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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 3 x is 3 . When we write x , the coefficient is 1 , since x = 1 x . [link] gives the coefficients for each of the terms in the left column.

Term Coefficient
7 7
9 a 9
y 1
5 x 2 5

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. [link] gives some examples of algebraic expressions with various numbers of terms. Notice that we include the operation before a term with it.

Expression Terms
7 7
y y
x + 7 x , 7
2 x + 7 y + 4 2 x , 7 y , 4
3 x 2 + 4 x 2 + 5 y + 3 3 x 2 , 4 x 2 , 5 y , 3

Identify each term in the expression 9 b + 15 x 2 + a + 6 . Then identify the coefficient of each term.

Solution

The expression has four terms. They are 9 b , 15 x 2 , a , and 6 .

The coefficient of 9 b is 9 .

The coefficient of 15 x 2 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 .

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Identify all terms in the given expression, and their coefficients:

4 x + 3 b + 2

The terms are 4 x , 3 b , and 2. The coefficients are 4, 3, and 2.

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Identify all terms in the given expression, and their coefficients:

9 a + 13 a 2 + a 3

The terms are 9 a , 13 a 2 , and a 3 , The coefficients are 9, 13, and 1.

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Some terms share common traits. Look at the following terms. Which ones seem to have traits in common?

5 x , 7 , n 2 , 4 , 3 x , 9 n 2

Which of these terms are like terms?

  • The terms 7 and 4 are both constant terms.
  • The terms 5 x and 3 x are both terms with x .
  • The terms n 2 and 9 n 2 both have n 2 .

Terms are called like terms    if they have the same variables and exponents. All constant terms are also like terms. So among the terms 5 x , 7 , n 2 , 4 , 3 x , 9 n 2 ,

7 and 4 are like terms.
5 x and 3 x are like terms.
n 2 and 9 n 2 are like terms.

Like terms

Terms that are either constants or have the same variables with the same exponents are like terms.

Identify the like terms:

  1. y 3 , 7 x 2 , 14 , 23 , 4 y 3 , 9 x , 5 x 2
  2. 4 x 2 + 2 x + 5 x 2 + 6 x + 40 x + 8 x y

Solution

y 3 , 7 x 2 , 14 , 23 , 4 y 3 , 9 x , 5 x 2

Look at the variables and exponents. The expression contains y 3 , x 2 , x , and constants.

The terms y 3 and 4 y 3 are like terms because they both have y 3 .

The terms 7 x 2 and 5 x 2 are like terms because they both have x 2 .

The terms 14 and 23 are like terms because they are both constants.

The term 9 x does not have any like terms in this list since no other terms have the variable x raised to the power of 1 .

4 x 2 + 2 x + 5 x 2 + 6 x + 40 x + 8 x y

Look at the variables and exponents. The expression contains the terms 4 x 2 , 2 x , 5 x 2 , 6 x , 40 x , and 8 x y

The terms 4 x 2 and 5 x 2 are like terms because they both have x 2 .

The terms 2 x , 6 x , and 40 x are like terms because they all have x .

The term 8 x y has no like terms in the given expression because no other terms contain the two variables x y .

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Identify the like terms in the list or the expression:

9 , 2 x 3 , y 2 , 8 x 3 , 15 , 9 y , 11 y 2

9, 15; 2 x 3 and 8 x 3 , y 2 , and 11 y 2

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Identify the like terms in the list or the expression:

4 x 3 + 8 x 2 + 19 + 3 x 2 + 24 + 6 x 3

4 x 3 and 6 x 3 ; 8 x 2 and 3 x 2 ; 19 and 24

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Simplify expressions by combining like terms

We can simplify an expression by combining the like terms    . What do you think 3 x + 6 x would simplify to? If you thought 9 x , you would be right!

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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