Observe the figure representing (a + b)². Which area corresponds to the term '2ab'?

The two identical rectangles with sides 'a' and 'b'

The entire large square of side (a+b)

Consider the geometric representation of (a + m)(b + n). Which region represents the product 'an'?

The top-right rectangle with sides 'a' and 'n'

The top-left rectangle with sides 'a' and 'b'

The top-right rectangle with sides 'a' and 'n'

The bottom-left rectangle with sides 'm' and 'b'

The bottom-right rectangle with sides 'm' and 'n'

In the geometric proof of (a + b)(a - b) = a² - b², the area of the square of side 'a' minus the area of the square of side 'b' is rearranged into which shape?

A rectangle with sides (a+b) and (a-b)

A rectangle with sides 'a' and 'b'

A rectangle with sides (a+b) and (a-b)

Two squares of sides 'a' and 'b'

Assertion (A): The expression (x + 3)(x - 3) simplifies to x² - 9. Reason (R): This is an application of the identity (a + b)(a - b) = a² - b².

A and R both correct, R explains A

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The main 'Game' mode of this quiz features 40 questions covering various aspects of the chapter, divided into easy, medium, and hard difficulty levels.

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What topics does this quiz cover?

This quiz covers the distributive property, expansion of products, special algebraic identities like (a+b)², (a-b)², and (a+b)(a-b), and their applications in simplifying expressions and quick calculations.

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Home › CBSE › Class 8 › Maths › WE DISTRIBUTE, YET THINGS MULTIPLY

CBSE · Class 8 · 🧮 Maths · Chapter 6

WE DISTRIBUTE, YET THINGS MULTIPLY

Distributive PropertyAlgebraic IdentitiesExpansion of ProductsSquaring BinomialsDifference of SquaresLike Terms

This chapter introduces students to the fundamental concept of the distributive property of multiplication over addition and subtraction. It delves into expanding products of algebraic expressions, leading to the derivation and application of key algebraic identities: (a+b)² = a² + 2ab + b², (a-b)² = a² - 2ab + b², and (a+b)(a-b) = a² - b². Students learn to use these identities for simplifying complex expressions, performing fast calculations, and understanding patterns in numbers. The chapter emphasizes the importance of algebra in representing general mathematical statements and solving problems.

Distributive Property ka Extension

Distributive property humein algebraic expressions ko expand karne mein help karti hai. Iska basic form hai a(b + c) = ab + ac. Ab hum isko thoda aur extend karte hain:

Single term ko multiple terms se multiply karna:
a(b + c + d) = ab + ac + ad
Example: 3x(2y - 5z + 1) = 6xy - 15xz + 3x

Two binomials ka product:
Jab do binomials (a + b) aur (c + d) ko multiply karte hain, toh har term ko dusre bracket ke har term se multiply karte hain.
(a + b)(c + d) = a(c + d) + b(c + d)
= ac + ad + bc + bd
Is method ko FOIL method bhi kehte hain (First, Outer, Inner, Last), but distributive property ka extension hi hai ye.

Integer values ke liye bhi valid:
Ye properties positive aur negative integers dono ke liye valid hain.
Example: (x + 1)(y - 1) = xy - x + y - 1

Product mein change:
Agar ab mein a ko m se aur b ko n se increase karte hain, toh naya product (a + m)(b + n) hota hai.
(a + m)(b + n) = ab + an + bm + mn
Increase = an + bm + mn
Agar decrease ho toh sign change ho jaayega. Jaise (a - m)(b + n) = ab + an - bm - mn

Like terms:
Terms jinke letter-numbers (variables aur unki powers) same hote hain, unhe like terms kehte hain.
Like terms ko hi add ya subtract kar sakte hain.
Example: 3x^2y aur 5x^2y like terms hain, 3x^2y + 5x^2y = 8x^2y.
3x^2y aur 5xy^2 like terms nahi hain.

Expansion Steps:

Har term ko dusre expression ke har term se multiply karo.
Signs ka dhyaan rakho (integer multiplication rules).
Like terms ko identify karo aur combine karo.
Final expression ko simplest form mein likho.

❗Important

Distributive Property: a(b + c) = ab + ac

This is the fundamental rule for multiplying a monomial by a polynomial. Isko extend karke hum kisi bhi polynomial ko multiply kar sakte hain.

💡Tip

Jab bhi expressions ko expand karo, signs ka bohot dhyaan rakho, especially negative terms ke saath. (-a) b = -ab aur (-a) (-b) = ab.

