We begin our discussion with the definition of a term.
Definition: Term
A term is a single number or variable, or it can be the product of a number (called its coefficient) and one or more variables (called its variable part). The terms in an algebraic expression are separated by addition symbols.
Example 1
Identify the terms in the algebraic expression
\[ 3x^2 + 5xy + 9y^2 + 12\nonumber \]
For each term, identify its coefficient and variable part.
Solution
In tabular form, we list each term of the expression 3x^2 + 5xy + 9y^2 + 12, its coefficient, and its variable part.
Term |
Coefficient |
Variable Part |
3x2 |
3 |
x2 |
5xy |
5 |
xy |
9y2 |
9 |
y2 |
12 |
12 |
None |
Exercise
How many terms are in the algebraic expression 3x2 + 2xy − 3y2?
- Answer
-
3
Example 2
Identify the terms in the algebraic expression
\[ a^3 − 3a^2b + 3ab^2 − b^3\nonumber \]
For each term, identify its coefficient and variable part.
Solution
The first step is to write each difference as a sum, because the terms of an expression are defined above to be those items separated by addition symbols.
\[ a^3 + (−3a^2b)+3ab^2 + (−b^3)\nonumber \]
In tabular form, we list each term of the expression \(a^3+(−3a^2b)+3ab^2+(−b^3)\), its coefficient, and its variable part.
Term |
Coefficient |
Variable Part |
a3 |
1 |
a3 |
-3a2b |
−3 |
a2b |
3ab2 |
3 |
ab2 |
−b3 |
−1 |
b3 |
Exercise
How many terms are in the algebraic expression \(11 − a^2 − 2ab + 3b^2a\)?
- Answer
-
4
Like Terms
We define what is meant by “like terms” and “unlike terms.”
Definition
Like and Unlike Terms. The variable parts of two terms determine whether the terms are like terms or unlike terms.
Like Terms. Two terms are called like terms if they have identical variable parts, which means that the terms must contain the same variables raised to the same exponential powers.
Unlike Terms. Two terms are called unlike terms if their variable parts are different.
Example 3
Classify each of the following pairs as either like terms or unlike terms: (a) 3x and −7x, (b) 2y and 3y2, (c) −3t and 5u, and (d) −4a3 and 3a3.
Solution
Like terms must have identical variable parts.
- 3x and −7x have identical variable parts. They are “like terms.”
- 2y and 3y2 do not have identical variable parts (the exponents differ). They are “unlike terms.”
- −3t and 5u do not have identical variable parts (different variables). They are “unlike terms.”
- −4a3 and 3a3 have identical variable parts. They are “like terms.”
Exercise
Are −3xy and 11xy like or unlike terms?
- Answer
-
Like terms
Combining Like Terms
When using the distributive property, it makes no difference whether the multiplication is on the left or the right, one still distributes the multiplication times each term in the parentheses.
Distributive Property
If a, b, and c are integers, then
a(b + c) = ab + ac and (b + c)a = ba + ca.
In either case, you distribute a times each term of the sum.
“Like terms” can be combined and simplified. The tool used for combining like terms is the distributive property. For example, consider the expression 3y + 7y, composed of two “like terms” with a common variable part. We can use the distributive property and write
\[3y+7y=(3+7)y\nonumber \]
Note that we are using the distributive property in reverse, “factoring out” the common variable part of each term. Checking our work, note that if we redistribute the variable part y times each term in the parentheses, we are returned to the original expression 3y + 7y.
Example 4
Use the distributive property to combine like terms (if possible) in each of the following expressions: (a) −5x2 − 9x2, (b) −5ab + 7ab, (c) 4y3 − 7y2, and (d) 3xy2 − 7xy2.
