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We Distribute, Yet Things Multiply

NCERT Class 8 · Mathematics Based on NCERT Class 8 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter explores how the distributive property of multiplication over addition can be used to understand patterns in multiplication, expand algebraic expressions, and derive important identities. The big idea is that algebra provides a compact language to describe and prove patterns, and the distributive property is the key tool for expanding products like (a+b)(c+d). By the end, a student should be able to expand any product of two binomials and apply special identities like (a+b)² = a² + 2ab + b² to solve problems quickly.
Key Concepts (10)
Distributive Property of Multiplication over Addition
For any three numbers a, b, and c: a(b + c) = ab + ac. This property allows us to multiply a number by a sum by multiplying each addend separately and then adding the products.
e.g. To find 23 × (27 + 1), we can compute 23 × 27 + 23 × 1 = 621 + 23 = 644.
Identity
A mathematical statement that expresses the equality of two algebraic expressions and holds true for all values of the variables involved.
e.g. a(b + 8) = ab + 8a is an identity because it is true for any integers a and b.
Like Terms
Terms that have exactly the same letter-numbers (variables raised to the same powers). They can be combined by adding or subtracting their coefficients.
e.g. In the expression a²b + 2a²b, both terms have the same variable part a²b, so they are like terms and can be added to give 3a²b.
Square of the Sum of Two Numbers
(a + b)² = a² + 2ab + b². This identity gives the expansion of the square of a binomial sum.
e.g. To find 65², write 65 = 60 + 5, then (60+5)² = 60² + 2×60×5 + 5² = 3600 + 600 + 25 = 4225.
Square of the Difference of Two Numbers
(a – b)² = a² – 2ab + b². This identity gives the expansion of the square of a binomial difference.
e.g. To find 55², write 55 = 60 – 5, then (60-5)² = 60² – 2×60×5 + 5² = 3600 – 600 + 25 = 3025.
Difference of Two Squares
(a + b)(a – b) = a² – b². This identity shows that the product of the sum and difference of two numbers equals the difference of their squares.
e.g. To find 98 × 102, write 98 = 100 – 2 and 102 = 100 + 2, then (100+2)(100-2) = 100² – 2² = 10000 – 4 = 9996.
General Product (a + m)(b + n)
(a + m)(b + n) = ab + an + mb + mn. This is the sum of the product of each term of (a+m) with each term of (b+n).
e.g. For a=23, b=27, m=1, n=1, we get (23+1)(27+1) = 23×27 + 23×1 + 1×27 + 1×1 = 621 + 23 + 27 + 1 = 672.
Commutative Property of Multiplication
The order of multiplication does not change the product: a × b = b × a.
e.g. 23 × 27 = 27 × 23.
Exponent Notation
A shorthand way to write repeated multiplication of the same factor. For example, a × a = a², a × a × a = a³.
e.g. 3a² × a = 3a³ because a² × a = a × a × a = a³.
Khaṇḍa-guṇanam (Multiplication by Parts)
An ancient Indian method described by Brahmagupta where the multiplier is broken into parts, each part multiplies the multiplicand, and the results are added. This is equivalent to the distributive property.
e.g. To multiply 3874 by 11, break 11 into 10 + 1: 3874 × 10 + 3874 × 1 = 38740 + 3874 = 42614.
Formulas & Equations (8)
a(b + c) = ab + ac
Reads as: a times bracket b plus c equals a b plus a c
Variables: a, b, c are any numbers (integers, fractions, etc.)
Use when: Expanding a product where one factor is a sum of two terms.
⚠ Forgetting to multiply a with both b and c, e.g., writing a(b+c) = ab + c.
(a + b)c = ac + bc
Reads as: bracket a plus b times c equals a c plus b c
Variables: a, b, c are any numbers
Use when: Expanding a product where the sum is the first factor.
⚠ Confusing this with a(b+c) and forgetting the commutative property.
(a + m)(b + n) = ab + an + mb + mn
Reads as: bracket a plus m times bracket b plus n equals a b plus a n plus m b plus m n
Variables: a, m, b, n are any numbers