Special Algebraic Identities

Kuch products aise hote hain jo baar-baar aate hain aur unke liye special formulas ya identities bana di gayi hain. Ye identities calculations ko fast aur accurate banati hain.

Identity 1: Square of a Sum

(a + b)^2 = a^2 + 2ab + b^2
Explanation: (a + b)^2 = (a + b)(a + b)
Using distributive property: a(a + b) + b(a + b)
= a^2 + ab + ba + b^2
= a^2 + 2ab + b^2 (kyunki ab = ba)
Geometric Interpretation: Ek square jiske side ki length (a + b) hai, uska area (a + b)^2 hota hai. Is square ko a^2, b^2 aur do ab rectangles mein divide kar sakte hain.

Identity 2: Square of a Difference

(a - b)^2 = a^2 - 2ab + b^2
Explanation: (a - b)^2 = (a - b)(a - b)
Using distributive property: a(a - b) - b(a - b)
= a^2 - ab - ba + b^2
= a^2 - 2ab + b^2 (kyunki ab = ba)
Alternative Derivation: (a - b)^2 = (a + (-b))^2
Identity 1 use karke: a^2 + 2a(-b) + (-b)^2
= a^2 - 2ab + b^2

Identity 3: Product of Sum and Difference

(a + b)(a - b) = a^2 - b^2
Explanation: (a + b)(a - b)
Using distributive property: a(a - b) + b(a - b)
= a^2 - ab + ba - b^2
= a^2 - b^2 (kyunki -ab + ba = 0)
Geometric Interpretation: Ek square a^2 mein se ek chhota square b^2 remove karne par jo area bachta hai, usko (a + b) aur (a - b) sides wale rectangle ke area ke barabar dikhaya ja sakta hai.

General Identity for Binomial Products:

(x + a)(x + b) = x^2 + (a + b)x + ab
Explanation: (x + a)(x + b)
Using distributive property: x(x + b) + a(x + b)
= x^2 + xb + ax + ab
= x^2 + (a + b)x + ab
Ye identity tab useful hai jab binomials mein ek term common ho.

Important Note: Ye identities sirf a aur b ke liye hi nahi, kisi bhi algebraic expression ke liye valid hain. Jaise (2x + 3y)^2 mein a = 2x aur b = 3y le sakte hain.

🧮Formula

Key Identities:

(a + b)^2 = a^2 + 2ab + b^2
(a - b)^2 = a^2 - 2ab + b^2
(a + b)(a - b) = a^2 - b^2
(x + a)(x + b) = x^2 + (a + b)x + ab

🚧Misconception

Students often make mistakes like (a + b)^2 = a^2 + b^2 ya (a - b)^2 = a^2 - b^2. Middle term 2ab ko bhoolna nahi hai! Ye bahut common mistake hai.

Identities ka Application

Identities sirf algebraic expressions ko simplify karne ke liye nahi, balki complex numerical calculations ko bhi easy banane ke liye use hoti hain.

Fast Multiplication:
Numbers ending in 11, 101, etc.:
3874 × 11 = 3874 × (10 + 1) = 38740 + 3874 = 42614
5678 × 101 = 5678 × (100 + 1) = 567800 + 5678 = 573478
Using (a + b)^2:
104^2 = (100 + 4)^2 = 100^2 + 2(100)(4) + 4^2 = 10000 + 800 + 16 = 10816
37^2 = (30 + 7)^2 = 30^2 + 2(30)(7) + 7^2 = 900 + 420 + 49 = 1369
Using (a - b)^2:
99^2 = (100 - 1)^2 = 100^2 - 2(100)(1) + 1^2 = 10000 - 200 + 1 = 9801
58^2 = (60 - 2)^2 = 60^2 - 2(60)(2) + 2^2 = 3600 - 240 + 4 = 3364
Using (a + b)(a - b):
98 × 102 = (100 - 2)(100 + 2) = 100^2 - 2^2 = 10000 - 4 = 9996
45 × 55 = (50 - 5)(50 + 5) = 50^2 - 5^2 = 2500 - 25 = 2475

Simplifying Complex Expressions:
Identities ka use karke bade expressions ko chhota kiya ja sakta hai.
Example: (x + 5)^2 - (x - 5)^2
= (x^2 + 10x + 25) - (x^2 - 10x + 25)
= x^2 + 10x + 25 - x^2 + 10x - 25
= 20x

Finding Area/Perimeter:
Geometrical shapes ke area ya perimeter nikalne mein bhi identities help karti hain.
Agar ek square ki side (m + n) hai, toh uska area (m + n)^2 hoga.
Agar ek square ki side (n - m) hai, toh uska area (n - m)^2 hoga.
Example: Shaded region ka area (jahan total area (m+n)^2 hai aur 4 rectangles mn each remove kiye hain) (m+n)^2 - 4mn = m^2 + 2mn + n^2 - 4mn = m^2 - 2mn + n^2 = (m-n)^2.

