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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 · 4 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.

Multiplication of Algebraic Expressions

Multiplying algebraic expressions involves using the distributive property, which states that a(b + c) = ab + ac. When multiplying two expressions, each term of the first expression is multiplied by each term of the second expression.

For example, to multiply (x + 3) and (x + 5), we apply distributive property: (x + 3)(x + 5) = x(x + 5) + 3(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15.

The steps for multiplication are: 1. Multiply each term of the first expression by each term of the second expression. 2. Use the laws of exponents for variables (e.g., x^m × x^n = x^(m+n)). 3. Combine like terms if any.

Multiplying monomials involves multiplying coefficients and adding exponents of like variables. For example, (3x²y) × (4xy²) = (3 × 4)(x^(2+1))(y^(1+2)) = 12x³y³.

Multiplying a monomial by a polynomial requires distributing the monomial to each term of the polynomial.

This section also introduces the concept of special products and identities, which are specific patterns of multiplication that simplify calculations.

📊 Diagram: Diagram showing multiplication of two binomials with arrows indicating distribution of each term.

🧪 Activity: Activity: Multiply given pairs of algebraic expressions using distributive property and verify results.

🔗 Connection: This section prepares students for understanding algebraic identities in the next section.

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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