6.1 Add and Subtract Polynomials
Learning Objectives
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
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 , where is a constant and is a whole number, it is called a monomial. Some examples of monomial are , and .
Monomials
A monomial is a term of the form , where is a constant and 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.
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.
EXAMPLE 1
Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial.
| Polynomial | Number of terms | Type | |
| a) | Trinomial | ||
| b) | Monomial | ||
| c) | Polynomial | ||
| d) | Binomial | ||
| e) | Monomial |
TRY IT 1.1
Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial:
a) b) c) d) e)
Show answer
a) monomial b) polynomial c) trinomial d) binomial e) monomial
TRY IT 1.2
Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial:
a) b) c) d) e)
Show answer
a) binomial b) trinomial c) monomial d) polynomial e) monomial
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.
Let’s see how this works by looking at several polynomials. We’ll take it step by step, starting with monomials, and then progressing to polynomials with more terms.
A polynomial is in standard form when the terms