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MathematicsClass 8Algebraic Expressions and Identities

Algebraic Expressions and Identities | Class 8 Mathematics Notes

By ConceptScroll Team · Published on 17 July 2026 · 5 min read

Algebraic Expressions and Identities – this guide gives you a concise, exam-ready overview of Algebraic Expressions and Identities from Class 8 Mathematics, written by ConceptScroll editors and reviewed against the latest NCERT textbook.

Introduction to Algebraic Expressions

Algebraic expressions are mathematical phrases that include variables, constants, and arithmetic operations such as addition, subtraction, multiplication, and division. In Class 8 Mathematics, understanding algebraic expressions is fundamental as it forms the basis for solving equations and understanding identities. An algebraic expression consists of terms, where each term is a product of constants and variables raised to whole number powers. For example, 3x + 5y - 7 is an algebraic expression with three terms: 3x, 5y, and -7. Here, 3 and 5 are coefficients, x and y are variables, and -7 is a constant term.

Variables represent unknown or changing quantities and are usually denoted by letters such as x, y, z. Constants are fixed numbers. The degree of a term is the sum of the exponents of the variables in that term. For example, in 4x²y, the degree is 2 + 1 = 3. The degree of an algebraic expression is the highest degree among its terms.

Algebraic expressions can be classified based on the number of terms: monomial (one term), binomial (two terms), trinomial (three terms), and polynomial (more than one term). For instance, 7x² is a monomial, 3x + 5 is a binomial, and x² + 5x + 6 is a trinomial.

Understanding algebraic expressions helps in simplifying expressions, performing arithmetic operations on them, and solving algebraic equations. It also aids in recognizing patterns and formulating identities, which are equations true for all values of variables involved.

This section introduces the basic language and components of algebraic expressions, preparing students to manipulate and work with them in subsequent sections.

📊 Diagram: Diagram illustrating an algebraic expression 3x + 5y - 7, highlighting terms, coefficients, variables, and constants.

🧪 Activity: Activity: Identify and classify algebraic expressions from given examples and real-life contexts.

🔗 Connection: This section lays the groundwork for the next section on 'Like and Unlike Terms' by introducing the components of algebraic expressions.

Frequently asked questions

1. Add the following. (i) $ab - bc, bc - ca, ca - ab$ (ii) $a - b + ab, b - c + bc, c - a + ac$ (iii) $2p^{2}q^{2} - 3pq + 4,5 + 7pq - 3p^{2}q^{2}$ (iv) $l^{2} + m^{2}, m^{2} + n^{2}, n^{2} + l^{2},$ $$ 2lm + 2mn + 2nl $$

Solution:

(i) Add: (ab - bc) + (bc - ca) + (ca - ab) = ab - bc + bc - ca + ca - ab = (ab - ab) + (-bc + bc) + (-ca + ca) = 0

(ii) Add: (a - b + ab) + (b - c + bc) + (c - a + ac) = a - b + ab + b - c + bc + c - a + ac = (a - a) + (-b + b) + (-c + c) + ab + bc + ac = ab + bc + ac

(iii) Add: (2p^{2}q^{2} - 3pq + 4) + (5 + 7pq - 3p^{2}q^{2}) = 2p^{2}q^{2} - 3pq + 4 + 5 + 7pq - 3p^{2}q^{2} = (2p^{2}q^{2} - 3p^{2}q^{2}) + (-3pq + 7pq) + (4 + 5) = (-p^{2}q^{2}) + (4pq) + 9

(iv) Add: (l^{2} + m^{2})

2. (a) Subtract $4a - 7ab + 3b + 12$ from $12a - 9ab + 5b - 3$ (b) Subtract $3xy + 5yz - 7zx$ from $5xy - 2yz - 2zx + 10xyz$ (c) Subtract $4p^{2}q - 3pq + 5pq^{2} - 8p + 7q - 10$ from $$ 18 - 3p - 11q + 5pq - 2pq^{2} + 5p^{2}q $$

Solution:

(a) Subtracting (4a - 7ab + 3b + 12) from (12a - 9ab + 5b - 3): = (12a - 9ab + 5b - 3) - (4a - 7ab + 3b + 12) = 12a - 9ab + 5b - 3 - 4a + 7ab - 3b - 12 = (12a - 4a) + (-9ab + 7ab) + (5b - 3b) + (-3 - 12) = 8a - 2ab + 2b - 15

(b) Subtracting (3xy + 5yz - 7zx) from (5xy - 2yz - 2zx + 10xyz): = (5xy - 2yz - 2zx + 10xyz) - (3xy + 5yz - 7zx) = 5xy - 2yz - 2zx + 10xyz - 3xy - 5yz + 7zx = (5xy - 3xy) + (-2yz - 5yz) + (-2zx + 7zx) + 10xyz = 2xy - 7yz + 5zx + 10xyz

(c) Subtracting (4p^{2}q -

TRY THESE Find $4x imes 5y imes 7z$ First find $4x imes 5y$ and multiply it by $7z$; or first find $5y imes 7z$ and multiply it by $4x$. Is the result the same? What do you observe? Does the order in which you carry out the multiplication matter?

Solution:

First method: 4x × 5y = 20xy Then, 20xy × 7z = 140xyz

Second method: 5y × 7z = 35yz Then, 4x × 35yz = 140xyz

Observation: The result is the same (140xyz) in both methods.

Conclusion: The order of multiplication does not matter (commutative property of multiplication).

Example 3: Complete the table for area of a rectangle with given length and breadth. Solution: | length | breadth | area | |--------|---------|-------| | 3x | 5y | 3x × 5y = 15xy | | 9y | 4y² | ... | | 4ab | 5bc | ... | | 2l²m | 3lm² | ... |

Solution:

Area = length × breadth

For 9y and 4y²: Area = 9y × 4y² = 36y^{3}

For 4ab and 5bc: Area = 4ab × 5bc = 20ab^{2}c

For 2l^{2}m and 3lm^{2}: Area = 2l^{2}m × 3lm^{2} = 6l^{3}m^{3}

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