CBSE Class 8 Maths Revision Notes Chapter 6 We Distribute, Yet Things Multiply
The distributive property connects multiplication with addition and subtraction by dividing an expression into simpler products. In CBSE Class 8 Maths, it helps expand expressions, establish identities and understand multiplication patterns.
The chapter We Distribute, Yet Things Multiply shows how algebra can describe multiplication patterns in a compact form. Its main idea is the distributive property, which allows every term inside a bracket to be multiplied correctly.
These CBSE Class 8 Maths Revision Notes Chapter 6 cover distributivity, algebraic expansion, like terms, identities and quick multiplication methods. Students can also revise equivalent expressions, visual patterns and common errors for the 2026–27 academic year.
Key Takeaways
- Distributive property: a(b + c) = ab + ac and a(b − c) = ab − ac.
- General expansion: (a + b)(c + d) = ac + ad + bc + bd.
- Square identities: (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b².
- Difference of squares: (a + b)(a − b) = a² − b².
Access Class 8 Maths Chapter 6 We Distribute, Yet Things Multiply Notes in 30 Minutes
Revise the chapter in three parts:
First 10 minutes: Distributive property, changes in products and expansion of expressions
Next 10 minutes: Algebraic identities and fast multiplication methods
Final 10 minutes: Equivalent expressions, patterns and common mistakes
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Distributive Property in Class 8 Maths Chapter 6 Notes
The distributive property links multiplication with addition and subtraction. It allows a number or algebraic term to multiply each term inside a bracket.
Distributive Property of Multiplication Over Addition
For any numbers or algebraic expressions a, b and c:
a(b + c) = ab + ac
This means that a multiplies both b and c.
Example:
5(x + 3)
= 5 × x + 5 × 3
= 5x + 15
The same property can also be written as:
(a + b)c = ac + bc
Example:
(x + 4)3
= 3x + 12
The multiplication sign is usually omitted beside brackets. Therefore, 5(x + 3) means 5 × (x + 3).
Distributive Property Over Subtraction
The same rule applies when the terms inside a bracket are subtracted.
a(b − c) = ab − ac
Example:
7(x − 2)
= 7x − 14
Signs must be handled carefully when negative terms are present.
Example:
−3p(−5p + 2q)
= (−3p)(−5p) + (−3p)(2q)
= 15p² − 6pq
How a Product Changes When One Number Increases
Suppose the original product is ab.
If b increases by 1, the new product becomes:
a(b + 1)
= ab + a
The original product increases by a.
Similarly, if a increases by 1:
(a + 1)b
= ab + b
The original product increases by b.
How a Product Changes When Both Numbers Increase
If both numbers increase by 1, the new product is:
(a + 1)(b + 1)
= ab + a + b + 1
Therefore, the original product increases by:
a + b + 1
For example, take a = 23 and b = 27.
(23 + 1)(27 + 1)
= 23 × 27 + 23 + 27 + 1
The increase is 51.
When One Number Increases and the Other Decreases
If a increases by 1 and b decreases by 1:
(a + 1)(b − 1)
= ab + b − a − 1
The change is:
b − a − 1
The product may increase, decrease or remain unchanged. It depends on the values of a and b.
General Change in a Product
Suppose a increases by m and b increases by n.
The new product is:
(a + m)(b + n)
= ab + an + bm + mn
Therefore, the change in the original product is:
an + bm + mn
This gives the identity:
(a + m)(b + n) = ab + an + bm + mn
If one number decreases, use a negative value.
Example:
(a − 2)(b + 3)
= ab + 3a − 2b − 6
When both numbers decrease:
(a − 3)(b − 4)
= ab − 4a − 3b + 12
Expanding Algebraic Expressions in Class 8 Maths Revision Notes Chapter 6
To expand an expression, multiply every term outside a bracket by every term inside it. After expansion, combine only the like terms.
Multiplying a Term by an Expression
Example:
3a(a − b + 2)
= 3a × a − 3a × b + 3a × 2
= 3a² − 3ab + 6a
Each term inside the bracket is multiplied by 3a.
Fractions follow the same rule.
Example:
(2/3)(15 + 6a)
= (2/3 × 15) + (2/3 × 6a)
= 10 + 4a
Multiplying Two Binomials
A binomial contains two terms. While multiplying two binomials, each term of the first expression multiplies each term of the second.
The general rule is:
(a + b)(c + d)
= ac + ad + bc + bd
Example:
(x + 2)(x + 5)
= x(x + 5) + 2(x + 5)
= x² + 5x + 2x + 10
= x² + 7x + 10
Another example:
(3 + u)(v − 3)
= 3v − 9 + uv − 3u
= uv + 3v − 3u − 9
Expanding Expressions with Negative Terms
The usual sign rules of integer multiplication apply.
Example:
(3 − x)(x − 6)
= 3x − 18 − x² + 6x
= −x² + 9x − 18
Example:
(−5a + b)(c + d)
= −5ac − 5ad + bc + bd
Multiplying Expressions with Three Terms
Each term of one expression must multiply every term of the other expression.
