Évariste Galois proved that nigh equations of degree higher than iv cannot be solved by radicals, and showed that for each equation, i may decide whether it is solvable by radicals, and, if it is, solve information technology. This upshot marked the start of Galois theory and grouping theory, 2 important branches of modern algebra. Galois himself noted that the computations unsaid by his method were impracticable. Notwithstanding, formulas for solvable equations of degrees five and 6 take been published (meet quintic function and sextic equation).

When there is no algebraic expression for the roots, and when such an algebraic expression exists but is too complicated to exist useful, the unique way of solving information technology is to compute numerical approximations of the solutions.[20]
There are many methods for that; some are restricted to polynomials and others may apply to whatsoever continuous function. The most efficient algorithms permit solving easily (on a computer) polynomial equations of degree higher than 1,000 (run into Root-finding algorithm).

For polynomials with more than one indeterminate, the combinations of values for the variables for which the polynomial function takes the value null are generally called
zeros
instead of “roots”. The study of the sets of zeros of polynomials is the object of algebraic geometry. For a set of polynomial equations with several unknowns, there are algorithms to decide whether they accept a finite number of circuitous solutions, and, if this number is finite, for computing the solutions. Run into Organisation of polynomial equations.

The special example where all the polynomials are of caste one is called a arrangement of linear equations, for which another range of unlike solution methods exist, including the classical Gaussian elimination.

A polynomial equation for which one is interested but in the solutions which are integers is called a Diophantine equation. Solving Diophantine equations is more often than not a very difficult job. Information technology has been proved that there cannot be any general algorithm for solving them, or even for deciding whether the fix of solutions is empty (meet Hilbert’s 10th problem). Some of the most famous problems that have been solved during the last 50 years are related to Diophantine equations, such as Fermat’s Terminal Theorem.

Polynomial expressions

[edit]


Polynomials where indeterminates are substituted for another mathematical objects are ofttimes considered, and sometimes have a special name.

Trigonometric polynomials

[edit]

A
trigonometric polynomial
is a finite linear combination of functions sin(nx) and cos(nx) with
n
taking on the values of i or more than natural numbers.[21]
The coefficients may exist taken as real numbers, for existent-valued functions.

If sin(nx) and cos(nx) are expanded in terms of sin(10) and cos(10), a trigonometric polynomial becomes a polynomial in the two variables sin(10) and cos(x) (using List of trigonometric identities#Multiple-angle formulae). Conversely, every polynomial in sin(x) and cos(x) may be converted, with Product-to-sum identities, into a linear combination of functions sin(nx) and cos(nx). This equivalence explains why linear combinations are called polynomials.

For complex coefficients, at that place is no deviation betwixt such a function and a finite Fourier series.

Trigonometric polynomials are widely used, for example in trigonometric interpolation practical to the interpolation of periodic functions. They are also used in the discrete Fourier transform.

Matrix polynomials

[edit]

A matrix polynomial is a polynomial with square matrices as variables.[22]
Given an ordinary, scalar-valued polynomial





P
(
x
)
=





i
=
0


north




a

i



10

i



=

a

0


+

a

1


10
+

a

2



x

2


+



+

a

n



x

northward


,


{\displaystyle P(x)=\sum _{i=0}^{due north}{a_{i}x^{i}}=a_{0}+a_{1}ten+a_{2}10^{2}+\cdots +a_{n}x^{n},}



this polynomial evaluated at a matrix
A
is





P
(
A
)
=





i
=
0


due north




a

i



A

i



=

a

0


I
+

a

1


A
+

a

2



A

ii


+



+

a

n



A

n


,


{\displaystyle P(A)=\sum _{i=0}^{due north}{a_{i}A^{i}}=a_{0}I+a_{1}A+a_{2}A^{two}+\cdots +a_{north}A^{northward},}



where
I
is the identity matrix.[23]

A
matrix polynomial equation
is an equality between ii matrix polynomials, which holds for the specific matrices in question. A
matrix polynomial identity
is a matrix polynomial equation which holds for all matrices
A
in a specified matrix band
Thoun
(R).

Exponential polynomials

[edit]

A bivariate polynomial where the second variable is substituted for an exponential office applied to the commencement variable, for case

P(x,
e

ten
)
, may be called an exponential polynomial.


