6.1: Add and Subtract Polynomials
- Last updated
- May 3, 2019
- Save as PDF
- Page ID
- 18962
( \newcommand{\kernel}{\mathrm{null}\,}\)
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
Quiz
Before you get started, take this readiness quiz.
- Simplify:
.8 𝑥 + 3 𝑥
If you missed this problem, review Example 1.3.37. - Subtract:
.( 5 𝑛 + 8 ) − ( 2 𝑛 − 1 )
If you missed this problem, review Example 1.10.52. - Write in expanded form:
.𝑎 5
If you missed this problem, review Example 1.3.7.
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
Definition: Monomials
A monomial is a term of the form
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.
Definitions: 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 6 . 1 . 1
Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial.
4 𝑦 2 − 8 𝑦 − 6 − 5 𝑎 4 𝑏 2 2 𝑥 5 − 5 𝑥 3 − 3 𝑥 + 4 1 3 − 5 𝑚 3 - q
Solution
Try It 6 . 1 . 2
Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial:
- 5b
8 𝑦 3 − 7 𝑦 2 − 𝑦 − 3 − 3 𝑥 2 − 5 𝑥 + 9 8 1 − 4 𝑎 2 − 5 𝑥 6
- Answer
-
- monomial
- polynomial
- trinomial
- binomial
- monomial
Try It 6 . 1 . 3
Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial:
2 7 𝑧 3 − 8 1 2 𝑚 3 − 5 𝑚 2 − 2 𝑚 5 6 8 𝑥 4 − 7 𝑥 2 − 6 𝑥 − 5 − 𝑛 4
- Answer
-
- binomial
- trinomial
- monomial
- polynomial
- 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, i.e., it has no variable.
Definition: 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 of a polynomial are written in descending order of degrees. Get in the habit of writing the term with the highest degree first.
Example 6 . 1 . 4
Find the degree of the following polynomials.
- 10y
4 𝑥 3 − 7 𝑥 + 5 - −15
− 8 𝑏 2 + 9 𝑏 − 2 8 𝑥 𝑦 2 + 2 𝑦
Solution
1 0 𝑦 T h e e x p o n e n t o f y i s o n e . 𝑦 = 𝑦 1 T h e d e g r e e i s 1 . 4 𝑥 3 − 7 𝑥 + 5 T h e h i g h e s t d e g r e e o f a l l t h e t e r m s i s 3 . T h e d e g r e e i s 3 . − 1 5 T h e d e g r e e o f a c o n s t a n t i s 0 . T h e d e g r e e i s 0 . − 8 𝑏 2 + 9 𝑏 − 2 T h e h i g h e s t d e g r e e o f a l l t h e t e r m s i s 2 . T h e d e g r e e i s 2 . 8 𝑥 𝑦 2 + 2 𝑦 T h e h i g h e s t d e g r e e o f a l l t h e t e r m s i s 3 . T h e d e g r e e i s 3 .
Try It 6 . 1 . 5
Find the degree of the following polynomials:
- −15b
1 0 𝑧 4 + 4 𝑧 2 − 5 1 2 𝑐 5 𝑑 4 + 9 𝑐 3 𝑑 9 − 7 3 𝑥 2 𝑦 − 4 𝑥 - −9
- Answer
-
- 1
- 4
- 12
- 3
- 0
Try It 6 . 1 . 6
Find the degree of the following polynomials:
- 52
𝑎 4 𝑏 − 1 7 𝑎 4 5 𝑥 + 6 𝑦 + 2 𝑧 3 𝑥 2 − 5 𝑥 + 7 − 𝑎 3
- Answer
-
- 0
- 5
- 1
- 2
- 3
Add and Subtract Monomials
You have learned how to simplify expressions by combining like terms. Remember, like terms must have the same variables with the same exponent. Since monomials are terms, adding and subtracting monomials is the same as combining like terms. If the monomials are like terms, we just combine them by adding or subtracting the coefficient.
Example 6 . 1 . 7
Add:
Solution
Try It 6 . 1 . 8
Add:
- Answer
-
21
𝑞 2
Try It 6 . 1 . 9
Add:
- Answer
-
− 7 𝑐 2
Example 6 . 1 . 1 0
Subtract: 16p−(−7p)
Solution
Try It 6 . 1 . 1 1
Subtract: 8m−(−5m).
- Answer
-
13m
Try It 6 . 1 . 1 2
Subtract:
- Answer
-
− 1 0 𝑧 3
Remember that like terms must have the same variables with the same exponents.
Example 6 . 1 . 1 3
Simplify:
Solution
Try It 6 . 1 . 1 4
Add:
- Answer
-
5 𝑦 2 + 3 𝑧 2
Try It 6 . 1 . 1 5
Add:
- Answer
-
− 4 𝑚 2 + 𝑛 2
Example 6 . 1 . 1 6
Simplify:
Solution
Try It 6 . 1 . 1 7
Simplify:
- Answer
-
There are no like terms to combine.
Try It 6 . 1 . 1 8
Simplify:
- Answer
-
There are no like terms to combine.
