polynomial

polynomial

\PMlinkescapephrase

occur in

A

polynomial

can be defined iteratively as follows:

•

Constants are polynomials.

•

Variables

(such as

x

) are polynomials.

•

Adding, subtracting, or multiplying two polynomials always yields a polynomial.

The above process always yields expressions in which variables only have exponents that are

positive

(or nonnegative) and in which variables never occur in

denominators

or within functions such as under

radicals

or inside

absolute values

.

It should be mentioned that, if the above process is used to create a polynomial, then the process

must

terminate since polynomials are not infinitely long.

For example,

x

2

⁢

y

3

+

1

2

⁢

x

2

⁢

y

2

+

y

3

⁢

x

2

⁢

2

is a polynomial. Note that fractions, radicals, and the like

can

occur in polynomials. It is only stipulated that no

variables

appear in denominators, under radicals, etc.

A

monomial

is a polynomial in which variables are being multiplied only. Within a polynomial, a monomial that is as large as possible is called a

term

of the polynomial. In the example above,

x

2

⁢

y

3

,

1

2

⁢

x

2

⁢

y

2

, and

y

3

⁢

x

2

⁢

2

are the terms of the polynomial. As alluded to earlier, every polynomial has a finite number of terms.

Terms of a polynomial are

like

if their variable expressions match. In the example above,

x

2

⁢

y

3

and

y

3

⁢

x

2

⁢

2

are like terms.

When students are first learning about polynomials, it is advisable to teach them to alphabetize the variables in each term. That way, students can more easily detect like terms.

Like terms can be

combined

by using the

distributive property

. For example,

x

2

⁢

y

3

+

1

2

⁢

x

2

⁢

y

2

+

y

3

⁢

x

2

⁢

2

=

x

2

⁢

y

3

+

x

2

⁢

y

3

⁢

2

+

1

2

⁢

x

2

⁢

y

2

=

(

1

+

2

)

⁢

x

2

⁢

y

3

+

1

2

⁢

x

2

⁢

y

2

.

A polynomial is

expanded

if no variable occurs within parentheses. For example,

(

x

-

3

)

⁢

(

x

+

2

)

is a polynomial since both

x

-

3

and

x

+

2

are polynomials. Expanding and combining like terms yields

(

x

-

3

)

⁢

(

x

+

2

)

=

x

2

+

2

⁢

x

-

3

⁢

x

-

6

=

x

2

-

x

-

6

.

In an expanded polynomial in which all like terms have been combined, the

constant term

is the term in which no variable appears (or all variables occur to the zero power). For example,

-

6

is the constant term of

x

2

-

x

-

6

. If no constant term appears, then the constant term is

0

.

The

degree

of a (nonzero) monomial is the sum of the exponents of its variables. Since

x

0

=

1

, the degree of a (nonzero) constant is

0

. Most do not define the degree of the polynomial

0

; some define the degree of the polynomial

0

to be

-

∞

.

The

degree

of a polynomial is the maximum of the degrees of its terms after the polynomial has been expanded. For example, the polynomial

(

1

+

2

)

⁢

x

2

⁢

y

3

+

1

2

⁢

x

2

⁢

y

2

has degree

5

.

The

coefficient

of a monomial is the numerical (non-variable) portion of the monomial. For example, the coefficient of

-

2

⁢

x

2

⁢

y

3

is

-

2

.

Occasionally, it may be stipulated that all of the coefficients of a polynomial be in a certain set. For example, most textbooks on elementary mathematics deal with polynomials with

integer

coefficients almost exclusively. Other sets that are commonly used as the

coefficients of polynomials

include the

rational numbers

, the

real numbers

, and the

complex numbers

.

For the of this entry, only polynomials in one variable will be discussed.

An expanded polynomial is in

descending order

if the degrees of the terms of the polynomial are strictly decreasing as the polynomial is read from left to right. Note that, for a polynomial to be written in descending order, all like terms have to be combined. For example,

x

2

-

x

-

6

is in descending order. Since

x

0

=

1

, the constant term always occurs last in a polynomial written in descending order. Note that an expanded polynomial is in

ascending order

if the degrees of the terms of the polynomial are strictly increasing as the polynomial is read from left to right.

In an expanded polynomial in which all like terms have been combined, the

leading coefficient

is the coefficient of the term that determines the degree of the polynomial. Therefore, if a polynomial is written in descending order, then the leading coefficient will be the leftmost coefficient.

More to come…

Title

polynomial

Canonical name

Polynomial

Date of creation

2013-03-22 17:53:38

Last modified on

2013-03-22 17:53:38

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

9

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 97D40

Classification

msc 26C99

Classification

msc 12-00

Related topic

OppositePolynomial

Related topic

PolynomialRing

Defines

monomial

Defines

term

Defines

like terms

Defines

combine like terms

Defines

combined like terms

Defines

combining like terms

Defines

expanded

Defines

expand

Defines

constant term

Defines

degree

Defines

coefficient

Defines

descending order

Defines

ascending order

Defines

leading coefficient

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