[edit]

Rational functions

[edit]

A rational fraction is the quotient (algebraic fraction) of two polynomials. Whatsoever algebraic expression that tin can exist rewritten as a rational fraction is a rational office.

While polynomial functions are defined for all values of the variables, a rational function is defined only for the values of the variables for which the denominator is non zero.

The rational fractions include the Laurent polynomials, merely do not limit denominators to powers of an indeterminate.

Laurent polynomials

[edit]

Laurent polynomials are like polynomials, but allow negative powers of the variable(southward) to occur.

Power series

[edit]

Formal power series are like polynomials, simply let infinitely many non-zero terms to occur, so that they do not have finite degree. Dissimilar polynomials they cannot in general be explicitly and fully written downwards (just like irrational numbers cannot), merely the rules for manipulating their terms are the same as for polynomials. Non-formal power serial also generalize polynomials, only the multiplication of two power serial may not converge.

Polynomial ring

[edit]

A
polynomial

f

over a commutative band

R

is a polynomial all of whose coefficients belong to

R
. It is straightforward to verify that the polynomials in a given set of indeterminates over

R

course a commutative ring, called the
polynomial band
in these indeterminates, denoted




R
[
x
]


{\displaystyle R[x]}




in the univariate case and




R
[

x

ane


,



,

x

due north


]


{\displaystyle R[x_{ane},\ldots ,x_{north}]}




in the multivariate example.

I has





R
[

x

i


,



,

x

n


]
=

(

R
[

x

ane


,



,

x

n



1


]

)

[

ten

northward


]
.


{\displaystyle R[x_{i},\ldots ,x_{due north}]=\left(R[x_{1},\ldots ,x_{n-1}]\right)[x_{n}].}



So, about of the theory of the multivariate case can be reduced to an iterated univariate instance.

The map from

R

to

R[x]

sending

r

to itself considered as a constant polynomial is an injective band homomorphism, by which

R

is viewed as a subring of

R[x]
. In detail,

R[x]

is an algebra over

R
.

One can retrieve of the ring

R[10]

as arising from

R

by adding one new chemical element
10
to
R, and extending in a minimal way to a ring in which

ten

satisfies no other relations than the obligatory ones, plus commutation with all elements of

R

(that is

xr
=
rx
). To do this, one must add all powers of

x

and their linear combinations besides.

Germination of the polynomial band, together with forming factor rings by factoring out ideals, are important tools for constructing new rings out of known ones. For instance, the ring (in fact field) of complex numbers, which can be constructed from the polynomial ring

R[ten]

over the real numbers by factoring out the platonic of multiples of the polynomial

x
2
+ one
. Another instance is the structure of finite fields, which proceeds similarly, starting out with the field of integers modulo some prime as the coefficient ring

R

(see modular arithmetic).

If

R

is commutative, then one tin associate with every polynomial

P

in

R[ten]

a
polynomial function

f

with domain and range equal to

R
. (More mostly, one can take domain and range to be any same unital associative algebra over

R
.) One obtains the value

f(r)

by commutation of the value

r

for the symbol

10

in

P
. One reason to distinguish between polynomials and polynomial functions is that, over some rings, different polynomials may give ascent to the same polynomial function (encounter Fermat’s little theorem for an example where

R

is the integers modulo

p
). This is not the case when

R

is the real or complex numbers, whence the two concepts are not e’er distinguished in analysis. An even more of import reason to distinguish between polynomials and polynomial functions is that many operations on polynomials (similar Euclidean segmentation) require looking at what a polynomial is composed of as an expression rather than evaluating information technology at some constant value for

x
.

Divisibility

[edit]

If

R

is an integral domain and

f

and

thousand

are polynomials in

R[ten]
, it is said that

f

divides

g

or

f

is a divisor of

g

if at that place exists a polynomial

q

in

R[x]

such that

f
q
=
1000
. If




a



R
,


{\displaystyle a\in R,}




then
a
is a root of
f
if and just




10



a


{\displaystyle x-a}




divides
f. In this case, the caliber tin can be computed using the polynomial long division.[24]
[25]

If

F

is a field and

f

and

g

are polynomials in

F[x]

with

g
≠ 0
, so there exist unique polynomials

q

and

r

in

F[x]

with





f
=
q

1000
+
r


{\displaystyle f=q\,g+r}



and such that the degree of

r

is smaller than the degree of

g

(using the convention that the polynomial 0 has a negative degree). The polynomials

q

and

r

are uniquely determined by

f

and

1000
. This is chosen
Euclidean division, division with remainder
or
polynomial long division
and shows that the band

F[x]

is a Euclidean domain.

