We Distribute Yet Things Multiply — Class 8 Mathematics (Ganita Prakash)
"When we multiply two binomials, we are not just doing arithmetic — we are revealing the deep structure of all algebra."
1. About the Chapter
This chapter's playful title captures a paradox: when we DISTRIBUTE one expression across another, the answer still MULTIPLIES to something larger. We are doing two opposite-sounding things at once.
The chapter teaches:
- Algebraic expressions and their parts
- Distributive property in algebra
- Multiplying polynomials (binomials and beyond)
- Visual reasoning with area diagrams
- Algebraic identities — (a+b)², (a−b)², a²−b²
- Factorisation (the reverse of multiplication)
2. Algebraic Expressions — Quick Review
Terms
- A term is a single number or variable, or a product like 3xy or −5x².
- Variable: a letter (x, y, z) standing for a number.
- Constant: a fixed number (like 5).
- Coefficient: the number multiplied by a variable (in 7x, the coefficient is 7).
Types
- Monomial: one term (e.g., 5x, −3y², 7)
- Binomial: two terms (e.g., x + 5, 3a − 2b)
- Trinomial: three terms (e.g., a² + 2a + 1)
- Polynomial: general term for any algebraic expression with multiple terms
Degree
The degree of a polynomial is the highest power of any single variable.
- 3x² + 5x − 7 has degree 2 (a quadratic)
- x³ + 2x has degree 3 (a cubic)
3. The Distributive Property (Heart of the Chapter)
Statement
a × (b + c) = a × b + a × c
This says: to multiply a by (b + c), multiply a by each term separately and add.
Visual Proof (Rectangle Method)
Think of a rectangle of dimensions a × (b + c):
- Total area = a × (b + c)
- Same rectangle = a × b plus a × c (two smaller rectangles)
- So a × (b + c) = a × b + a × c
Examples
- 3 × (5 + 2) = 3 × 5 + 3 × 2 = 15 + 6 = 21 ✓
- 4 × (x + 7) = 4x + 28
- 2x × (3y + 5) = 6xy + 10x
Extending to Subtraction
a × (b − c) = a × b − a × c
Extending to More Terms
a × (b + c + d) = ab + ac + ad
4. Multiplying Two Binomials
The Distributive Property Applied Twice
(a + b) × (c + d) = a × (c + d) + b × (c + d) = ac + ad + bc + bd
FOIL Method
A mnemonic for binomial multiplication:
- First terms: a × c
- Outer terms: a × d
- Inner terms: b × c
- Last terms: b × d
Add them all.
Examples
Example 1: (x + 2)(x + 3)
- F: x × x = x²
- O: x × 3 = 3x
- I: 2 × x = 2x
- L: 2 × 3 = 6
- Sum: x² + 5x + 6
Example 2: (2a + 5)(3a − 4)
- F: 2a × 3a = 6a²
- O: 2a × (−4) = −8a
- I: 5 × 3a = 15a
- L: 5 × (−4) = −20
- Sum: 6a² + 7a − 20
Visual Proof: Area of Rectangle
A rectangle of (a+b) × (c+d) is divided into 4 sub-rectangles:
- ac, ad, bc, bd The total area = sum of these four.
5. Famous Algebraic Identities (MASTER ALL)
These appear repeatedly in algebra. Memorise them.
Identity 1: (a + b)²
(a + b)² = a² + 2ab + b²
Three terms: a², 2ab, b².
Examples:
- (x + 5)² = x² + 10x + 25
- (2y + 3)² = 4y² + 12y + 9
Identity 2: (a − b)²
(a − b)² = a² − 2ab + b²
Same as identity 1, but middle term is negative.
Examples:
- (x − 4)² = x² − 8x + 16
- (3p − 2q)² = 9p² − 12pq + 4q²
Identity 3: a² − b² (Difference of Squares)
a² − b² = (a + b)(a − b)
Examples:
- x² − 25 = (x + 5)(x − 5)
- 16y² − 9 = (4y + 3)(4y − 3)
Identity 4: (a + b)(a − b) = a² − b²
Same as Identity 3, written differently.
Identity 5: (x + a)(x + b)
(x + a)(x + b) = x² + (a + b)x + ab
Useful for quadratic factorisation.
6. Worked Examples
Example 1: Distribute
Simplify: 3x × (2x − 5y + 4)
- = 3x × 2x − 3x × 5y + 3x × 4
- = 6x² − 15xy + 12x
Example 2: Multiply Binomials
Multiply: (3x − 7)(2x + 5)
- F: 3x × 2x = 6x²
- O: 3x × 5 = 15x
- I: −7 × 2x = −14x
- L: −7 × 5 = −35
- Sum: 6x² + x − 35
Example 3: Apply (a+b)² Identity
Expand (4x + 7)².
- (a + b)² = a² + 2ab + b², where a = 4x, b = 7
- = (4x)² + 2(4x)(7) + 7²
- = 16x² + 56x + 49
Example 4: Apply (a−b)² Identity
Expand (5p − 3q)².