Checking Equality of Expressions:
Agar do expressions same pattern ko describe karte hain, toh unhe simplify karke check kar sakte hain ki kya wo equivalent hain.
Example: (k + 1)^2 - 1 aur k(k + 2) dono k^2 + 2k ke barabar hote hain.

💡Tip

Numerical calculations mein identities ka use karne se steps kam ho jaate hain aur calculation errors ke chances bhi kam ho jaate hain. Practice these a lot!

⭐Remember

Jab bhi (a + b)(a - b) form dikhe, turant a^2 - b^2 apply karo. Ye bahut time bachata hai.

Algebraic Patterns aur Problem Solving

Mathematics mein patterns ko samajhna aur unhe algebraic expressions mein convert karna ek important skill hai. Identities ismein bahut help karti hain.

Pattern Recognition:
Numerical sequences ya geometric arrangements mein hidden patterns ko observe karna.
Example: 2(2^2 + 1^2) = 3^2 + 1^2
Is pattern ko generalize kiya ja sakta hai 2(a^2 + b^2) = (a + b)^2 + (a - b)^2.
Proof: (a + b)^2 + (a - b)^2 = (a^2 + 2ab + b^2) + (a^2 - 2ab + b^2) = 2a^2 + 2b^2 = 2(a^2 + b^2).

Calendar Patterns:
Calendar mein 2x2 square ke numbers ke diagonal products mein pattern observe karna.
Agar top-left number a hai, toh square ke numbers honge:

a, a+1 a+7, a+8

Diagonal products: a(a+8) aur (a+1)(a+7)
a(a+8) = a^2 + 8a
(a+1)(a+7) = a^2 + 7a + a + 7 = a^2 + 8a + 7
Observation: (a+1)(a+7) is always 7 more than a(a+8).

Multiple Methods for Same Problem:
Ek hi problem ko solve karne ke kai tareeke ho sakte hain. Algebra ka use karke check kar sakte hain ki kya different methods se same result mil raha hai.
Example: Shaded area problem (as discussed in NCERT section 6.4)
Method 1: (m + n)^2 - 4mn
Method 2: (n - m)^2
Both simplify to m^2 - 2mn + n^2, proving they are equivalent.

Problem Solving Strategy:

Problem ko carefully read karo aur samajho.
Variables (letter-numbers) assign karo unknown quantities ko.
Algebraic expressions form karo given conditions ke according.
Identities ya distributive property ka use karke expressions ko simplify ya solve karo.
Result ko check karo.

❗Important

Algebraic identities sirf numbers ke liye hi nahi, balki kisi bhi algebraic expression ke liye true hoti hain. Ye unki universality hai.

💡Tip

Calendar patterns ya number patterns wale questions HOTS (Higher Order Thinking Skills) category mein aate hain. Inko solve karne ke liye algebraic representation bahut zaroori hai.

Easy (15)