Analogously,
prime polynomials
(more correctly,
irreducible polynomials) can be divers as
non-naught polynomials which cannot be factorized into the product of two non-abiding polynomials. In the case of coefficients in a ring,
“non-constant”
must be replaced by
“non-constant or non-unit”
(both definitions hold in the case of coefficients in a field). Whatever polynomial may be decomposed into the product of an invertible constant past a production of irreducible polynomials. If the coefficients belong to a field or a unique factorization domain this decomposition is unique up to the order of the factors and the multiplication of whatever non-unit cistron by a unit of measurement (and division of the unit of measurement factor by the same unit). When the coefficients vest to integers, rational numbers or a finite field, in that location are algorithms to test irreducibility and to compute the factorization into irreducible polynomials (encounter Factorization of polynomials). These algorithms are not practicable for hand-written computation, but are available in any computer algebra system. Eisenstein’due south criterion can also be used in some cases to determine irreducibility.

Applications

[edit]

Positional notation

[edit]

In modernistic positional numbers systems, such as the decimal system, the digits and their positions in the representation of an integer, for instance, 45, are a shorthand notation for a polynomial in the radix or base of operations, in this example,
4 × ten1
+ 5 × 100
. As another case, in radix five, a string of digits such as 132 denotes the (decimal) number
1 × 52
+ 3 × 51
+ two × 50

= 42. This representation is unique. Let
b
be a positive integer greater than 1. Then every positive integer
a
tin be expressed uniquely in the grade





a
=

r

m



b

m


+

r

m



ane



b

1000



1


+



+

r

1


b
+

r

0


,


{\displaystyle a=r_{yard}b^{m}+r_{m-1}b^{m-1}+\dotsb +r_{1}b+r_{0},}



where
m
is a nonnegative integer and the
r’south are integers such that

0 <
r

m

<
b

and
0 ≤
r

i

<
b

for

i
= 0, 1, . . . ,
m
− 1
.[26]

Interpolation and approximation

[edit]

The unproblematic construction of polynomial functions makes them quite useful in analyzing general functions using polynomial approximations. An important case in calculus is Taylor’s theorem, which roughly states that every differentiable role locally looks like a polynomial function, and the Rock–Weierstrass theorem, which states that every continuous function defined on a meaty interval of the real axis can be approximated on the whole interval as closely as desired past a polynomial function. Practical methods of approximation include polynomial interpolation and the use of splines.[27]

Other applications

[edit]

Polynomials are frequently used to encode information about another object. The characteristic polynomial of a matrix or linear operator contains information nigh the operator’s eigenvalues. The minimal polynomial of an algebraic element records the simplest algebraic relation satisfied by that element. The chromatic polynomial of a graph counts the number of proper colourings of that graph.

The term “polynomial”, equally an describing word, can also be used for quantities or functions that can be written in polynomial form. For case, in computational complication theory the phrase
polynomial time
means that the time it takes to complete an algorithm is bounded by a polynomial function of some variable, such as the size of the input.

History

[edit]

Determining the roots of polynomials, or “solving algebraic equations”, is amid the oldest problems in mathematics. However, the elegant and practical notation we use today only developed beginning in the 15th century. Earlier that, equations were written out in words. For example, an algebra problem from the Chinese Arithmetic in 9 Sections, circa 200 BCE, begins “3 sheafs of good crop, two sheafs of mediocre crop, and ane sheaf of bad ingather are sold for 29 dou.” Nosotros would write
threex
+ 2y
+
z
= 29
.

