Class 8 Mathematics — Algebraic Expressions and Identities

📝 Section 1 — Classroom Notes

🌎 Real-World Connection: Imagine you buy x notebooks at ₹15 each and y pens at ₹8 each. The total cost is 15x + 8y — an algebraic expression! You do not need to know the exact values of x and y to write this formula. Algebraic expressions let us describe real-life situations using letters, making it easy to calculate for any values. Shopkeepers, scientists, and engineers use them every day.

🔢 Topic 1: What Is an Algebraic Expression?

It is made of terms.
A term has a number and a variable.
The number part is the coefficient.
The letter part is the variable.
Example: 3x + 5y − 2 has 3 terms.
A constant is a term with no variable.
In 3x + 5y − 2, the constant is −2.

🔢 Topic 2: Types of Expressions

Monomial = only 1 term. Example: 5x²
Binomial = exactly 2 terms. Example: 3x + 7
Trinomial = exactly 3 terms.
Example of trinomial: x² + 2x + 1
Polynomial = one or more terms.
All the above are polynomials.
Count the terms to find the type.

🔢 Topic 3: Like and Unlike Terms

Like terms have the same variable part.
Example: 3x and 7x are like terms.
Example: 5x² and −2x² are like terms.
Unlike terms have different variable parts.
Example: 3x and 4y are unlike.
Example: 2x and 2x² are unlike.
We can only add or subtract like terms.

🔢 Topic 4: Adding Algebraic Expressions

Write the expressions one below the other.
Line up the like terms in columns.
Add the coefficients of like terms.
Keep the variable part the same.
Example: 3x + 5 added to 2x + 4.
Like terms: 3x + 2x = 5x.
Constants: 5 + 4 = 9. Answer: 5x + 9.

🔢 Topic 5: Subtracting Algebraic Expressions

Change the signs of the second expression.
Then add like terms as before.
+ becomes − and − becomes +.
Example: (5x + 3) − (2x + 1).
Change signs: −2x − 1.
Add: 5x − 2x = 3x.
Constants: 3 − 1 = 2. Answer: 3x + 2.

🔢 Topic 6: Monomial × Monomial

Multiply the coefficients (numbers).
Multiply the variables using exponent rules.
Rule: xᵃ × xᵇ = xᵃ⁺ᵇ
Example: 3x × 4x = 12x².
Example: 2x² × 5x³ = 10x⁵.
Example: (−3a) × (4b) = −12ab.
Multiply numbers first, then letters.

🔢 Topic 7: Monomial × Polynomial

Use the distributive property.
Multiply the monomial with each term.
a × (b + c) = ab + ac.
Example: 2x × (3x + 4).
= 2x × 3x + 2x × 4.
= 6x² + 8x.
Do each multiplication one at a time.

🔢 Topic 8: Polynomial × Polynomial

Multiply each term of the first
with every term of the second.
Then add all the results together.
Combine any like terms at the end.
Example: (x + 2)(x + 3).
= x² + 3x + 2x + 6.
= x² + 5x + 6.

🔢 Topic 9: Identity (a + b)²

This is a standard identity.
(a + b)² = (a + b)(a + b).
= a² + ab + ab + b².
= a² + 2ab + b².
Example: (x + 3)² = x² + 6x + 9.
Here a = x and b = 3.
Always remember the middle term 2ab.

(a + b)² = a² + 2ab + b²

🔢 Topic 10: Identity (a − b)²

This is another standard identity.
(a − b)² = (a − b)(a − b).
= a² − ab − ab + b².
= a² − 2ab + b².
Example: (x − 4)² = x² − 8x + 16.
Here a = x and b = 4.
The middle term is −2ab (negative).

(a − b)² = a² − 2ab + b²

🎨 Section 2 — Easy Illustrations

Picture 1: Parts of an Expression

Every expression is made of terms. Each term has a coefficient and a variable.

Picture 2: Like vs Unlike Terms

Only like terms can be added or subtracted. Check the variable part!

Picture 3: Area Model for (a + b)²

The big square has 4 parts. Add them all to get the identity!

Picture 4: Area Model for (a − b)²

Start with the big square a². Remove two strips of ab. Add back the b² overlap.

Picture 5: Multiplication Grid

Use a grid to multiply each term. Then add all the products together.

✏️ Section 3 — Worked Examples

Example 1: Adding Two Expressions

Question: Add (3x² + 5x − 2) and (4x² − 3x + 7).

1 Write both expressions. Line up like terms.

2 x² terms: 3x² + 4x² = 7x².

3 x terms: 5x + (−3x) = 5x − 3x = 2x.

4 Constants: −2 + 7 = 5.

5 Put it all together.

7x² + 2x + 5

Always add only like terms! ✓

Example 2: Multiplying Two Binomials

Question: Multiply (2x + 3)(x − 4).

1 Multiply each term of the first with each term of the second.

2 2x × x = 2x².

3 2x × (−4) = −8x.

4 3 × x = 3x.

5 3 × (−4) = −12.

6 Add all: 2x² − 8x + 3x − 12.

7 Combine like terms: −8x + 3x = −5x.

2x² − 5x − 12

Example 3: Using the Identity (a + b)²

Question: Expand (5p + 3)² using the identity.

1 Use the identity: (a + b)² = a² + 2ab + b².

2 Here a = 5p and b = 3.

3 a² = (5p)² = 25p².