Which property states that a(b + c) = ab + ac?
- Commutative property
- Associative property
- Distributive property (correct)
- Identity property
Why: The property a(b + c) = ab + ac is known as the distributive property of multiplication over addition.
What is the expanded form of (a + b)²?
- a² + b²
- a² - b²
- a² + 2ab + b² (correct)
- a² - 2ab + b²
Why: The identity for the square of a sum is (a + b)² = a² + 2ab + b².
What is the expanded form of (a - b)²?
- a² + b²
- a² - b²
- a² + 2ab + b²
- a² - 2ab + b² (correct)
Why: The identity for the square of a difference is (a - b)² = a² - 2ab + b².
What is the expanded form of (a + b)(a - b)?
- a² + b²
- a² - b² (correct)
- a² + 2ab + b²
- a² - 2ab + b²
Why: The identity for the product of a sum and a difference is (a + b)(a - b) = a² - b².
Terms having the same letter-numbers are called what?
- Unlike terms
- Constant terms
- Like terms (correct)
- Variable terms
Why: Terms with identical variable parts (letter-numbers) are defined as like terms.
What is the result of expanding (x + 3)(y + 2)?
- xy + 6
- xy + 2x + 3y + 6 (correct)
- xy + 3x + 2y + 6
- x + y + 5
Why: Using the distributive property, (x + 3)(y + 2) expands to xy + 2x + 3y + 6.
Which of the following is an identity?
- x + 5 = 10
- 2x = 8
- (a + b)² = a² + 2ab + b² (correct)
- x² - 4 = 0
Why: An identity is an equation that is true for all possible values of its variables.
The product of (x + 1) and (x + 2) is x² + 3x + 2. Is this statement true or false?
Why: Expanding (x+1)(x+2) using the distributive property yields x² + 3x + 2.
The expression (5x)² is equal to 5x². Is this statement true or false?
Why: (5x)² means (5x) multiplied by (5x), which is 25x², not 5x².
The distributive property can be used for fast multiplication. Is this statement true or false?
Why: The distributive property is indeed used for fast multiplication, especially with numbers like 11, 101, etc.
The product of (a + m) and (b + n) is ab + mb + an + ______.
Why: The complete expansion of (a + m)(b + n) is ab + mb + an + mn.
The value of 104² can be calculated using the identity (a + b)² by taking 'a' as 100 and 'b' as ______.
Why: 104 can be written as 100 + 4, so b = 4 when using (a+b)².
The product of (x + 5) and (x - 5) is x² - ______.
Why: Using the identity (a+b)(a-b) = a² - b², (x+5)(x-5) = x² - 5² = x² - 25.
The expansion of (2x + 3)² is 4x² + 12x + ______.
Why: Using (a+b)² = a² + 2ab + b², where a=2x and b=3, the last term is 3² = 9.
The product of two numbers a and b increases by a + b + 1 when each of a and b are increased by 1. Is this statement true or false?
Why: The new product is (a+1)(b+1) = ab + a + b + 1, so the increase is a + b + 1.

Medium (15)

Expand the expression (3x + 4y)². Which of the following is the correct expansion?
- 9x² + 16y²
- 9x² + 12xy + 16y²
- 9x² + 24xy + 16y² (correct)
- 9x² + 6xy + 16y²
Why: Using (a+b)² = a² + 2ab + b², with a=3x and b=4y, gives 9x² + 24xy + 16y².
Simplify the expression (5p - 2q)². Which of the following is the correct simplification?
- 25p² - 4q²
- 25p² - 10pq + 4q²
- 25p² - 20pq + 4q² (correct)
- 25p² + 20pq + 4q²
Why: Using (a-b)² = a² - 2ab + b², with a=5p and b=2q, gives 25p² - 20pq + 4q².
Evaluate 99² using a suitable identity.
- 9801 (correct)
- 9901
- 9899
- 10000
Why: 99² can be written as (100 - 1)², which expands to 100² - 2(100)(1) + 1² = 10000 - 200 + 1 = 9801.
Which of the following is equivalent to (2x + 3)(2x - 3)?
- 4x² + 9
- 4x² - 9 (correct)
- 4x² + 12x + 9
- 4x² - 12x + 9
Why: Using the identity (a+b)(a-b) = a² - b², (2x+3)(2x-3) = (2x)² - 3² = 4x² - 9.
If the product of two numbers a and b is ab, and 'a' is increased by 1 while 'b' is decreased by 1, what is the new product?
- ab + a - b - 1
- ab + b - a - 1 (correct)
- ab + a + b - 1
- ab - a - b + 1
Why: The new product is (a+1)(b-1) = ab - a + b - 1, which can be written as ab + b - a - 1.
Match the algebraic expressions with their expanded forms.
Why: Each expression matches its correct expansion based on algebraic identities and distributive property.
Match the historical figures with their contributions to the distributive property.
Why: Brahmagupta explicitly stated the property, while Euclid and Āryabhaṭa used it implicitly. Sridharacharya developed a quick squaring method.
Match the operations with their results when applied to like terms.
Why: Like terms can be added/subtracted by combining coefficients. Multiplication/division involves both coefficients and variables.
What is the term for mathematical statements that express the equality of two algebraic expressions, true for all variable values?
Why: Such statements are called identities, as they hold true universally for their variables.
What is the name given to the property a(b + c) = ab + ac?
Why: The property a(b + c) = ab + ac is the distributive property.
Arrange the steps to expand (x + 2)(x + 3) using the distributive property.
Why: The steps involve applying the distributive property twice, then combining like terms to simplify.
Arrange the steps to simplify the expression 2(x - 1) + 3(x + 4).
Why: First distribute, then group and combine like terms to simplify the expression.
Observe the figure representing (a + b)². Which area corresponds to the term '2ab'?
- The large square of side 'a'
- The small square of side 'b'
- The two identical rectangles with sides 'a' and 'b' (correct)
- The entire large square of side (a+b)
Why: The two rectangles in the (a+b)² diagram each have area 'ab', so together they represent '2ab'.
Consider the geometric representation of (a + m)(b + n). Which region represents the product 'an'?
- The top-left rectangle with sides 'a' and 'b'
- The top-right rectangle with sides 'a' and 'n' (correct)
- The bottom-left rectangle with sides 'm' and 'b'
- The bottom-right rectangle with sides 'm' and 'n'
Why: In the (a+m)(b+n) diagram, the top-right rectangle has dimensions 'a' by 'n', representing 'an'.
In the geometric proof of (a + b)(a - b) = a² - b², the area of the square of side 'a' minus the area of the square of side 'b' is rearranged into which shape?
- A square of side (a+b)
- A rectangle with sides 'a' and 'b'
- A rectangle with sides (a+b) and (a-b) (correct)
- Two squares of sides 'a' and 'b'
Why: The geometric proof of a² - b² involves rearranging the remaining area into a rectangle of dimensions (a+b) by (a-b).

