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By the end of this section, you will be able to:
  • Identify polynomials, monomials, binomials, and trinomials
  • Determine the degree of polynomials
  • Add and subtract monomials
  • Add and subtract polynomials
  • Evaluate a polynomial for a given value

Before you get started, take this readiness quiz.

  1. Simplify: 8 x + 3 x .
    If you missed this problem, review [link] .
  2. Subtract: ( 5 n + 8 ) ( 2 n 1 ) .
    If you missed this problem, review [link] .
  3. Write in expanded form: a 5 .
    If you missed this problem, review [link] .

Identify polynomials, monomials, binomials and trinomials

You have learned that a term is a constant or the product of a constant and one or more variables. When it is of the form a x m , where a is a constant and m is a whole number, it is called a monomial. Some examples of monomial are 8 , −2 x 2 , 4 y 3 , and 11 z 7 .

Monomials

A monomial    is a term of the form a x m , where a is a constant and m is a positive whole number.

A monomial, or two or more monomials combined by addition or subtraction, is a polynomial. Some polynomials have special names, based on the number of terms. A monomial is a polynomial with exactly one term. A binomial has exactly two terms, and a trinomial has exactly three terms. There are no special names for polynomials with more than three terms.

Polynomials

polynomial    —A monomial, or two or more monomials combined by addition or subtraction, is a polynomial.

  • monomial —A polynomial with exactly one term is called a monomial.
  • binomial    —A polynomial with exactly two terms is called a binomial.
  • trinomial    —A polynomial with exactly three terms is called a trinomial.

Here are some examples of polynomials.

Polynomial b + 1 4 y 2 7 y + 2 4 x 4 + x 3 + 8 x 2 9 x + 1 Monomial 14 8 y 2 −9 x 3 y 5 −13 Binomial a + 7 4 b 5 y 2 16 3 x 3 9 x 2 Trinomial x 2 7 x + 12 9 y 2 + 2 y 8 6 m 4 m 3 + 8 m z 4 + 3 z 2 1

Notice that every monomial, binomial, and trinomial is also a polynomial. They are just special members of the “family” of polynomials and so they have special names. We use the words monomial , binomial , and trinomial when referring to these special polynomials and just call all the rest polynomials .

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial.

  1. 4 y 2 8 y 6
  2. −5 a 4 b 2
  3. 2 x 5 5 x 3 9 x 2 + 3 x + 4
  4. 13 5 m 3
  5. q

Solution

Polynomial Number of terms Type (a) 4 y 2 8 y 6 3 Trinomial (b) −5 a 4 b 2 1 Monomial (c) 2 x 5 5 x 3 9 x 2 + 3 x + 4 5 Polynomial (d) 13 5 m 3 2 Binomial (e) q 1 Monomial

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Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial:

5 b 8 y 3 7 y 2 y 3 −3 x 2 5 x + 9 81 4 a 2 −5 x 6

monomial polynomial trinomial binomial monomial

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Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial:

27 z 3 8 12 m 3 5 m 2 2 m 5 6 8 x 4 7 x 2 6 x 5 n 4

binomial trinomial monomial polynomial monomial

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Determine the degree of polynomials

The degree of a polynomial and the degree of its terms are determined by the exponents of the variable.

A monomial    that has no variable, just a constant, is a special case. The degree of a constant is 0—it has no variable.

Degree of a polynomial

The degree of a term    is the sum of the exponents of its variables.

The degree of a constant    is 0.

The degree of a polynomial    is the highest degree of all its terms.

Practice Key Terms 8

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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