Class 7 Mathematics – Algebraic Expressions

1. Variables and Constants

🔍 Discovering Variables

✏️ Activity

What is the price of 1 kg of tomatoes at your local bazaar today? Will it be the same price next week?
How many students are present in your class right now? Will the number be exactly the same every day?
Think of any number. Is it the same number your friend is thinking of?

What do you notice? These answers keep changing — the tomato price changes, attendance changes, and everyone thinks of a different number. These are unknown or changing quantities.

✅ Conclusion

In mathematics, when a quantity changes or is unknown, we write it using a letter like $x$, $y$, $a$, or $b$. Because its value can vary, we call it a variable.

📖 Definition — Variable

A variable is a letter that stands for an unknown or changing number. It does not have one fixed value.
Examples: $x,\ y,\ a,\ b,\ m,\ n$

🔍 Discovering Constants

✏️ Activity

How many days are in a week?
How many sides does a triangle have?
If you have a Rs 10 note, will it become Rs 20 tomorrow?

What do you notice? 7 days is always 7 days. A triangle always has 3 sides. The number 10 is always 10. These values never change.

✅ Conclusion

A quantity that never changes is called a constant. Numbers like $3,\ 7,\ -5,\ \frac{1}{2}$ are always constants.

📖 Definition — Constant

A constant is a fixed number whose value never changes.
Examples: $4,\ -9,\ 0,\ \frac{1}{2},\ 100$

⭐ Quick Way to Remember

Variable = letter = value varies | Constant = number = value is constant (fixed)

2. Terms and Algebraic Expressions

What Is a Term?

Think of a term as one complete unit in an algebra sentence — like one word in an English sentence. A term can be:

$7$number only

$x$letter only

$4x$number × letter

Inside a single term, parts are multiplied together — never added or subtracted. The moment you see a $+$ or $-$ sign, you are looking at two separate terms.

Anatomy of a Term — $5x^2$

$5$

Coefficient

$x$

Variable

$2$

Power

📖 Definition — Term

A term is a single building block of algebra. It is a number, a letter, or numbers and letters multiplied together. Parts of one term are never separated by $+$ or $-$.

What Is an Algebraic Expression?

When we join two or more terms using $+$ or $-$, we get an algebraic expression. Think of it as a complete algebra phrase made of several term-circles joined together:

$4x^2$term 1

$-$

$3x$term 2

$+$

$9$term 3

$=\ 4x^2 - 3x + 9$

📖 Definition — Algebraic Expression

An algebraic expression is one or more terms joined by $+$ or $-$ signs.
Simple examples: $x + 5$, $3a - 2$, $x^2 + 4x + 4$

3. Types of Algebraic Expressions

We name an expression by how many terms it has:

Monomial

1 term

$5x$
$-9a^2b$
$24$

Binomial

2 terms

$x + 6$
$5a - 3b$
$8x^2 + y$

Trinomial

3 terms

$x^2 + 4x + 4$
$a - b + 3c$
$5x^2 - 2y + z$

⭐ Memory Trick

Mono = one (like a monocycle has 1 wheel) | Bi = two (like a bicycle has 2 wheels) | Tri = three (like a tricycle has 3 wheels)

4. Multiplication of Algebraic Expressions

Two rules to always remember:

⭐ Key Rules

Rule 1 — Numbers: Multiply the number parts (coefficients) as usual.
Rule 2 — Same letters: Add the powers. $x^m \times x^n = x^{m+n}$
Example: $x^2 \times x^3 = x^5$ (because $2 + 3 = 5$)

A. Monomial × Monomial

Multiply the number parts together, then multiply the letter parts together (add powers for the same letter).

Example 1 — Multiply $2x$ by $3y$

(2x) \times (3y) = (2 \times 3) \times (x \times y) = 6xy

Example 2 — Multiply $3a^2$ by $4a^3$

Numbers: $3 \times 4 = 12$
Same letter $a$: $a^2 \times a^3 = a^{2+3} = a^5$
Answer: $12a^5$

\[(3a^2)(4a^3) = 12a^5\]

Example 3 — Multiply $2xy$ by $5x^2z$

Numbers: $2 \times 5 = 10$
Letter $x$: $x^1 \times x^2 = x^3$
Letters $y$ and $z$: just write $yz$
Answer: $10x^3yz$

\[(2xy)(5x^2z) = 10x^3yz\]

B. Monomial × Binomial — The Distributive Property

The outside term must be multiplied into each term inside the bracket. This is called distributing.

