Algebraic Terms

Here we will learn about algebraic terms, including how to distinguish them, state their variables and coefficients, and identify like terms.

There are also algebraic terms worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What are algebraic terms?

Algebraic terms are individual letters, groups of letters, or groups of letters and numbers separated by addition or subtraction in algebraic expressions. 

For example, this is an algebraic expression with three algebraic terms.

Term 1 \;\;\; Term 2 \;\;\; Term 3

Individual algebraic terms can look very different to one another, so here is a list of some of the types you may come across.

Note that individual numbers are known as constant terms, such as 8 or \frac{4}{9}.

Coefficients and variables

Every algebraic term has a coefficient (the number in the term) and a variable (the letter or letters in the term).

Here are some examples,

For the term 3x

The variable is x.

The coefficient of this ‘ x -term’ is 3.

For the term -5ab

The variables are a and b.

The coefficient of this ‘ ab -term’ is -5.

For the term 7h^2

The variable is h^2.

The coefficient of this ‘ h^2 -term’ is 7.

For the term \frac{y}{4}

The variable is y.

The coefficient of this ‘ y -term’ is \frac{1}{4}.

(Note that \frac{y}{4} can also be written as \frac{1}{4}y.)

For the term t

The variable is t.

The coefficient of this ‘ t -term’ is 1.

(Note that the coefficient of 1 is never written next to the variable in an algebraic term.) 

Like terms

Like terms are terms that have the same variable or variables. They can however have different coefficients. 

For example, 7h, \ -2h and h are all ‘like terms’ because they all have the same variable ‘ h ’.

For example, 3a^5, \ 12a^5 and \frac{a^5}{7} are all ‘like terms’ because they all have the same variable ‘ a^5 ‘.

Note that 3a and 7ab are not like terms. Although they both have variable a, to be like terms all variables must match.

Note that all constant terms are like terms. 

Once you can identify like terms the next step is to simplify algebraic expressions by collecting like terms.

Step-by-step guide: Collecting like terms

Algebraic expressions

Algebraic expressions are made up of one or more algebraic terms which are separated by addition or subtraction signs. Algebraic expressions may also include constant terms.

Monomial expressions have one term.

For example, 6n.

Binomial expressions are made up of two terms.

For example, 3x-12.

Trinomial expressions are made of three terms. 

For example, x^2 + 2x - 3.

Step-by-step guide: Algebraic expressions

Note that algebraic expressions do not have equal signs. If algebraic expressions or individual terms are written as equal to each other then the result would be an algebraic equation.

For example, 5x-8=12.

Step-by-step guide: Solving equations

See also: Rearranging equations

Keyword Summary

Here is a summary of the key words often used when working with algebraic terms.

Additional notes about algebraic terms

• In algebra, when numbers and letters are written next to each other it indicates that they are multiplied together. For example, the algebraic term 5a means 5 \times a.
• When letters, or numbers and letters, are written in fraction form, it means one is being divided by the other. For example, the algebraic term \frac{x}{4} means x \div 4.
• If an algebraic term is made up of a letter and a number, the number always goes at the start of the term. For example, for a \times 4 you should write 4a and not a4.
• If an algebraic term has more than one letter then the letters should be placed in alphabetical order. For example, 5ba is incorrect and should be written as 5ab.
• When a term is subtracted in an algebraic expression, that term must be considered ‘negative’ when identifying separate terms. In other words, the sign before the term belongs to that term. For example, 5a-7b. The first term is 5a and the second term is -7b, not just 7b.
• Multiplication is commutative; therefore reordering letters and numbers that are written next to each other does not change the overall value of the algebraic term.

Step-by-step guide: Simplifying expressions

What are algebraic terms?

How to identify and describe algebraic terms

In order to identify and describe algebraic terms in algebraic expressions:

1. Count the number of individual terms that are separated by addition/subtraction in the algebraic expression.
2. State the coefficient (the number) and variable/s (the letter/s) of each algebraic term identified in step 1. Also note any constant terms.
3. Check if any of the terms have identical variables and if so, state that they are like terms.

Explain how to identify and describe algebraic terms

Algebraic terms examples

Example 1: linear algebraic expression with integer coefficients

For this algebraic expression, state the number of terms, the coefficient and variable/s of each term and identify if there are any like terms.

1. Count the number of individual terms that are separated by addition/subtraction in the algebraic expression.

There are three terms in this expression.

2State the coefficient (the number) and variable/s (the letter/s) of each algebraic term identified in step 1. Also note any constant terms.

Term 5a. Variable is a. Coefficient is 5.

Term 7ab. Variables are a and b. Coefficient is 7.

Term -6. This is a constant term.

3Check if any of the terms have identical variables and if so, state that they are like terms.

None of the terms have identical variables so there are no like terms.

Example 2: linear algebraic expression with integer coefficients

For this algebraic expression, state the number of terms, the coefficient and variable/s of each term and identify if there are any like terms. 

Example 3: non-linear algebraic expression with integer coefficients

For this algebraic expression, state the number of terms, the coefficient and variable/s of each term and identify if there are any like terms.

Example 4: linear algebraic expression including non-integer coefficients

For this algebraic expression, state the number of terms, the coefficient and variable/s of each term and identify if there are any like terms.

Example 5: non-linear algebraic expression including non-integer coefficients

For this algebraic expression, state the number of terms, the coefficient and variable/s of each term and identify if there are any like terms.

Example 6: algebraic expression involving brackets and fractions

For this algebraic expression, state the number of terms, the coefficient and variable/s of each term and identify if there are any like terms.

Step-by-step guide: Expanding brackets

Common misconceptions

• Forgetting to state the coefficient as negative when a term is subtracted in an algebraic expression

For example, for the expression 7a-8b students may incorrectly state the coefficient of the b- term as 8. Remember that the sign written before a term belongs to that term. The correct coefficient for the b- term is -8.

• Thinking the coefficient is missing when actually it is 1

Students can sometimes think the coefficient is missing or perhaps it is 0 when there is no number written in front of the variable/s, for example, terms abc and n^{2}.

However both of these terms actually have coefficients of 1.

Also, watch out for negative terms, for example the term -x has a coefficient of -1.

• Incorrectly identify like terms because terms have a letter in common

For example, students may think 5x and 3xy are like terms because they both have the letter x. This is incorrect. Variables must be identical for terms to be considered like terms.

Another common error involves variables with exponents.

For example some students may think a^2 and a are like terms because they both have the letter a and no other letters/variables. However, a^{2} and a are distinct variables and are not like terms.

The next lessons are

• Expanding brackets
• Factorising
• Solving equations

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