Hard (10)

If (x + 1/x)² = 25, find the value of x² + 1/x².
Why: Expand (x + 1/x)² and use the given value to find x² + 1/x².
If a - b = 5 and ab = 6, what is the value of a² + b²?
Why: Use the identity (a - b)² = a² - 2ab + b² and substitute the given values.
Calculate the value of 101 × 99 using an algebraic identity.
Why: Use the identity (a+b)(a-b) = a² - b² by writing 101 as (100+1) and 99 as (100-1).
If (a + b) = 7 and (a - b) = 3, what is the value of a² - b²?
Why: Use the identity (a + b)(a - b) = a² - b² and substitute the given values.
Assertion (A): The expression (x + 3)(x - 3) simplifies to x² - 9. Reason (R): This is an application of the identity (a + b)(a - b) = a² - b².
Why: The assertion is correct, and the reason accurately explains the identity used for simplification.
Assertion (A): The expansion of (2x + y)² is 4x² + y². Reason (R): The square of a sum (a + b)² is a² + b².
Why: Both the assertion and the reason are incorrect. (2x+y)² is 4x²+4xy+y², and (a+b)² is a²+2ab+b².
A rectangular garden has a length of (3x + 2) meters and a width of (x + 5) meters. If a square flower bed of side 'x' meters is built inside the garden, what is the area of the remaining garden space?
- 2x² + 17x + 10 (correct)
- 3x² + 17x + 10
- 2x² + 10
- 3x² + 10
Why: Total garden area is (3x+2)(x+5). Subtract the flower bed area x² to get the remaining area.
A square plot has a side length of (5a - 3b) units. If a smaller square region of side (2a - b) units is removed from it, what is the area of the remaining plot?
- 21a² - 26ab + 8b² (correct)
- 21a² - 20ab + 8b²
- 29a² - 36ab + 10b²
- 21a² - 18ab + 10b²
Why: Calculate the area of the large square and subtract the area of the small square using (a-b)² identity.
Consider the figure showing a large square of side (m + n) with four congruent rectangles of dimensions m × n at the corners, leaving an interior shaded square. What is the area of this interior shaded square?
- (m + n)² - mn
- (m + n)² - 2mn
- (m + n)² - 4mn (correct)
- (m - n)² (correct)
Why: The area of the interior square is the total area (m+n)² minus the area of the four rectangles (4mn), which simplifies to (m-n)². Both options are correct.
A square of side 'x' has a smaller square of side 'y' removed from its corner. The remaining L-shaped region is then cut and rearranged to form a rectangle. What are the dimensions of this new rectangle?
- x by y
- (x + y) by (x - y) (correct)
- x by (x - y)
- (x - y) by (x - y)
Why: The geometric proof of x² - y² shows that the remaining area can be rearranged into a rectangle with sides (x+y) and (x-y).