Like Terms

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Like Terms Definition

Like terms can be defined as terms that contain the same variables raised to the same power. Only the numerical coefficients are different. In an expression, only we can combine like terms. We combine like terms to short and simplify the algebraic expressions, so we can work with them more easily.


What are Like Terms?

Like terms contain the same variable which is raised to the same power.

For example, 5x + 10x is an algebraic expression with like terms. When we have to simplify this algebraic expression, we can add the like terms. Thus, the simplification of the given expression is 15x. In a similar way, we can perform all the arithmetic operations on the like terms.


Like Terms Examples

Consider terms as 5x, 6x, 2x, and -3x

Here all four terms are like terms because x is the common variable.

Consider another example 2xy², \[\frac{1}{2}\]xy², 7xy² and \[\frac{xy^{2}}{2}\] 

Here also all four terms are like terms because xy ²  is the common variable.


Combining Like Terms

To combine like terms, we have to add the coefficients and keep the variables the same.

We add like terms to make one term. 

Example: 7x+3x both are like terms we can combine together and write 10x.

Here we have 10 terms. Let’s combine all the like terms together.

2xy², 3xy³, 1/2xyz, 9xy², -3/4xy², -6xy³, -8xyz³, 10xy, -4xy, 5xy 


2xy², 9xy², \[\frac{-3}{4}\]xy²

All these terms have xy²

\[\frac{1}{2}\]xyz

This is only xyz term

3xy³, -6xy³

All these terms have xy³

-8xyz³

This is only xyz³ term

10xy, -4xy, 5xy 

All these terms have xy


Thus, from the above table, we can say that algebraic terms with the same variables are added to each other. The addition of certain terms was possible only because the variables in both these cases are the same even if the numerical coefficients are different which we can add as normal numbers and the variable factor remains as it is. Now, the terms which have the same variables are called like terms.


Simplify by Combining Like Terms

Let’s understand this with an example of 2xy + 5x² + 6xy + 9x² + 20y² 

Notice the terms 2xy and 6xy, as well as 5x² and 9x² have common factors. Only the numerical coefficients are different. Apart from that, all the variable factors are the same, so we can add these terms.

On adding like terms,

2xy + 6xy = 8xy, 

5x² + 9x² = 14x²

These terms have variable factors in common and an arithmetic operation can be performed on them, as they are called like terms. After adding the like terms, above the expression can be written again.

2xy + 5x² + 6xy + 9x² + 20y² = 8xy + 14x² + 20y²   

If we observe the expression now, none of the terms has any common factors and we cannot perform any further arithmetic operation on them. These terms are called, unlike terms.


Solved Examples

1. Write the Final Expression after Combining Like Terms. The Expression is 4x³ + 3x² + 7x² - 4x² - 10 + 5.

Ans: Here, we need to combine like terms. Write like terms in a bracket and simplify.

⇒(4x³) + (3x² + 7x² - 4x²) + (-10 + 5)

On simplification, we get 

⇒ 4x³ + 6x² - 5


2. Find Like Terms in the Expression n(n + 1) + 6 (n – 1).

Ans: First write the expression in the simplest form then we can identify the like items. After solving bracket terms we get, n² + n + 6n - 6. In this n and 6n are like terms so we can combine them together.

Final expression n² + 7n - 6. Hence the like terms are n and 6n.


Did you know?

  • The number in front of a term is known as the coefficient. 

  •  We generally write “x” instead of “1x” because it is simpler to write “x” and also "1x" looks strange.

FAQ (Frequently Asked Questions)

1. Why Do We Combine Like Terms?

Ans: We combine like terms in algebraic expressions because after the combination of like terms, expressions comes in the simplest form and no further calculation is required. After combining like terms, expression can be solved easily.

2. Are pq and qp Like Terms?

Ans: Like terms follow the associative property of multiplication. So, pq and qp are like terms. Another like term can be xy2, 2y2x.