$a(b + c) = ab + ac$

📌 Picture it — $3x(2x + 5)$

$3x$ $\times\Big($ $2x$ $+$ $5$ $\Big)$

$3x \times 2x$

↓

$3x \times 5$

↓

$6x^2$ $+$ $15x$

Example 1 — Multiply $3x$ by $(2x + 5)$

3x(2x + 5) = (3x \times 2x) + (3x \times 5) = 6x^2 + 15x

Example 2 — Multiply $4a$ by $(2a - 3)$

Step 1 — Distribute: $4a \times 2a = 8a^2$ and $4a \times (-3) = -12a$
Step 2 — Write answer: $8a^2 - 12a$

\[4a(2a - 3) = 8a^2 - 12a\]

Example 3 — Multiply $2x^2y$ by $(x - 3y^2)$

2x^2y\,(x - 3y^2) = 2x^3y - 6x^2y^3

C. Monomial × Trinomial

Same idea — distribute the outside term into all three terms inside the bracket.

$a(b + c + d) = ab + ac + ad$

Example 1 — Multiply $2x$ by $(x^2 - 3x + 4)$

$2x \times x^2 = 2x^3$
$2x \times (-3x) = -6x^2$
$2x \times 4 = 8x$

\[2x(x^2 - 3x + 4) = 2x^3 - 6x^2 + 8x\]

Example 2 — Multiply $3a^2$ by $(a^2 - 2a + 1)$

\[3a^2(a^2 - 2a + 1) = 3a^4 - 6a^3 + 3a^2\]

5. Advanced Multiplication — The "Split & Repeat" Method

💡 Big Idea — Split & Repeat

When we multiply two brackets together, we split the first bracket into its individual terms, then repeat the entire second bracket for each term. Finally, we distribute and simplify.

A. Binomial × Binomial

$(a + b)(c + d) = a(c + d) + b(c + d)$

📌 Split & Repeat — $(x + 4)(x + 7)$

Step 1 — Split the first bracket:

$\Big($ $x$ $+$ $4$ $\Big)$ $(x + 7)$

$x$ gets $(x+7)$

↓

$4$ gets $(x+7)$

↓

Step 2 — Distribute each part:

$x(x+7)$ $+$ $4(x+7)$

\[= (x^2 + 7x) + (4x + 28) = x^2 + 11x + 28\]

Example 1 — Multiply $(x + 4)$ by $(x + 7)$

Step 1 — Split & Repeat: $\[x(x + 7) + 4(x + 7)\]$
Step 2 — Distribute: $\[(x^2 + 7x) + (4x + 28)\]$
Step 3 — Collect like terms $(7x + 4x)$: $\[x^2 + 11x + 28\]$

📌 Split & Repeat — $(2x + 3)(x + 5)$

$2x$ $+$ $3$ $\times$ $(x + 5)$

$2x \times (x+5)$

↓

$3 \times (x+5)$

↓

$2x^2 + 10x$ $+$ $3x + 15$

\[= 2x^2 + 13x + 15\]

Example 2 — Multiply $(3x - 5)$ by $(2x + 3)$

Step 1 — Split & Repeat: $\[3x(2x + 3) - 5(2x + 3)\]$
Step 2 — Distribute (careful with the minus sign on $-5$): $\[(6x^2 + 9x) + (-10x - 15)\]$
Step 3 — Collect like terms: $\[6x^2 - x - 15\]$

B. Binomial × Trinomial

Exactly the same method — split the binomial and repeat the trinomial for each part.

$(a + b)(c + d + e) = a(c + d + e) + b(c + d + e)$

📌 Split & Repeat — $(x + 2)(x^2 + 3x + 4)$

$x$ $+$ $2$ $\times$ $(x^2 + 3x + 4)$

$x$ repeats the trinomial

↓

$2$ repeats the trinomial

↓

$x^3 + 3x^2 + 4x$ $+$ $2x^2 + 6x + 8$

\[= x^3 + 5x^2 + 10x + 8\]

Example 1 — Multiply $(x + 3)$ by $(x^2 + 4x + 5)$

Step 1 — Split & Repeat: $\[x(x^2 + 4x + 5) + 3(x^2 + 4x + 5)\]$
Step 2 — Distribute: $\[(x^3 + 4x^2 + 5x) + (3x^2 + 12x + 15)\]$
Step 3 — Collect like terms $(4x^2 + 3x^2)$ and $(5x + 12x)$: $\[x^3 + 7x^2 + 17x + 15\]$

📋 Chapter Summary

Variable	A letter for an unknown/changing value ($x, y, a$)
Constant	A fixed number ($3, -7, 100$)
Term	One unit: $5,\ x,\ 3x^2,\ 4xy$
Expression	Terms joined by $+$ or $-$
Indices Rule	$x^m \times x^n = x^{m+n}$
Split & Repeat	Split first bracket → multiply each part into full second bracket

Algebraic Expressions & Operations