Add and Subtract Polynomials
We can think of adding and subtracting polynomials as just adding and subtracting a series of monomials. Look for the like terms—those with the same variables and the same exponent. The Commutative Property allows us to rearrange the terms to put like terms together.
Example 6 . 1 . 1 9
Find the sum:
Solution
Identify like terms. | ![]() |
Rearrange to get the like terms together. | ![]() |
Combine like terms. | ![]() |
Try It 6 . 1 . 2 0
Find the sum:
- Answer
-
8 𝑥 2 − 1 1 𝑥 + 1
Try It 6 . 1 . 2 1
Find the sum:
- Answer
-
1 7 𝑦 2 + 1 4 𝑦 + 1
Example 6 . 1 . 2 2
Find the difference:
Solution
![]() |
|
Distribute and identify like terms. | ![]() |
Rearrange the terms. | ![]() |
Combine like terms. | ![]() |
Try It 6 . 1 . 2 3
Find the difference:
- Answer
-
1 5 𝑥 2 + 3 𝑥 − 5
Try It 6 . 1 . 2 4
Find the difference:
- Answer
-
6 𝑏 2 + 3
Example 6 . 1 . 2 5
Subtract:
Solution
![]() |
|
![]() |
|
Distribute and identify like terms. | ![]() |
Rearrange the terms. | ![]() |
Combine like terms. | ![]() |
Try It 6 . 1 . 2 6
Subtract:
- Answer
-
2 𝑧 2 + 1 2 𝑧 − 2
Try It 6 . 1 . 2 7
Subtract:
- Answer
-
5 𝑥 2 + 1 4 𝑥 + 7
Example 6 . 1 . 2 8
Find the sum:
Solution
Try It 6 . 1 . 2 9
Find the sum:
- Answer
-
5 𝑥 2 − 5 𝑥 𝑦 + 5 𝑦 2
Try It 6 . 1 . 3 0
Find the sum:
- Answer
-
7 𝑥 2 − 6 𝑥 𝑦 − 2 𝑦 2
Example 6 . 1 . 3 1
Find the difference:
Solution
Try It 6 . 1 . 3 2
Find the difference:
- Answer
-
− 5 𝑎 𝑏 − 5 𝑏 2
Try It 6 . 1 . 3 3
Find the difference:
- Answer
-
4 𝑛 2 + 7 𝑚 𝑛
Example 6 . 1 . 3 4
Simplify:
Solution
Try It 6 . 1 . 3 5
Simplify:
- Answer
-
𝑥 3 − 𝑦 3
Try It 6 . 1 . 3 6
Simplify:
- Answer
-
𝑝 3 − 2 𝑝 2 𝑞 + 𝑞 3
Evaluate a Polynomial for a Given Value
We have already learned how to evaluate expressions. Since polynomials are expressions, we’ll follow the same procedures to evaluate a polynomial. We will substitute the given value for the variable and then simplify using the order of operations.
Example 6 . 1 . 3 7
Evaluate
- x=4
- x=−2
- x=0
Solution
1. x=4 | |
![]() |
|
![]() |
![]() |
Simplify the exponents. | ![]() |
Multiply. | ![]() |
Simplify. | ![]() |
2. x=−2 | |
![]() |
|
![]() |
![]() |
Simplify the exponents. | ![]() |
Multiply. | ![]() |
Simplify. | ![]() |
3. x=0 | |
![]() |
|
![]() |
![]() |
Simplify the exponents. | ![]() |
Multiply. | ![]() |
Simplify. | ![]() |
Try It 6 . 1 . 3 8
Evaluate:
- x=3
- x=−5
- x=0
- Answer
-
- 18
- 50
- −15
Try It 6 . 1 . 3 9
Evaluate:
- z=−2
- z=0
- z=2
- Answer
-
- 18
- −4
- 14
Example 6 . 1 . 4 0
The polynomial
Solution
Try It 6 . 1 . 4 1
The polynomial
- Answer
-
250
Try It 6 . 1 . 4 2
The polynomial
- Answer
-
106
Example 6 . 1 . 4 3
The polynomial
Solution
![]() |
|
![]() |
![]() |
Simplify. | ![]() |
Simplify. | ![]() |
Simplify. | ![]() |
The cost of producing the box is $456. |
Try It 6 . 1 . 4 3
The polynomial
- Answer
-
$576
Try It 6 . 1 . 4 4
The polynomial
- Answer
-
$750
Key Concepts
- Monomials
- A monomial is a term of the form
, where aa is a constant and mm is a whole number𝑎 𝑥 𝑚
- A monomial is a term of the form
- 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.
- 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.
Glossary
- binomial
- A binomial is a polynomial with exactly two terms.
- degree of a constant
- The degree of any constant is 0.
- degree of a polynomial
- The degree of a polynomial is the highest degree of all its terms.
- degree of a term
- The degree of a term is the exponent of its variable.
- monomial
- A monomial is a term of the form
, where a is a constant and m is a whole number; a monomial has exactly one term.𝑎 𝑥 𝑚
- polynomial
- A polynomial is a monomial, or two or more monomials combined by addition or subtraction.
- standard form
- A polynomial is in standard form when the terms of a polynomial are written in descending order of degrees.
- trinomial
- A trinomial is a polynomial with exactly three terms.