Introduction to Algebraic Expressions
Algebraic expressions — addition, subtraction and multiplication of monomials.
Algebraic Expressions — Introduction (Class 8 Maths)
Chapter 11, Algebraic Expressions, builds the vocabulary and the basic rules you will use throughout algebra — for CBSE, Telangana and Andhra Pradesh Class 8 students. Before solving equations, factorising, or working with polynomials in later classes, you first need to be comfortable with words like variable, constant, term and expression, and with simple operations such as adding, subtracting and multiplying expressions. This lesson explains every idea from the introduction with clear examples, tables and step-by-step working.
What You Will Learn in This Lesson
- The meaning of variable, constant, term and algebraic expression, with simple examples of each.
- How to tell an algebraic expression apart from a numerical expression.
- How to count the number of terms in an expression and classify it as a monomial, binomial, trinomial or multinomial.
- How to find the degree of a monomial and the degree of an algebraic expression.
- How to identify like terms and unlike terms.
- How to add, subtract and multiply algebraic expressions step by step.
The Building Blocks of Algebra
Every algebraic expression is built from just a few simple pieces. A variable is a letter or symbol that stands for a value we don't know yet, or one that can change — common examples are x, y, a, b. A constant, on the other hand, is a fixed value that never changes — it is either a plain number or a symbol that always represents the same known number, such as 3, −9, 8.5 or π.
When a number and one or more variables are multiplied together, the result is called a term. A term can be just a number (like 5), just a variable (like t), or a product such as 2x, 3.5pq or 3⁄2lmn. Finally, when one or more terms are joined together using + or − signs, the result is an algebraic expression — for example, x + 5y − 1 or 2a² + 3a + 5.
| Building Block | What It Means | Examples |
|---|---|---|
| Variable | A letter or symbol that stands for an unknown or changing value | x, y, a, b |
| Constant | A fixed value that never changes — a number or a known symbol | 3, −9, 8.5, π |
| Term | A number, a variable, or a product of numbers and variables | 5, t, 2x, 3.5pq, 3⁄2lmn |
| Expression | One or more terms joined by + or − signs | x+5y−1, 2a²+3a+5 |
Algebraic Expressions vs Numerical Expressions
Not every expression involving numbers is "algebraic". An expression that contains at least one variable is called an algebraic expression, while an expression made up of only numbers (with no variable at all) is called a numerical expression.
| Expression | Contains a Variable? | Type |
|---|---|---|
| x + 5y − 1 | Yes — x and y | Algebraic expression |
| 2a² + 3a + 5 | Yes — a | Algebraic expression |
| 8m³ − 2⁄3mn + √3 | Yes — m and n | Algebraic expression |
| 3 + 5 − 0.5 + √3 | No — only numbers | Numerical expression |
Try These: Counting the Number of Terms
To count the terms in an expression, look for the + and − signs — each one marks the start of a new term, and the sign in front of a term is treated as part of that term. Let's count the terms in four expressions and see how each one is colour-coded below:
| Expression | Terms | Count | Name |
|---|---|---|---|
| 5xy² | 5xy² | 1 | Monomial |
| 5xy³ − 9x | 5xy³−9x | 2 | Binomial |
| 3xy + 4y − 8 | 3xy+4y−8 | 3 | Trinomial |
| 9x² + 2x + pq + q | 9x²+2x+pq+q | 4 | Multinomial |
Classifying Expressions: Monomial, Binomial, Trinomial, Multinomial
Based on how many terms it has, every algebraic expression gets a name. These names are used very often in later chapters such as Factorisation and Polynomials, so it helps to learn them well now:
| Name | Number of Terms | Example |
|---|---|---|
| Monomial | Exactly 1 | 5xy² |
| Binomial | Exactly 2 | 5xy³ − 9x |
| Trinomial | Exactly 3 | 3xy + 4y − 8 |
| Multinomial | More than 3 | 9x² + 2x + pq + q |
Degree of a Monomial
The degree of a monomial is found by adding up the exponents (powers) of all the variables in it. Remember that a variable written without a visible exponent, like x or a, has an exponent of 1.
| Monomial | Exponents of Variables | Sum of Exponents (Degree) |
|---|---|---|
| 5xy² | x¹, y² | 1 + 2 = 3 |
| −3l³m²n | l³, m², n¹ | 3 + 2 + 1 = 6 |
| pq | p¹, q¹ | 1 + 1 = 2 |
| 3a | a¹ | 1 |
| 1.5 | — (no variable) | 0 |
| 0 | — | not defined |
Degree of an Algebraic Expression
Once you can find the degree of a single monomial, finding the degree of a whole expression is easy: just find the degree of every term, and the highest of these degrees is the degree of the expression.
