Polynomials and Polynomial Functions
Add and Subtract Polynomials
Learning Objectives
By the end of this section, you will be able to:
- Determine the degree of polynomials
- Add and subtract polynomials
- Evaluate a polynomial function for a given value
- Add and subtract polynomial functions
Before you get started, take this readiness quiz.
Determine the Degree of Polynomials
We have learned that a term is a constant or the product of a constant and one or more variables. A monomial is an algebraic expression with one term. When it is of the form where a is a constant and m is a whole number, it is called a monomial in one variable. Some examples of monomial in one variable are. Monomials can also have more than one variable such as and
A monomial is an algebraic expression with one term.
A monomial in one variable is a term of the form where a is a constant and m is a 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.
polynomial—A monomial, or two or more algebraic terms 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 | ||||
| Monomial | 14 | |||
| Binomial | ||||
| Trinomial |
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.
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.
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.
Let’s start by looking at a monomial. The monomial has two variables a and b. To find the degree we need to find the sum of the exponents. The variable a doesn’t have an exponent written, but remember that means the exponent is 1. The exponent of b is 2. The sum of the exponents, is 3 so the degree is 3.
Here are some additional examples.
Working with polynomials is easier when you list the terms in descending order of degrees. When a polynomial is written this way, it is said to be