Use when: Finding the product of two binomials or understanding how a product changes when both numbers are increased.
⚠ Missing the cross terms an and mb, or incorrectly adding instead of multiplying.
(a + b)² = a² + 2ab + b²
Reads as: a plus b whole squared equals a squared plus two a b plus b squared
Variables: a, b are any numbers
Use when: Expanding the square of a sum, or computing squares of numbers like 65, 104, etc.
⚠ Writing (a+b)² = a² + b², forgetting the middle term 2ab.
(a – b)² = a² – 2ab + b²
Reads as: a minus b whole squared equals a squared minus two a b plus b squared
Variables: a, b are any numbers
Use when: Expanding the square of a difference, or computing squares like 55, 99, etc.
⚠ Writing (a-b)² = a² – b², or forgetting that the middle term is negative.
(a + b)(a – b) = a² – b²
Reads as: a plus b times a minus b equals a squared minus b squared
Variables: a, b are any numbers
Use when: Finding the product of sum and difference of two numbers, or computing products like 98×102.
⚠ Writing (a+b)(a-b) = a² + b², or forgetting the minus sign.
a² = (a + b)(a – b) + b²
Reads as: a squared equals bracket a plus b times bracket a minus b plus b squared
Variables: a, b are any numbers
Use when: Computing squares of numbers using a convenient b, e.g., 31² = (31+1)(31-1) + 1² = 32×30 + 1 = 961.
⚠ Choosing b such that (a+b) or (a-b) is not easy to compute mentally.
2(a² + b²) = (a + b)² + (a – b)²
Reads as: two times bracket a squared plus b squared equals bracket a plus b whole squared plus bracket a minus b whole squared
Variables: a, b are any numbers
Use when: Proving patterns like 2(5²+6²) = (6+5)² + (6-5)².
⚠ Forgetting the factor 2 on the left side.
How It Works
Expanding a product using the distributive property
  1. Step 1: Identify the expression to expand, e.g., (a + 1)(b + 1).
  2. Step 2: Treat the first bracket as a single term and multiply it by each term in the second bracket: (a+1)(b+1) = (a+1)b + (a+1)1.
  3. Step 3: Apply the distributive property again to each term: (a+1)b = ab + 1·b = ab + b, and (a+1)1 = a·1 + 1·1 = a + 1.
  4. Step 4: Add all terms: ab + b + a + 1.
  5. Step 5: Combine like terms if any (here none).
💡 Always multiply each term of the first bracket with each term of the second bracket. A common mistake is to multiply only the first term of the first bracket with the second bracket.
Finding the increase in product when numbers are changed
  1. Step 1: Start with the original product ab.
  2. Step 2: If a is increased by m and b by n, the new product is (a+m)(b+n).
  3. Step 3: Expand using the distributive property: (a+m)(b+n) = ab + an + mb + mn.
  4. Step 4: The increase is the new product minus the original: (ab + an + mb + mn) – ab = an + mb + mn.
💡 When one number is increased and the other decreased, use the general formula with negative values for m or n. For example, (a+1)(b-1) = ab + b – a – 1.
Multiplying a number by 11 using the distributive property
  1. Step 1: Write 11 as (10 + 1).
  2. Step 2: Multiply the number by 10 (add a zero at the end).
  3. Step 3: Add the original number to the result.
  4. Step 4: For a multi-digit number, the digits can be added in a pattern: the units digit of the product is the units digit of the original number; the tens digit is the sum of the tens and units digits of the original number (carry over if needed), and so on.
💡 The rule for multiplying by 11: write the first digit, then the sum of consecutive digits from left to right, and finally the last digit. For example, 3874 × 11 = 42614.
Multiplying a number by 101 using the distributive property
  1. Step 1: Write 101 as (100 + 1).
  2. Step 2: Multiply the number by 100 (add two zeros at the end).
  3. Step 3: Add the original number to the result.
  4. Step 4: For a 4-digit number dcba, the product is d c (b+d) (a+c) b a.
💡 The pattern for multiplying by 101 involves adding adjacent digits with a carry, similar to multiplying by 11 but with an extra zero in between.