4 2ab = 2 × 5p × 3 = 30p.

5 b² = 3² = 9.

6 Put it together: 25p² + 30p + 9.

(5p + 3)² = 25p² + 30p + 9

Always find a², 2ab, and b² separately! ✓

Term	What It Means	Key Fact
Expression	Terms joined by + or −	3x + 5y − 2
Monomial	Just 1 term	5x², 7y, −3
Binomial	Exactly 2 terms	3x + 7
Trinomial	Exactly 3 terms	x² + 2x + 1
Like Terms	Same variable part	3x and 7x
Unlike Terms	Different variable part	3x and 4y
Coefficient	Number in front	In 5x, it is 5
Constant	Term with no variable	−7, 4, 0
(a+b)²	Square of a sum	a² + 2ab + b²
(a−b)²	Square of a difference	a² − 2ab + b²
Distributive	a(b+c) = ab + ac	Multiply each term
Exponent Rule	xᵃ × xᵇ = xᵃ⁺ᵇ	Add the powers

📄 Section 5 — Take-Home Handout

📝 A. Fill in the Blanks

1. An expression with one term is called a _________.
2. In 7x², the coefficient is _________.
3. 3x and 5x are _________ terms.
4. (a + b)² = a² + _________ + b².
5. When we subtract, we change the _________ of the second expression.

✓ B. True or False

1. 5x and 5x² are like terms. _______
2. A binomial has exactly two terms. _______
3. (a − b)² = a² − b². _______
4. 3x × 4x = 12x². _______
5. A constant term has a variable. _______

✏️ C. Solve These

1. Add: (5x² + 3x − 1) and (2x² − 7x + 4).

Show your work here

2. Expand (3m + 2)² using the identity.

Show your work here

🔗 D. Match the Columns

Column A	Column B
1. 5x²	a. Trinomial
2. 3x + 7	b. Monomial
3. x² + 2x + 1	c. a² − 2ab + b²
4. (a − b)²	d. Binomial

🎨 E. Multiply These

Find the product: (x + 5)(x − 2).

Show your steps here

✅ Detailed Solutions — Section 5 Handout

A. Fill in the Blanks — Answers

1. An expression with one term is called a monomial. (e.g. 5x or 3x².)

2. In 7x², the coefficient is 7. (The number in front of the variable.)

3. 3x and 5x are like terms. (Same variable x to the same power.)

4. (a + b)² = a² + 2ab + b². (Standard identity.)

5. When we subtract, we change the sign of the second expression.

B. True or False — Answers

1. 5x and 5x² are like terms. — False. Like terms need the same power. 5x has power 1; 5x² has power 2.

2. A binomial has exactly two terms. — True. (e.g. 3x + 7. ✓)

3. (a − b)² = a² − b². — False. (a − b)² = a² − 2ab + b². The identity a² − b² = (a+b)(a−b).

4. 3x × 4x = 12x². — True. (3 × 4 = 12, x × x = x². ✓)

5. A constant term has a variable. — False. A constant (like 5 or −3) has no variable.

C1. Add: (5x² + 3x − 1) and (2x² − 7x + 4)

1 Group like terms together.

2 x² terms: 5x² + 2x² = 7x².

3 x terms: 3x − 7x = −4x.

4 Constants: −1 + 4 = 3.

5 Result: 7x² − 4x + 3.

7x² − 4x + 3

C2. Expand (3m + 2)² using the identity

1 Use identity: (a + b)² = a² + 2ab + b².

2 Here a = 3m and b = 2.

3 a² = (3m)² = 9m².

4 2ab = 2 × 3m × 2 = 12m.

5 b² = 2² = 4.

6 Result: 9m² + 12m + 4.

(3m + 2)² = 9m² + 12m + 4

D. Match the Columns — Answers

1. 5x² → b. Monomial (one term).

2. 3x + 7 → d. Binomial (two terms).

3. x² + 2x + 1 → a. Trinomial (three terms).

4. (a − b)² → c. a² − 2ab + b² (standard identity).

E. Find the product: (x + 5)(x − 2)

1 Multiply each term in the first bracket by each term in the second (FOIL method).

2 x × x = x².

3 x × (−2) = −2x.

4 5 × x = 5x.

5 5 × (−2) = −10.

6 Combine: x² − 2x + 5x − 10 = x² + 3x − 10.

(x + 5)(x − 2) = x² + 3x − 10

🃏 Section 6 — Revision Flashcards

👉

🔢

What is an algebraic expression?

A combination of terms joined by + or − signs. Each term has a number and a variable.

🔎

What is a monomial?

An expression with only one term.

Examples: 5x, 7y², −3ab

📈

What are like terms?

Terms that have the same variable part. Only like terms can be added or subtracted.

3x and 7x are like. 3x and 3y are not.

🔗

What is the distributive property?

a × (b + c) = ab + ac

Multiply the outside term with each inside term.

❓

What is (a + b)² equal to?

a² + 2ab + b²

The middle term is always +2ab.

💡

What is (a − b)² equal to?

a² − 2ab + b²

The middle term is always −2ab.

🔧

How do you multiply two monomials?

Multiply the numbers together. Then multiply the variables using exponent rules.

3x × 4x = 12x²

🎯

How do you subtract expressions?

Change the signs of the second expression. Then add the like terms.

+ becomes −, and − becomes +.