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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. Operations with algebraic expressions and numerical evaluations are introduced in this chapter. Coefficients are described rather than merely defined. Special binomial products have both literal and symbolic explanations and since they occur so frequently in mathematics, we have been careful to help the student remember them. In each example problem, the student is "talked" through the symbolic form.Objectives of this module: be familar with polynomials, be able classify polynomials and polynomial equations.

Overview

  • Polynomials
  • Classification of Polynomials
  • Classification of Polynomial Equations

Polynomials

Polynomials

Let us consider the collection of all algebraic expressions that do not contain variables in the denominators of fractions and where all exponents on the variable quantities are whole numbers. Expressions in this collection are called polynomials.

Some expressions that are polynomials are

2 5 x 2 y 6 .

A fraction occurs, but no variable appears in the denominator.

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5 x 3 + 3 x 2 2 x + 1

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Some expressions that are not polynomials are

3 x 16 .

A variable appears in the denominator.

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4 x 2 5 x + x 3 .

A negative exponent appears on a variable.

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Classification of polynomials

Polynomials can be classified using two criteria: the number of terms and degree of the polynomial.

Number of Terms Name Example Comment
One Monomial 4 x 2 mono means “one” in Greek.
Two Binomial 4 x 2 7 x bi means “two” in Latin.
Three Trinomial 4 x 2 7 x + 3 tri means “three” in Greek.
Four or more Polynomial 4 x 3 7 x 2 + 3 x 1 poly means “many” in Greek.

Degree of a term containing one variable

The degree of a term containing only one variable is the value of the exponent of the variable. Exponents appearing on numbers do not affect the degree of the term. We consider only the exponent of the variable. For example:

5 x 3 is a monomial of degree 3.

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60 a 5 is a monomial of degree 5.

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21 b 2 is a monomial of degree 2.

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8 is a monomial of degree 0. We say that a nonzero number is a term of 0 degree since it could be written as 8 x 0 . Since x 0 = 1 ( x 0 ) , 8 x 0 = 8 . The exponent on the variable is 0 so it must be of degree 0. (By convention, the number 0 has no degree.)

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4 x is a monomial of the first degree. 4 x could be written as 4 x 1 . The exponent on the variable is 1 so it must be of the first degree.

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Degree of a term containing several variables

The degree of a term containing more than one variable is the sum of the exponents of the variables, as shown below.

4 x 2 y 5 is a monomial of degree 2 + 5 = 7 . This is a 7th degree monomial.

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37 a b 2 c 6 d 3 is a monomial of degree 1 + 2 + 6 + 3 = 12 . This is a 12th degree monomial.

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5 x y is a monomial of degree 1 + 1 = 2 . This is a 2nd degree monomial.

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Degree of a polynomial

The degree of a polynomial is the degree of the term of highest degree; for example:

2 x 3 + 6 x 1 is a trinomial of degree 3. The first term, 2 x 3 , is the term of the highest degree. Therefore, its degree is the degree of the polynomial.

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7 y 10 y 4 is a binomial of degree 4.

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a 4 + 5 a 2 is a trinomial of degree 2.

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2 x 6 + 9 x 4 x 7 8 x 3 + x 9 is a polynomial of degree 7.

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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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