| Expression | Degree of Each Term | Degree of Expression |
|---|---|---|
| 3x² − 5x + 6 | 2, 1, 0 | 2 |
| 5pqr − q² | 3, 2 | 3 |
| 3x² − x³ + x + 1 | 2, 3, 1, 0 | 3 |
| 7 + a | 0, 1 | 1 |
| 3x³y + 5xy − 4 | 4, 2, 0 | 4 |
Like Terms and Unlike Terms
Two or more terms are called like terms if they have exactly the same variables raised to exactly the same exponents — their numerical coefficients can be completely different. If the variable parts don't match exactly, the terms are unlike terms.
To add two algebraic expressions, write one below the other so that like terms line up in the same column. Then simply add the numerical coefficients of each column to get the coefficient of that term in the answer — the variable part stays the same.
Worked Example
Add 5a² − 3ab + 7b² and 8a² + 2ab + 3b².
| 5a² − 3ab + 7b² | |
| + | 8a² + 2ab + 3b² |
| 13a² − ab + 10b² |
- Line up the three groups of like terms in columns: the a² terms, the ab terms, and the b² terms.
- Add the a² coefficients: 5 + 8 = 13, giving 13a².
- Add the ab coefficients: −3 + 2 = −1, giving −ab.
- Add the b² coefficients: 7 + 3 = 10, giving 10b².
Subtracting an expression is the same as adding its opposite. So before adding column by column, flip the sign of every single term in the expression that is being subtracted — not just the first one. This is the step students forget most often.
Worked Example
Subtract 3x² + y² − xy from 2xy + x².
Write the second expression with all signs flipped: 3x² + y² − xy becomes −3x² − y² + xy. Now add this to the first expression, lining up the x², xy and y² columns:
| x² + 2xy + 0y² | |
| + | −3x² + xy − y² |
| −2x² + 3xy − y² |
- Write the expression we are subtracting from: x² + 2xy (we can think of it as x² + 2xy + 0y² so every column has an entry).
- Write the expression being subtracted: 3x² + y² − xy.
- Flip the sign of every term in it: 3x² + y² − xy → −3x² − y² + xy.
- Add column by column — x²: 1 + (−3) = −2; xy: 2 + 1 = 3; y²: 0 + (−1) = −1.
To multiply two monomials, multiply their numerical coefficients together first. Then, for every variable that appears in both monomials, use the law of exponents — multiply the powers by adding their exponents. If a variable appears in only one monomial, it is simply carried over to the answer as it is.
am × an = am+nExample A — Different Variables
5p × 7q
- Multiply the coefficients: 5 × 7 = 35.
- p and q are different variables, so they are simply written together: p × q = pq.
Example B — Repeated Variables
−2a²b × 4a × 8b²
- Multiply all the coefficients together: (−2) × 4 × 8 = −64.
- Combine the powers of a using am×an=am+n: a² × a¹ = a³.
- Combine the powers of b in the same way: b¹ × b² = b³.
Common Mistakes to Avoid
- Confusing a "term" with an "expression": A term itself never has a + or − joining two separate parts; an expression is made of two or more terms joined by + or −.
- Ignoring the sign in front of a term: In 5xy³ − 9x, the second term is −9x, not just "9x" — the minus sign belongs to that term.
- Calling 3x² and 3x "like terms": They look similar, but the exponents (2 and 1) are different, so they are unlike terms and cannot be combined.
- Forgetting to flip ALL signs while subtracting: Every term of the expression being subtracted must have its sign reversed — not just the first one.
- Multiplying exponents instead of adding them: a² × a should give a³ (since 2 + 1 = 3), not a² (and definitely not a²×¹).
- Saying the degree of 0 is 0: By convention, the degree of the number 0 is not defined, while the degree of any other constant is 0.
What This Lesson Prepares You For
This introduction sets up all the vocabulary and the addition, subtraction and multiplication rules you will use throughout Exercise 11.1, where many more practice problems on adding and subtracting algebraic expressions are solved in full detail.
Multiplying monomials uses the same law of exponents, am × an = am+n, that you will study in much more depth in Exponents and Powers. Once you are confident adding, subtracting and multiplying expressions, you will be ready for Factorisation, where these same expressions are broken back down into their building blocks.