Expanding (a+b)(a²+2ab+b²) using the distributive property
  1. Step 1: Treat (a+b) as a single term and multiply by each term in the second bracket: (a+b)a² + (a+b)×2ab + (a+b)b².
  2. Step 2: Expand each part: a·a² + b·a² = a³ + a²b; (a+b)×2ab = 2a²b + 2ab²; (a+b)b² = ab² + b³.
  3. Step 3: Combine like terms: a²b + 2a²b = 3a²b; 2ab² + ab² = 3ab².
  4. Step 4: Write the final expression: a³ + 3a²b + 3ab² + b³.
💡 Always simplify exponents: a × a² = a³. Also, combine like terms carefully by adding their coefficients.
Exam Traps — Don't Make These Mistakes
⚠ Forgetting the middle term in (a+b)²
✗ (a+b)² = a² + b²
→ The student forgets the cross term 2ab. The correct expansion is a² + 2ab + b².
✓ (a+b)² = a² + 2ab + b²
⚠ Incorrect sign in (a-b)²
✗ (a-b)² = a² – b²
→ The student confuses this with the difference of squares identity. The correct expansion is a² – 2ab + b².
✓ (a-b)² = a² – 2ab + b²
⚠ Mistaking (a+b)(a-b) for a² + b²
✗ (a+b)(a-b) = a² + b²
→ The cross terms cancel: ab – ab = 0, leaving a² – b², not a² + b².
✓ (a+b)(a-b) = a² – b²
⚠ Adding unlike terms
✗ 5w² + 6w = 11w²
→ w² and w are not like terms because they have different exponents. They cannot be added.
✓ 5w² + 6w cannot be simplified further; it remains as 5w² + 6w.
⚠ Incorrect distribution with negative signs
✗ –3p(–5p + 2q) = –3p + 5p – 2q = 2p – 2q
→ The student incorrectly distributes –3p only to the first term and then changes signs incorrectly. The correct expansion is (–3p)×(–5p) + (–3p)×(2q) = 15p² – 6pq.
✓ –3p(–5p + 2q) = 15p² – 6pq
⚠ Misapplying the distributive property to multiplication inside brackets
✗ 3a(2b × 3c) = 6ab × 9ac = 54a²bc
→ The distributive property applies to addition, not multiplication. Inside the bracket, 2b × 3c = 6bc, so 3a(6bc) = 18abc.
✓ 3a(2b × 3c) = 3a(6bc) = 18abc
Compare & Contrast
Equality
(a+b)²
a² + b²
(a+b)² = a² + 2ab + b², which is always greater than a² + b² unless ab = 0. Students often wrongly think they are equal.
Value
(a-b)²
(b-a)²
They are equal because (a-b)² = (b-a)². Both expand to a² – 2ab + b². The square of a negative number is positive.
Application
Distributive property a(b+c)
Distributive property (a+b)c
Both are equivalent due to commutativity of multiplication, but a(b+c) = ab + ac, while (a+b)c = ac + bc. The order of multiplication does not matter.
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 SA (3 marks) on expansion, 1 LA (5 marks) on applying identities to solve a problem or pattern
Most asked: Expanding (a+b)², (a-b)², (a+b)(a-b) · Using identities to compute squares and products (e.g., 104², 98×102) · Expanding products of binomials using the distributive property · Identifying and correcting common mistakes in expansions
Don't skip: Long answer question: Expand (a+b)(a²+ab+b²) and simplify, or find the area of a shaded region using algebraic expressions.
One-Line Revision (read aloud before the exam)
  1. The distributive property states a(b + c) = ab + ac.
  2. An identity is an equation that is true for all values of the variables.
  3. Like terms have the same variable part and can be combined by adding coefficients.
  4. (a + b)² = a² + 2ab + b².
  5. (a – b)² = a² – 2ab + b².
  6. (a + b)(a – b) = a² – b².
  7. To multiply a number by 11, add the original number to ten times itself.
  8. To multiply a number by 101, add the original number to one hundred times itself.
  9. The product (a+m)(b+n) expands to ab + an + mb + mn.
  10. When expanding (a+b)(c+d), multiply each term of the first bracket with each term of the second.
  11. Squares of even numbers are multiples of 4.
  12. Squares of odd numbers are 1 more than a multiple of 8.
  13. The difference of two squares identity can be used to compute squares: a² = (a+b)(a-b) + b².
  14. Always combine like terms after expanding an expression.
  15. The distributive property can be visualised using area models with rectangles.

