Like Terms

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 \[2xy^{2}, \frac{1}{2}xy^{2}, 7xy^{2}, and \frac{xy^{2}}{2} \] 

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 

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.

• Algebraic expressions in their earliest mentions did not make use of symbols (variables) as we are familiar with today. Instead, an algebraic expression was constructed in three stages:

• The first stage was the "Rhetorical algebra". During this stage, equations were written in full sentences. For example, the rhetorical form of a+2=3 is "The thing plus two equals three".  The foundations of rhetorical algebra were first laid by the ancient Babylonians and remained a dominant form of algebraic expression up to the 16th century.

• In the second stage, the Syncopated algebra, some symbolism developed in the expression of algebraic terms. However, these did not contain all of the characteristics of symbolic algebra. For example, there may be a limit on the number of times an operation can be used on one side of an equation, which is not the case with symbolic algebra. First mentions of a syncopated algebraic expression is found in Diophantus' Arithmetica (3rd century AD), and later in the works of Brahmagupta's Brahma Sphuta Siddhanta (7th century).

• The last stage is how we know most of the algebra, called the Symbolic algebra, in which full symbolism came into existence. Early steps toward this stage can be noticed in the contributions of several Islamic mathematicians such as Ibn al-Banna (13th-14th centuries) and al-Qalasadi (15th century), even though full use of symbolism in algebra was laid by François Viète (16th century). In the later period, René Descartes (17th century) introduced the modern notation and elaborated on the fact that the problems occurring in geometry can be expressed and solved in terms of algebra ( known to us as Cartesian geometry).

• Some authors also mention these stages into four conceptual ones. They are:

• The first stage was the "Geometric stage", where the concepts of algebra are largely represented through geometry. This was first known by the Babylonians and continued with the Greeks.

• The second conceptual stage was the Static equation-solving stage, where the primary aim was to find numbers satisfying certain relationships. 

• In the third stage, called the Dynamic function stage, a motion was considered the underlying idea. The idea of a function began emerging with Sharaf al-Dīn al-Tūsī, but algebra did not clearly move to the dynamic function stage until the work of Gottfried Leibniz.

• The last stage known as the Abstract stage was where the mathematical structure was fully realized through symbolism. It played a key role. Abstract algebra is mostly considered to be a product of the 19th and 20th centuries.