- = (5p)² − 2(5p)(3q) + (3q)²
- = 25p² − 30pq + 9q²
Example 5: Apply Difference of Squares
Factorise: x² − 64
- = x² − 8² = (x + 8)(x − 8)
Example 6: Apply x² + (a+b)x + ab
Factorise: x² + 7x + 12
- We need two numbers whose product is 12 and sum is 7. Try 3 and 4: 3 × 4 = 12 ✓, 3 + 4 = 7 ✓
- So x² + 7x + 12 = (x + 3)(x + 4)
Example 7: Compute Using Identity
Compute 102² using identity.
- 102² = (100 + 2)² = 100² + 2(100)(2) + 2² = 10000 + 400 + 4 = 10404
Example 8: Compute Using Identity
Compute 998² using identity.
- 998² = (1000 − 2)² = 1000² − 2(1000)(2) + 2² = 1000000 − 4000 + 4 = 996004
Example 9: Difference of Squares for Computation
Compute 105 × 95.
- = (100 + 5)(100 − 5) = 100² − 5² = 10000 − 25 = 9975
7. Introduction to Factorisation
What is Factorisation?
Factorisation is the reverse of multiplication. We express a polynomial as a product of simpler polynomials.
Method 1: Common Factor
- 6x + 9y = 3(2x + 3y) (3 is common)
- 4xy + 2x = 2x(2y + 1) (2x common)
Method 2: Identity-based
- a² + 2ab + b² = (a + b)²
- a² − 2ab + b² = (a − b)²
- a² − b² = (a + b)(a − b)
Method 3: Splitting the Middle Term (for x² + bx + c)
Find p, q such that p + q = b and p × q = c.
- x² + 7x + 12 → p + q = 7, p × q = 12 → p = 3, q = 4
- x² + 7x + 12 = (x + 3)(x + 4)
Method 4: Grouping
Sometimes terms can be grouped to find a common factor.
- 2x² + 4x + 3x + 6 = 2x(x + 2) + 3(x + 2) = (x + 2)(2x + 3)
8. Common Mistakes
-
Sign errors in (a − b)²
- (a − b)² = a² − 2ab + b² (the middle term is NEGATIVE)
- NOT a² − 2ab − b² (wrong sign on b²!)
- NOT a² + 2ab − b²
- Triple-check signs.
-
Forgetting middle term
- (x + 3)² ≠ x² + 9 ❌
- (x + 3)² = x² + 6x + 9 ✓
-
(a + b)² ≠ a² + b²
- This is a classic error.
- Always remember the middle term 2ab.
-
Distributing wrong
- x(y + z) = xy + xz (correct)
- x(y + z) = xy + z ❌
-
Factorising backwards
- x² − 9 = (x − 3)(x + 3) (not (x + 9)(x − 1))
9. Mental-Math Power of Identities
Compute 51 × 49
- = (50 + 1)(50 − 1)
- = 50² − 1²
- = 2500 − 1 = 2499
Compute 47²
- = (50 − 3)²
- = 50² − 2(50)(3) + 3²
- = 2500 − 300 + 9 = 2209
Compute 102 × 98
- = (100 + 2)(100 − 2)
- = 100² − 4 = 9996
These algebraic identities turn into mental-math shortcuts!
10. Real-World Applications
Area Calculation
A square plot of side (x + 5) m has area (x + 5)² = x² + 10x + 25 m². Useful in real-estate planning.
Physics
Kinematic equations use the identity (a + b)²:
- s = ut + ½at² uses these expansions implicitly.
Engineering
- Stress and strain calculations
- Structural design uses polynomial expansions
- Signal processing decomposes signals using identities
Computing
- Fast multiplication algorithms use the identity (a+b)(a−b) = a² − b²
- Karatsuba's algorithm (used in libraries) is based on similar tricks
11. Historical Context
Brahmagupta's Identity
The Indian mathematician Brahmagupta (7th century CE) developed identities for products of binomials. His 'Brahma-Sphuta-Siddhanta' contained many algebraic results.
Lilavati
Bhaskara II's 'Lilavati' (12th c. CE) had numerous problems involving binomial multiplication, often disguised as poems and stories.
Modern Influence
European mathematicians (15th-17th c.) learnt algebraic identities through Arabic translations of Indian texts. Al-Khwarizmi's work (9th c.) propagated these to Europe.
The identities you learn today are part of an unbroken chain of mathematical heritage stretching back over 1,500 years.
12. Tips for Mastery
For Identities
- Write each identity 10 times until you can reproduce them from memory
- Practise applying each identity to 5 examples
- Practise BACKWARDS: given x² + 6x + 9, factorise to (x + 3)²
For Computation
- Whenever you see a number near a multiple of 10 or 100, try identities:
- 102 = 100 + 2
- 97 = 100 − 3
- 51 × 49 = (50 + 1)(50 − 1)
For Factorisation
- First, find common factors
- Second, check if it's a perfect square trinomial
- Third, check if it's a difference of squares
- Fourth, try splitting the middle term
13. Conclusion
'We Distribute Yet Things Multiply' bridges arithmetic and algebra. The distributive property and the algebraic identities are tools you'll use in:
- Quadratic equations (Class 10)
- Polynomial calculus (Class 11+)
- Coordinate geometry (Class 9+)
- Physics and engineering problems
Master these identities now, and the rest of algebra becomes much easier. The visual area-method (rectangle decomposition) gives you a geometric intuition for what algebra is doing — never forget that algebra and geometry are two sides of the same coin.