Solution
If the terms are “like terms,” you can use the distributive property to “factor out” the common variable part.
a) Factor out the common variable part x2.
\[ \begin{aligned} -5x^2 -9x^2 =(-5-9)x^2 ~ & \textcolor{red}{ \text{ Use the distributive property.}} \\ = -14x^2 ~ & \textcolor{red}{ \text{ Simplify: } -5-9=-5+(-9) = -14.} \end{aligned}\nonumber \]
b) Factor out the common variable part ab.
\[ \begin{aligned} -5ab +7ab = (-5+7)ab ~ & \textcolor{red}{ \text{ Use the distributive property.}} \\ =2ab ~ & \textcolor{red}{ \text{ Simplify: } -5+7 = 2.} \end{aligned}\nonumber \]
c) The terms in the expression 4y3 − 7y2 have different variable parts (the exponents are different). These are “unlike terms” and cannot be combined.
d) Factor out the common variable part xy2.
\[ \begin{aligned} 3xy^2 - 7xy^2 =(3-7)xy^2 ~ & \textcolor{red}{ \text{ Use the distributive property.}} \\ =-4xy^2 ~ & \textcolor{red}{ \text{ Simplify: } 3-7=3+(-7)=-4.} \end{aligned}\nonumber \]
Exercise
Simplify: −8z − 11z
- Answer
-
−19z
Speeding Things Up a Bit
Once you’ve written out all the steps for combining like terms, like those shown in Example 4, you can speed things up a bit by following this rule:
Combining Like Terms
To combine like terms, simply add their coefficients and keep the common variable part.
Thus for example, when presented with the sum of two like terms, such as in 5x+ 8x, simply add the coefficients and repeat the common variable part; that is, 5x + 8x = 13x.
Example 5
Combine like terms:
- −9y − 8y,
- −3y5 + 4y5 and
- −3u2 + 2u2.
Solution
a) Add the coefficients and repeat the common variable part. Therefore,
\[−9y − 8y = −17y.\nonumber \]
b) Add the coefficients and repeat the common variable part. Therefore,
\[−3y^5 + 4y^5 = 1y^5.\nonumber \]
However, note that 1y5 = y5. Following the rule that the final answer should use as few symbols as possible, a better answer is −3y5 + 4y5 = y5.
c) Add the coefficients and repeat the common variable part. Therefore,
\[−3u^2 + 2u^2 = (−1)u^2.\nonumber \]
However, note that (−1)u2 = −u2. Following the rule that the final answer should use as few symbols as possible, a better answer is −3u2+ 2u2 = −u2.
Simplify
A frequently occurring instruction asks the reader to simplify an expression.
Simplify
The instruction simplify is a generic term that means “try to write the expression in its most compact form, using the fewest symbols possible.”
One way you can accomplish this goal is by combining like terms when they are present.
Example 6
Simplify: 2x + 3y − 5x + 8y.
Solution
Use the commutative property to reorder terms and the associative and distributive properties to regroup and combine like terms.
\[ \begin{aligned} 2x + 3y - 5x + 8y = (2x - 5x) + (3y + 8y) ~ & \textcolor{red}{ \text{ Reorder and regroup.}} \\ = -3x + 11y ~ & \textcolor{red}{ \text{ Combine like terms:}} \\ ~ & \textcolor{red}{ 2x - 5x = -3x \text{ and } 3y + 8y = 11y.} \end{aligned}\nonumber \]
Alternate solution
Of course, you do not need to show the regrouping step. If you are more comfortable combining like terms in your head, you are free to present your work as follows:
\[2x + 3y − 5x + 8y = −3x + 11y.\nonumber \]
Exercise
Simplify: −3a + 4b − 7a − 9b
- Answer
-
−10a − 5b
Example 7
Simplify: −2x − 3 − (3x + 4).