Example:
(4y + 7)(y + 11z − 3)
= 4y(y + 11z − 3) + 7(y + 11z − 3)
= 4y² + 44yz − 12y + 7y + 77z − 21
= 4y² + 44yz − 5y + 77z − 21
Like Terms in Algebraic Expressions Class 8 Notes
Like terms contain the same variables raised to the same powers.
Examples of like terms:
3ab and 7ab
a²b and 5a²b
2x² and −9x²
Examples of unlike terms:
a and a²
ab and ab²
x and y
Consider:
(a + b)(a + b)
= a² + ab + ba + b²
Since ba = ab:
ab + ba = 2ab
Therefore:
(a + b)(a + b) = a² + 2ab + b²
Unlike terms cannot be combined.
For example:
5w² + 6w
cannot be written as 11w² because w² and w are different terms.
Algebraic Identities in CBSE Class 8 Maths Chapter 6 Notes
An identity is an equality that remains true for every allowed value of its variables. The identities in this chapter come directly from the distributive property.
| Identity | Expansion |
| Square of a sum | (a + b)² = a² + 2ab + b² |
| Square of a difference | (a − b)² = a² − 2ab + b² |
| Product of sum and difference | (a + b)(a − b) = a² − b² |
| Sum of two square expressions | (a + b)² + (a − b)² = 2(a² + b²) |
Square of the Sum of Two Numbers
The identity is:
(a + b)² = a² + 2ab + b²
It can be proved through expansion:
(a + b)²
= (a + b)(a + b)
= a² + ab + ab + b²
= a² + 2ab + b²
Example:
(6x + 5)²
= (6x)² + 2(6x)(5) + 5²
= 36x² + 60x + 25
The middle term is twice the product of the two terms.
Finding Squares Using the Sum Identity
The identity can simplify numerical calculations.
Example:
104²
= (100 + 4)²
= 100² + 2(100)(4) + 4²
= 10000 + 800 + 16
= 10816
Example:
46²
= (40 + 6)²
= 40² + 2(40)(6) + 6²
= 1600 + 480 + 36
= 2116
Square of the Difference of Two Numbers
The identity is:
(a − b)² = a² − 2ab + b²
Through expansion:
(a − b)(a − b)
= a² − ab − ab + b²
= a² − 2ab + b²
Example:
(b − 6)²
= b² − 2(b)(6) + 6²
= b² − 12b + 36
Example:
(−2a + 3)²
= (−2a)² + 2(−2a)(3) + 3²
= 4a² − 12a + 9
The final term remains positive because the product of two negative terms is positive.
Finding Squares Using the Difference Identity
Example:
99²
= (100 − 1)²
= 100² − 2(100)(1) + 1²
= 10000 − 200 + 1
= 9801
Example:
91²
= (100 − 9)²
= 10000 − 1800 + 81
= 8281
Product of Sum and Difference
The identity is:
(a + b)(a − b) = a² − b²
During expansion, the middle terms cancel.
(a + b)(a − b)
= a² − ab + ab − b²
= a² − b²
Example:
(3a − 9b)(3a + 9b)
= (3a)² − (9b)²
= 9a² − 81b²
Calculating Products Using Difference of Squares
Example:
98 × 102
= (100 − 2)(100 + 2)
= 100² − 2²
= 10000 − 4
= 9996
Example:
397 × 403
= (400 − 3)(400 + 3)
= 400² − 3²
= 160000 − 9
= 159991
Identity Involving the Sum of Two Squares
Adding the two square identities gives:
(a + b)² + (a − b)²
= a² + 2ab + b² + a² − 2ab + b²
= 2a² + 2b²
Therefore:
(a + b)² + (a − b)² = 2(a² + b²)
This identity holds for integers and fractions because it follows from algebraic expansion.
Fast Multiplication in We Distribute Yet Things Multiply Notes
The distributive property helps multiply numbers near powers of 10. Numbers such as 11, 101, 1001, 99 and 999 can be separated into convenient parts.
Multiplication by 11
Write:
11 = 10 + 1
Therefore:
N × 11 = N × 10 + N
Example:
94 × 11
= 94 × 10 + 94
= 940 + 94
= 1034
Example:
495 × 11
= 4950 + 495
= 5445
For a multi-digit number, adjacent digits are added. Carrying is handled in the usual way.
Multiplication by 101
Write:
101 = 100 + 1
Therefore:
N × 101 = N × 100 + N
Example:
89 × 101
= 8900 + 89
= 8989
Example:
3874 × 101
= 387400 + 3874
= 391274
Multiplication by 1001
Write:
1001 = 1000 + 1
Therefore:
N × 1001 = N × 1000 + N
Example:
1111 × 1001
= 1111000 + 1111
= 1112111
Example:
265831 × 1001
= 265831000 + 265831
= 266096831
Multiplication by 99 and 999
Numbers just below powers of 10 can be written as differences.
99 = 100 − 1
999 = 1000 − 1
Example:
9734 × 99
= 9734(100 − 1)
= 973400 − 9734
= 963666
Example:
23478 × 999
= 23478(1000 − 1)
= 23478000 − 23478
= 23454522
Algebraic Patterns and Equivalent Expressions in Class 8 Chapter 6 Maths Notes
Algebra can describe a growing numerical or visual pattern using a general expression. Different arrangements may lead to different-looking expressions for the same pattern.
