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A polynomialMathworldPlanetmathPlanetmathPlanetmath 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 radicalsPlanetmathPlanetmathPlanetmathPlanetmath or inside absolute valuesMathworldPlanetmathPlanetmathPlanetmathPlanetmath.

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, x2y3+12x2y2+y3x22 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 monomialMathworldPlanetmathPlanetmath 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, x2y3, 12x2y2, and y3x22 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, x2y3 and y3x22 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,

x2y3+12x2y2+y3x22 =x2y3+x2y32+12x2y2

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) =x2+2x-3x-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 x2-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 x0=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)x2y3+12x2y2 has degree 5.

The coefficient of a monomial is the numerical (non-variable) portion of the monomial. For example, the coefficient of -2x2y3 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 numbersMathworldPlanetmathPlanetmath.

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, x2-x-6 is in descending order. Since x0=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