Solution
First, distribute the negative sign.
\[ \begin{aligned} -2x-3-(3x+4)= -2x-3-3x-4 ~ & \textcolor{red}{-(3x+4)=-3x-4.} \end{aligned}\nonumber \]
Next, use the commutative property to reorder, then the associative property to regroup. Then combine like terms.
\[ \begin{aligned} =(-2x-3x)+(-3-4) ~ & \textcolor{red}{ \text{ Reorder and regroup.}} \\ =-5x+(-7) ~ & \textcolor{red}{ \text{ Combine like terms:}} \\ ~ & \textcolor{red}{ -2x-3x=-5x.} \\ =-5x-7 ~ & \textcolor{red}{ \text{ Simplify:}} \\ ~ & \textcolor{red}{-5x+(-7)=-5x-7.} \end{aligned}\nonumber \]
Alternate solution
You may skip the second step if you wish, simply combining like terms mentally. That is, it is entirely possible to order your work as follows:
\[ \begin{aligned} -2x-3-(3x+4) = -2x-3-3x-4 ~ & \textcolor{red}{ \text{ Distribute negative sign.}} \\ =-5x-7 ~ & \textcolor{red}{ \text{ Combine like terms.}} \end{aligned}\nonumber \]
Exercise
Simplify: −9a − 4 − (4a − 8)
- Answer
-
−13a + 4
Example 8
Simplify: 2(5 − 3x) − 4(x + 3).
Solution
Use the distributive property to expand, then use the commutative and associative properties to group the like terms and combine them.
\[ \begin{aligned} 2(5-3x)-4(x+3) = 10-6x-4x-12 ~ & \textcolor{red}{ \text{ Use the distributive property.}} \\ =(-6x-4x)+(10-12) ~ & \textcolor{red}{ \text{ Group like terms.}} \\ =-10x-2 ~ & \textcolor{red}{ \text{ Combine like terms: }} \\ ~ & \textcolor{red}{-6x-4x=-10x \text{ and} \\ ~ & \textcolor{red}{10-12=-2.} \end{aligned}\nonumber \]
Alternate solution
You may skip the second step if you wish, simply combining like terms mentally. That is, it is entirely possible to order your work as follows:
\[ \begin{aligned} 2(5-3x)-4(x+3) = 10-6x-4x-12 ~ & \textcolor{red}{ \text{ Distribute.}} \\ =-10x-2 ~& \textcolor{red}{ \text{ Combine like terms.}} \end{aligned}\nonumber \]
Exercise
Simplify: −2(3a − 4) − 2(5 − a)
- Answer
-
−4a − 2
Example 9
Simplify: −8(3x2y − 9xy) − 8(−7x2 − 8xy)
Solution
We will proceed a bit quicker with this solution, using the distributive property to expand, then combining like terms mentally.
\[ \begin{aligned} -8(3x^2y-9xy) -8(-7x^2y-8xy)=-24x^2y+72xy+56x^2y+64xy \\ = 32x^2y+136xy \end{aligned}\nonumber \]
Exercise
Simplify: (a2 − 2ab) − 2(3ab + a2)
- Answer
-
−a2 − 8ab
Exercises
Combine like terms and simplify.
1. 17xy2 + 18xy2 + 20xy2
7. 12r − 12r
8. 16s − 5s
11. −5q + 7q
13. r − 13r − 7r
14. 19m + m + 15m
16. 13x2y + 2x2y
17. −8 + 17n + 10 + 8n
19. −2x3 − 19x2y − 15x2y + 11x3
23. −13 + 16m + m + 16
29. 9n + 10 + 7 + 15n
31. 3y +1+6y + 3
35. 3(−4x2 + 10y2) + 10(4y2 − x2)
39. −10q − 10 − (−3q + 5)
41. 7(8y + 7) − 6(8 − 7y)
45. −2(6 + 4n) + 4(−n − 7)
47. 8 − (4 + 8y)
53. 7(1 + 7r) + 2(4 − 5r)
55. −2(−5 − 8x2) − 6(6)
Answers
1. 55xy2
7. 0
8. 11s
11. 2q
13. −19r
14. 35m
16. 15x2y
17. 2 + 25n
19. 9x3 − 34x2y
23. 3 + 17m
29. 24n + 17
31. 9y + 4
35. −22x2 + 70y2
39. −7q − 15
41. 98y + 1
45. −40 − 12n
47. 4 − 8y
53. 15 + 39r
55. −26 + 16x2