Learning Outcomes
- Identify the variables and constants in a term
- Identify the coefficient of a variable term
- Identify and combine like terms in an expression
Identify Terms, Coefficients, and Like Terms
Algebraic expressions are made up of terms. A term is a constant or the product of a constant and one or more variables. Some examples of terms are 7,y,5x2,9a,and 13xy7,y,5x2,9a,and 13xy.
The constant that multiplies the variable(s) in a term is called the coefficient. We can think of the coefficient as the number in front of the variable. The coefficient of the term 3x3x is 33. When we write xx, the coefficient is 11, since x=1⋅xx=1⋅x. The table below gives the coefficients for each of the terms in the left column.
Term | Coefficient |
---|---|
77 | 77 |
9a9a | 99 |
yy | 11 |
5x25x2 | 55 |
An algebraic expression may consist of one or more terms added or subtracted. In this chapter, we will only work with terms that are added together. The table below gives some examples of algebraic expressions with various numbers of terms. Notice that we include the operation before a term with it.
Expression | Terms |
---|---|
77 | 77 |
yy | yy |
x+7x+7 | x,7x,7 |
2x+7y+4 | 2x,7y,4 |
3x2+4x2+5y+3 | 3x2,4x2,5y,3 |
example
Identify each term in the expression 9b+15x2+a+6. Then identify the coefficient of each term.
Solution:
The expression has four terms. They are 9b,15x2,a, and 6.
- The coefficient of 9b is 9.
- The coefficient of 15x2 is 15.
- Remember that if no number is written before a variable, the coefficient is 1. So the coefficient of a is 1.
- The coefficient of a constant is the constant, so the coefficient of 6 is 6.
try it
Some terms share common traits. Look at the following terms. Which ones seem to have traits in common?
5x,7,n2,4,3x,9n2
Which of these terms are like terms?
- The terms 7 and 4 are both constant terms.
- The terms 5x and 3x are both terms with x.
- The terms n2 and 9n2 both have n2.
Terms are called like terms if they have the same variables and exponents. All constant terms are also like terms. So among the terms 5x,7,n2,4,3x,9n2,
- 7 and 4 are like terms.
- 5x and 3x are like terms.
- n2 and 9n2 are like terms.
Like Terms
Terms that are either constants or have the same variables with the same exponents are like terms.
example
Identify the like terms:
- y3,7x2,14,23,4y3,9x,5x2
- 4x2+2x+5x2+6x+40x+8xy
try it
Simplify Expressions by Combining Like Terms
We can simplify an expression by combining the like terms. What do you think 3x+6x would simplify to? If you thought 9x, you would be right!
We can see why this works by writing both terms as addition problems.
Add the coefficients and keep the same variable. It doesn’t matter what x is. If you have 3 of something and add 6 more of the same thing, the result is 9 of them. For example, 3 oranges plus 6 oranges is 9 oranges. We will discuss the mathematical properties behind this later.
The expression 3x+6x has only two terms. When an expression contains more terms, it may be helpful to rearrange the terms so that like terms are together. The Commutative Property of Addition says that we can change the order of addends without changing the sum. So we could rearrange the following expression before combining like terms.
Now it is easier to see the like terms to be combined.
Combine like terms
- Identify like terms.
- Rearrange the expression so like terms are together.
- Add the coefficients of the like terms.
example
Simplify the expression: 3x+7+4x+5.
try it
example
Simplify the expression: 8x+7x2+x2+4x.
try it
In the following video, we present more examples of how to combine like terms given an algebraic expression.
Candela Citations
- Simplify Expressions by Combining Like Terms (No Negatives). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. Located at: https://youtu.be/KMUCQ_Pwt7o. License: CC BY: Attribution
- Ex 1: Combining Like Terms. Authored by: James Sousa (Mathispower4u.com). Located at: https://youtu.be/JIleqbO8Tf0. License: CC BY: Attribution
- Ex 2: Combining Like Terms. Authored by: James Sousa (Mathispower4u.com). Located at: https://youtu.be/b9-7eu29pNM. License: CC BY: Attribution
- Question ID: 144899, 144900, 144905,146540. Authored by: Alyson Day. License: CC BY: Attribution. License Terms: IMathAS Community License CC-BY + GPL
- Prealgebra. Provided by: OpenStax. License: CC BY: Attribution. License Terms: Download for free at http://cnx.org/contents/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757