Like Terms in Algebra – Definition and Examples

Like terms in algebraic expressions are characterized by having the same variables with the same exponents. Also, constant terms are considered like terms. These terms can be simplified by adding their coefficients.

Here, we will learn about like terms using several examples. Then, we will solve some practice problems.

ALGEBRA
Examples of like terms

Relevant for

Learning about like terms in algebra.

See definition

ALGEBRA
Examples of like terms

Relevant for

Learning about like terms in algebra.

See definition

Definition of like terms in algebraic expressions

Like terms are algebraic terms that contain the same variables raised to the same powers. These terms can be added when simplifying algebraic expressions.

Two or more like terms may or may not have the same coefficients. For two algebraic terms to be like, they must have the same variables with the same exponents.

Let’s recall that variables are the “letters” of an algebraic term and the coefficients are the numbers that multiply the variables.

parts of an algebraic expression

Constant terms, that is, terms that have no variables, are also considered like terms.

The following are some examples of like and unlike terms:

Like terms

  • $latex 2x^2~$ and $latex ~4x^2$
  • $latex 4x^2y~$ and $latex ~2x^2y$
  • $latex x^3z~$, $latex ~4x^3z~$ and $latex ~3x^3z~$

Unlike terms

  • $latex 2x^2~$ and $latex ~2x$
  • $latex 3x^2y~$ and $latex ~2x^2$
  • $latex x^2z$ ~and $latex ~4x^3z$

Steps to combine like terms in algebraic expressions

Combining like terms is a technique for simplifying algebraic expressions since it allows us to combine terms with the same variables and the same exponents.

To combine like terms, we can follow these steps:

1. Identify terms that have the same variables with the same exponents.

For example, $latex 4x^2y~$ and $latex ~2x^2y$ have the same variables with the same exponents, as do $latex 2xy^2z^3~$ and $latex ~3xy^2z^3$.

2. Add the coefficients of the like terms identified in step 1.

For example, $latex 4x^2y~$ and $latex ~2x^2y$ add up to get $latex 6x^2y$. Similarly, $latex 2xy^2z^3~$ and $latex ~3xy^2z^3$ add up to get $latex 5xy^2z^3~$.

3. Add the constant terms.

All constant terms are like terms. Therefore, we just have to add them together to simplify.


Solved examples of like terms

EXAMPLE 1

Which of the following are like terms of $latex 4x^2$?

$latex 3x^2~~$ $latex ~~2x~~$ $latex ~~4x^3~~$ $latex ~~5x^2$

The given term is $latex 4x^2$. This term has the variable x raised to the exponent 2. Therefore, we can analyze the given terms one by one:

  • $latex 3x^2$ has the same variable x, raised to the exponent 2. Yes, it is a like term.
  • $latex 2x$ has the same variable x, but it is raised to 1. No, it is not a like term.
  • $latex 4x^3$ has the same variable x, but it is raised to the power of 3. No, it is not a like term.
  • $latex 5x^2$ has the same variable x, raised to the exponent 2. Yes, it is a like term.

The like terms of $latex 4x^2$ are $latex 3x^2~$ and $latex ~5x^2$.

EXAMPLE 2

Identify the like terms of $latex 2xy^2z$ from the following terms.

$latex xy^2~~$ $latex ~~2xy^2z~~$ $latex ~~4xy^3z~~$ $latex ~~xy^2z$

The given term is $latex 2xy^2z$. Analyzing the given terms one by one, we have:

  • $latex xy^2$ doesn’t have the same variables (it doesn’t have the z). It is not a like term.
  • $latex 2xy^2z$ has the same variables with the same powers. Yes, it is a like term.
  • $latex 4xy^3z$ has the same variables, but the y is raised to 3. It is not a like term.
  • $latex xy^2z$ has the same variables with the same powers. Yes, it is a like term.

Like terms for $latex 2xy^2z$ are $latex 2xy^2z~$ and $latex ~xy^2z$.

EXAMPLE 3

Simplify the like terms in the following expression:

$$2x^2+4x+5+x^2+2+x$$

To simplify this expression, we need to start by identifying like terms. Thus, we see that the following are like terms:

  • $latex 2x^2~$ and $latex ~x^2$
  • $latex 4x~$ and $latex ~x$
  • $latex 5~$ and $latex ~2$

Now, we add the coefficients of the like terms and the constant terms to obtain the following expression:

$$(2x^2+x^2)+(4x+x)+(5+2)$$

$$3x^2+5x+7$$

EXAMPLE 4

Combine the like terms of the following expression:

$$xy^2+2x^2y+5xy^2+4+3x^2y+2+2xy^2+4$$

Let’s start by identifying the like terms of the given expression:

  • $latex xy^2~$, $latex ~5xy^2~$ and $latex ~2xy^2$
  • $latex 2x^2y~$ and $latex ~3x^2y~$
  • $latex 4~$, $latex ~2~$ and $latex ~4$

Adding the like terms, we have:

$$(xy^2+5xy^2+2xy^2)+(2x^2y+3x^2y)+(4+2+4)$$

$$8xy^2+5x^2y+10$$

EXAMPLE 5

Find the most simplified form of the following expression:

$$2xyz+x^2y+3xy^2+4xyz+3x^2y+3xyz+5xy^2+4x^2y$$

We start by identifying the like terms of the expression:

  • $latex 2xyz~$, $latex ~4xyz~$ and $latex ~3xyz$
  • $latex x^2y~$, $latex ~3x^2y~$ and $latex ~4x^2y$
  • $latex 3xy^2~$ and $latex ~5xy^2$

Adding the coefficients of like terms, we have:

$$(2xyz+4xyz+3xyz)+(x^2y+3x^2y+4x^2y)+(3xy^2+5xy^2)$$

$$9xyz+8x^2y+8xy^2$$

You can explore more solved examples on this topic in our article: Like Terms in Algebra – Examples and Practice problems.


Like terms – Practice problems

Like terms quiz
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What is the coefficient of $latex xy^2$ when we combine like terms? $$2xy^2+2xy+3x^2y-4xy^2-2x^2y-xy^2+6xy$$

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