History of the annotation

[edit]

The earliest known use of the equal sign is in Robert Recorde’due south
The Whetstone of Witte, 1557. The signs + for addition, − for subtraction, and the use of a letter for an unknown announced in Michael Stifel’s
Arithemetica integra, 1544. René Descartes, in
La géometrie, 1637, introduced the concept of the graph of a polynomial equation. He popularized the use of letters from the showtime of the alphabet to denote constants and messages from the end of the alphabet to denote variables, equally can exist seen higher up, in the general formula for a polynomial in i variable, where the

a
‘s denote constants and

x

denotes a variable. Descartes introduced the utilise of superscripts to announce exponents likewise.[28]

See also

[edit]

  • List of polynomial topics

Notes

[edit]


  1. ^

    Come across “polynomial” and “binomial”,
    Compact Oxford English Dictionary
  2. ^


    a




    b




    Weisstein, Eric W. “Polynomial”.
    mathworld.wolfram.com
    . Retrieved
    2020-08-28
    .


  3. ^


    a




    b




    “Polynomials | Brilliant Math & Science Wiki”.
    brilliant.org
    . Retrieved
    2020-08-28
    .


  4. ^


    a




    b




    c



    Barbeau 2003, pp. 1–ii

  5. ^



    Weisstein, Eric W. “Nada Polynomial”.
    MathWorld.




  6. ^

    Edwards 1995, p. 78
  7. ^


    a




    b




    c




    Edwards, Harold Thousand. (1995).
    Linear Algebra. Springer. p. 47. ISBN978-0-8176-3731-6.



  8. ^


    Salomon, David (2006).
    Coding for Information and Reckoner Communications. Springer. p. 459. ISBN978-0-387-23804-3.


  9. ^


    a




    b





    Introduction to Algebra. Yale University Press. 1965. p. 621.
    Any 2 such polynomials tin can exist added, subtracted, or multiplied. Furthermore , the result in each instance is some other polynomial



  10. ^


    Kriete, Hartje (1998-05-20).
    Progress in Holomorphic Dynamics. CRC Press. p. 159. ISBN978-0-582-32388-nine.
    This class of endomorphisms is closed under composition,



  11. ^


    Marecek, Lynn; Mathis, Andrea Honeycutt (six May 2020).
    Intermediate Algebra 2e. OpenStax. §7.1.



  12. ^


    Haylock, Derek; Cockburn, Anne D. (2008-x-14).
    Understanding Mathematics for Immature Children: A Guide for Foundation Stage and Lower Primary Teachers. SAGE. p. 49. ISBN978-1-4462-0497-nine.
    We observe that the set of integers is non closed nether this functioning of segmentation.


  13. ^


    a




    b



    Marecek & Mathis 2020, §5.4]

  14. ^


    Selby, Peter H.; Slavin, Steve (1991).
    Practical Algebra: A Self-Teaching Guide
    (2d ed.). Wiley. ISBN978-0-471-53012-one.



  15. ^


    Weisstein, Eric Due west. “Ruffini’south Dominion”.
    mathworld.wolfram.com
    . Retrieved
    2020-07-25
    .



  16. ^

    Barbeau 2003, pp. 80–2

  17. ^

    Barbeau 2003, pp. 64–5

  18. ^


    Proskuryakov, I.Five. (1994). “Algebraic equation”. In Hazewinkel, Michiel (ed.).
    Encyclopaedia of Mathematics. Vol. 1. Springer. ISBN978-1-55608-010-4.



  19. ^


    Leung, Kam-tim; et al. (1992).
    Polynomials and Equations. Hong Kong Academy Press. p. 134. ISBN9789622092716.



  20. ^


    McNamee, J.Thousand. (2007).
    Numerical Methods for Roots of Polynomials, Function 1. Elsevier. ISBN978-0-08-048947-half-dozen.



  21. ^


    Powell, Michael J. D. (1981).
    Approximation Theory and Methods. Cambridge University Press. ISBN978-0-521-29514-7.



  22. ^


    Gohberg, Israel; Lancaster, Peter; Rodman, Leiba (2009) [1982].
    Matrix Polynomials. Classics in Applied Mathematics. Vol. 58. Lancaster, PA: Society for Industrial and Practical Mathematics. ISBN978-0-89871-681-8. Zbl 1170.15300.