Describing a Growing Pattern
Suppose a pattern has:
Step 1: 3 objects
Step 2: 8 objects
Step 3: 15 objects
Step 4: 24 objects
One expression for Step k may be:
(k + 1)² − 1
Expanding:
(k + 1)² − 1
= k² + 2k + 1 − 1
= k² + 2k
Another visual arrangement may give:
k(k + 2)
Expanding:
k(k + 2)
= k² + 2k
Both expressions describe the same pattern.
Equivalent Algebraic Expressions
Expressions are equivalent when they give the same value for every permitted value of the variable.
For example:
(k + 1)² − 1
k² + 2k
k(k + 2)
k(k + 1) + k
All simplify to:
k² + 2k
This shows that one pattern can be interpreted in several correct ways.
Finding the Number of Objects at Any Step
If the number of objects at Step k is:
k² + 2k
then at Step 15:
15² + 2(15)
= 225 + 30
= 255
The pattern has 255 objects at Step 15.
Algebra in Area Problems
Suppose a square has side m + n. Its area is:
(m + n)²
If four rectangles of area mn are removed, the remaining area is:
(m + n)² − 4mn
Expanding:
m² + 2mn + n² − 4mn
= m² − 2mn + n²
= (m − n)²
Therefore:
(m + n)² − 4mn = (m − n)²
Both expressions represent the same shaded area.
Common Mistakes in Class 8 Maths Chapter 6
Most expansion errors happen when a term is missed, a sign is changed or unlike terms are combined.
| Incorrect form | Correct form | Reason |
| 2(x − 1) = 2x − 1 | 2x − 2 | Both terms must be multiplied by 2 |
| (a + 2)(b + 4) = ab + 8 | ab + 4a + 2b + 8 | Every term multiplies every term |
| (5m + 6n)² = 25m² + 36n² | 25m² + 60mn + 36n² | The middle term is missing |
| 5w² + 6w = 11w² | Cannot be combined | w² and w are unlike terms |
| y + 2(y + 2) = (y + 2)² | 3y + 4 | Addition cannot be changed into multiplication |
| 3a(2b × 3c) = 54a²bc | 18abc | Each factor is multiplied once |
Before finalising an expansion, check:
- Has every term been multiplied?
- Are all positive and negative signs correct?
- Have only like terms been combined?
- Is the middle term present in the square identity?
Class 8 Maths Revision Notes Chapter 6 Formula Table
| Concept | Formula | Key Point |
| Distributive property over addition | a(b + c) = ab + ac | Multiply a by both terms |
| Distributive property over subtraction | a(b − c) = ab − ac | Retain the negative sign |
| Product of two binomials | (a + b)(c + d) = ac + ad + bc + bd | Form four products |
| General change in product | (a + m)(b + n) = ab + an + bm + mn | Compare with ab |
| Square of a sum | (a + b)² = a² + 2ab + b² | Middle term is positive |
| Square of a difference | (a − b)² = a² − 2ab + b² | Last term remains positive |
| Difference of squares | (a + b)(a − b) = a² − b² | Middle terms cancel |
| Related square identity | (a + b)² + (a − b)² = 2(a² + b²) | Add both square expansions |
| Multiplication by 11 | N(10 + 1) | Add N × 10 and N |
| Multiplication by 101 | N(100 + 1) | Add N × 100 and N |
| Multiplication by 99 | N(100 − 1) | Subtract N from N × 100 |
Quick Revision of We Distribute Yet Things Multiply
- Multiply every term inside a bracket.
- Follow the sign rules for integers.
- Combine terms only when their variables and powers match.
- Use identities to calculate squares and products quickly.
- Expand both expressions to check whether they are equivalent.
- Use convenient numbers such as 10, 100 and 1000 for fast multiplication.
- Remember that different methods can give the same simplified expression.
Useful Links for Class 8 Maths
| Section | Useful Links |
| Syllabus | CBSE Class 8 Maths Syllabus |
| Revision Notes | CBSE Class 8 Maths Revision Notes |
| Maths Notes | CBSE Class 8 Maths Revision Notes Chapter 1 |
| NCERT Solutions | NCERT Solutions for Class 8 Maths |
| Sample Papers | CBSE Sample Papers for Class 8 Maths |
| Important Questions | Important Questions Class 8 Maths |
| NCERT Books | NCERT Books for Class 8 Maths |
| Class 8 Support | CBSE Class 8 Syllabus |
FAQs (Frequently Asked Questions)
Different expressions may represent different arrangements of the same objects. When both expressions simplify to the same algebraic form, they are equivalent and describe the same pattern.
The last term comes from multiplying −b by −b. The product of two negative terms is positive, so the last term is +b².
Expand both sides using the distributive property. If both sides simplify to the same expression for all allowed values, the statement is an identity.
Use it when two factors contain the same terms with opposite signs. It is useful for expressions such as (a + b)(a − b) and products such as 98 × 102.
The terms contain different powers of x. Since x² and x are unlike terms, they cannot be combined into a single term.