MCQs — 10 Questions with Answers

Q1. What is the distributive property of multiplication over addition?

  • A. a(b + c) = ab + ac ✓
  • B. a(b + c) = ab + c
  • C. a(b + c) = a + b + ac
  • D. (a + b)(c) = ac + bc only when a and b are positive

Answer: A — The distributive property states that a(b + c) = ab + ac, which allows us to multiply the number outside the bracket with each term inside.

Q2. By how much does the product ab increase when a is increased by 1?

  • A. By a
  • B. By b ✓
  • C. By 1
  • D. By a + b

Answer: B — Using the distributive property: a(b + 1) = ab + a. When we compare with the original product ab, the increase is a. Wait—the increase is a, so the answer should be A. Actually, when the first number a is increased by 1 to become (a+1), we get (a+1)b = ab + b, so the increase is b.

Q3. When both numbers a and b in the product ab are increased by 1, what is the increase in the product?

  • A. a + b
  • B. a + b + 1 ✓
  • C. ab + 1
  • D. 2ab

Answer: B — Expanding (a + 1)(b + 1) = ab + b + a + 1, so comparing with original product ab, the increase is a + b + 1.

Q4. A shopkeeper has a shop with dimensions 23 metres by 27 metres. If he increases both dimensions by 1 metre each, what will be the increase in the shop's area?

  • A. 50 square metres
  • B. 51 square metres ✓
  • C. 23 + 27 = 50 square metres
  • D. 1 square metre

Answer: B — Using the identity (a + 1)(b + 1) = ab + a + b + 1, the increase is 23 + 27 + 1 = 51 square metres.

Q5. What is the expansion of (a + 2)(b + 3) using Identity 1: (a + m)(b + n) = ab + mb + an + mn?

  • A. ab + 3a + 2b + 6 ✓
  • B. ab + 2a + 3b + 6
  • C. ab + 5 + 6
  • D. ab + 2b + 3a + 5

Answer: A — Using Identity 1 with m = 2 and n = 3: (a + 2)(b + 3) = ab + 2b + a(3) + 2(3) = ab + 2b + 3a + 6 = ab + 3a + 2b + 6.

Q6. Which of the following is NOT a correct identity?

  • A. (a + 1)(b - 1) = ab + b - a - 1
  • B. (a - 2)(b + 3) = ab + 3a - 2b - 6
  • C. (a + 4)(b + 2) = ab + 4b + 2a + 8
  • D. (a - 1)(b - 1) = ab - a - b - 1 ✓

Answer: D — For (a - 1)(b - 1), we expand as ab - a - b + 1 (not ab - a - b - 1), because the product of the two negative terms (-1)(-1) = +1.

Q7. In a factory, a rectangular container has dimensions (x + 5) metres by (x + 3) metres. Which expression correctly represents its area?

  • A. x² + 8x + 15 ✓
  • B. x² + 5x + 3x + 15
  • C. x² + 15
  • D. (x + 5)(x + 3) only—cannot be simplified further

Answer: A — Using (a + m)(b + n) = ab + mb + an + mn with a = x, m = 5, b = x, n = 3: (x + 5)(x + 3) = x² + 3x + 5x + 15 = x² + 8x + 15.

Q8. If one number is decreased by 2 and another number b is increased by 3, what is the increase in their product using Identity 1?

  • A. 3a - 2b - 6 ✓
  • B. -2a + 3b - 6
  • C. 3a - 2b + 6
  • D. a + b - 2 + 3

Answer: A — Using (a + m)(b + n) = ab + mb + an + mn with m = -2 and n = 3: the increase is mb + an + mn = (-2)b + a(3) + (-2)(3) = -2b + 3a - 6 = 3a - 2b - 6.

Q9. The distributive property works for which types of numbers?

  • A. Only positive integers
  • B. Only whole numbers
  • C. All integers, including negative integers ✓
  • D. Only natural numbers greater than 1

Answer: C — The study material explicitly states that integers satisfy the distributive property, and this property holds when letter-numbers are replaced by any integers, including negative ones.

Q10. Two algebraic expressions are equal identities if and only if they:

  • A. look the same when written out
  • B. take on the same values when their letter-numbers are replaced by the same numbers ✓
  • C. have the same variables
  • D. can be rearranged using multiplication only

Answer: B — An identity is defined as a mathematical statement where two algebraic expressions take on the same values for any replacement of their letter-numbers with numbers.

Flashcards

State the distributive property of multiplication over addition for three numbers a, b, and c.

a(b + c) = ab + ac, meaning the number outside the bracket multiplies with each term inside.

By how much does the product ab increase if b is increased by 1?

The product increases by a, because a(b + 1) = ab + a.

What is the expansion of (a + 1)(b + 1)?

(a + 1)(b + 1) = ab + a + b + 1, so the product increases by a + b + 1.

Write Identity 1: the expansion of (a + m)(b + n).

(a + m)(b + n) = ab + mb + an + mn, where each term of the first bracket multiplies each term of the second.

What is the increase in product (a + m)(b + n) compared to ab?

The increase is an + bm + mn, obtained from Identity 1 by subtracting ab.

Define like terms in algebra.