  23. ^

    Horn & Johnson 1990, p. 36.
    sfn error: no target: CITEREFHornJohnson1990 (aid)


  24. ^


    Irving, Ronald S. (2004).
    Integers, Polynomials, and Rings: A Form in Algebra. Springer. p. 129. ISBN978-0-387-20172-half-dozen.



  25. ^


    Jackson, Terrence H. (1995).
    From Polynomials to Sums of Squares. CRC Printing. p. 143. ISBN978-0-7503-0329-3.



  26. ^

    McCoy 1968, p. 75

  27. ^


    de Villiers, Johann (2012).
    Mathematics of Approximation. Springer. ISBN9789491216503.



  28. ^


    Eves, Howard (1990).
    An Introduction to the History of Mathematics
    (6th ed.). Saunders. ISBN0-03-029558-0.



  1. ^

    The coefficient of a term may be whatsoever number from a specified set. If that set is the prepare of real numbers, nosotros speak of “polynomials over the reals”. Other common kinds of polynomials are polynomials with integer coefficients, polynomials with complex coefficients, and polynomials with coefficients that are integers modulo some prime number

    p
    .

  2. ^

    This terminology dates from the time when the distinction was non clear between a polynomial and the function that it defines: a constant term and a abiding polynomial define abiding functions.[
    citation needed
    ]


  3. ^

    In fact, as a homogeneous function, information technology is homogeneous of
    every
    degree.[
    citation needed
    ]


  4. ^

    Some authors use “monomial” to hateful “monic monomial”. See
    Knapp, Anthony W. (2007).
    Advanced Algebra: Along with a Companion Volume Bones Algebra. Springer. p. 457. ISBN978-0-8176-4522-nine.



  5. ^

    This paragraph assumes that the polynomials accept coefficients in a field.

References

[edit]

  • Barbeau, E.J. (2003).
    Polynomials. Springer. ISBN978-0-387-40627-five.

  • Bronstein, Manuel; et al., eds. (2006).
    Solving Polynomial Equations: Foundations, Algorithms, and Applications. Springer. ISBN978-iii-540-27357-eight.

  • Cahen, Paul-Jean; Chabert, Jean-Luc (1997).
    Integer-Valued Polynomials. American Mathematical Society. ISBN978-0-8218-0388-ii.

  • Lang, Serge (2002),
    Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN978-0-387-95385-4, MR 1878556

    . This classical book covers most of the content of this article.
  • Leung, Kam-tim; et al. (1992).
    Polynomials and Equations. Hong Kong University Press. ISBN9789622092716.

  • Mayr, K. (1937). “Über die Auflösung algebraischer Gleichungssysteme durch hypergeometrische Funktionen”.
    Monatshefte für Mathematik und Physik.
    45: 280–313. doi:10.1007/BF01707992. S2CID 197662587.

  • McCoy, Neal H. (1968),
    Introduction To Modernistic Algebra, Revised Edition, Boston: Allyn and Bacon, LCCN 68015225

  • Prasolov, Victor V. (2005).
    Polynomials. Springer. ISBN978-3-642-04012-2.

  • Sethuraman, B.A. (1997). “Polynomials”.

    Rings, Fields, and Vector Spaces: An Introduction to Abstract Algebra Via Geometric Constructibility
    . Springer. ISBN978-0-387-94848-5.

  • Umemura, H. (2012) [1984]. “Resolution of algebraic equations by theta constants”. In Mumford, David (ed.).
    Tata Lectures on Theta II: Jacobian theta functions and differential equations. Springer. pp. 261–. ISBN978-0-8176-4578-6.

  • von Lindemann, F. (1884). “Ueber die Auflösung der algebraischen Gleichungen durch transcendente Functionen”.
    Nachrichten von der Königl. Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen.
    1884: 245–8.

  • von Lindemann, F. (1892). “Ueber die Auflösung der algebraischen Gleichungen durch transcendente Functionen. II”.
    Nachrichten von der Königl. Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen.
    1892: 245–8.

External links

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  • “Polynomial”,
    Encyclopedia of Mathematics, EMS Printing, 2001 [1994]

  • “Euler’s Investigations on the Roots of Equations”. Archived from the original on September 24, 2012.



A Polynomial Added to a Polynomial is a Polynomial

Source: https://en.wikipedia.org/wiki/Polynomial