Terms are like terms if they have exactly the same letter-numbers; for example, 3ab and 5ab are like terms and combine to 8ab.

Expand 3a²(a − b + 1/5) using distributivity.

3a²(a − b + 1/5) = 3a³ − 3a²b + 3a/10 by multiplying 3a² with each term inside.

Expand (a − u)(b − v) using the distributive property.

(a − u)(b − v) = ab − ub − av + uv, where signs follow integer multiplication rules.

What is an algebraic identity?

An identity is a statement that two algebraic expressions are equal for all values of the variables, such as a(b + 8) = ab + 8a.

Expand (a + b)(a + b) and simplify.

(a + b)(a + b) = a² + 2ab + b² after combining like terms ba and ab into 2ab.

Important Board Questions

State the distributive property of multiplication over addition. [1 mark]

Write the property as a(b + c) = ... or use the phrase 'a single number multiplies each term inside the brackets'.

Expand 2a(3b − 5) and simplify. Show your working. [2 marks]

Apply distributivity: 2a multiplies 3b and −5 separately. Then simplify each term using rules of multiplication.

Using Identity 1, find (a + 3)(b − 2). Verify your answer by taking m = 3 and n = −2 in the formula (a + m)(b + n) = ab + mb + an + mn. [3 marks]

First substitute into Identity 1: ab + 3b + a(−2) + 3(−2). Then expand by distributivity and simplify with correct signs; verify both methods give the same result.

Expand (x + 2)(x² + 3x + 5) using the distributive property. Combine like terms and write the final simplified expression. Explain how you identified which terms were like terms. [5 marks]

Distribute (x + 2) to each term in the second bracket: x times each term, then 2 times each term. Identify like terms (same letter-numbers, such as x² terms, x terms) and add their coefficients. Write the expanded form in descending order of exponents.

Assertion-Reason Questions for We Distribute, Yet Things Multiply

Q1. Assertion (A):

The distributive property a(b + c) = ab + ac holds true when a, b, and c are negative integers.

Reason (R):

Algebraic identities are true only for positive numbers because they use multiplication.

Answer: C — A is true but R is false — The assertion is true—identities work for all integers including negatives (verified by substituting negative values). However, the reason is false because identities are not limited to positive numbers. The distributive property is a fundamental property that holds universally.

Q2. Assertion (A):

When both numbers in a product ab are increased by 1, the product increases by exactly a + b + 1.

Reason (R):

This is because (a + 1)(b + 1) = ab + a + b + 1 according to the expansion of the expression.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. The expansion (a + 1)(b + 1) = ab + a + b + 1 shows that the increase from the original product ab is indeed a + b + 1. The reason correctly explains why the assertion is true.

Q3. Assertion (A):

The identity (a + m)(b + n) = ab + mb + an + mn can be visualized using a rectangle divided into four smaller rectangles.

Reason (R):

The four rectangles have areas ab, mb, an, and mn representing different parts of the total area.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — Both statements are true. The rectangle visualization in the textbook shows this clearly with four rectangular sections. However, the reason, while true, is not the explanation for why the identity holds algebraically. The identity is proven through the distributive property, not through visualization alone. Visualization is just an aid to understanding.

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Case-Based Questions (CBQ) for We Distribute, Yet Things Multiply

Case Study:

A construction company is planning to build a rectangular park. The original plan has dimensions of 60 meters by 40 meters. However, due to increased community demand, they decide to extend the length by 8 meters and the width by 5 meters. The architect uses the expression (60 + 8)(40 + 5) to calculate the new area and needs to determine how much the area increases from the original plan.

Q1. What is the original area of the park in square meters?

  • A. 2400 ✓
  • B. 100
  • C. 2500
  • D. 3000

Answer: A

Q2. Using the distributive property, expand (60 + 8)(40 + 5) to find all components of the new area.

  • A. 2400 + 300 + 320 + 40 ✓
  • B. 2400 + 320 + 300 + 40
  • C. 2400 + 240 + 400 + 40
  • D. 2400 + 200 + 400 + 8

Answer: A

Q3. What is the total increase in area after the expansion?

  • A. 800 square meters
  • B. 540 square meters
  • C. 720 square meters
  • D. 660 square meters ✓

Answer: D

Q4. Which term in the expansion represents the small corner rectangle created by both extensions?

  • A. The entire new area
  • B. 60 × 5 = 300 square meters
  • C. 40 × 8 = 320 square meters
  • D. 8 × 5 = 40 square meters ✓

Answer: D

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