Class 9 Maths Chapter 4: Exploring Algebraic Identities Concept Infographic (Ganita Manjari)

← Back to Class List

✦ NCF 2023 ALIGNED • VISUAL LEARNING SERIES ✦

Chapter 4: Exploring Algebraic Identities

Discover the visual geometry behind algebraic structures, mapping variable multiplication to spatial realities.

📖 Core Vocabulary

Algebraic Identity:
An algebraic identity is an equation that is true for all values of the variables occurring in it. (Unlike a normal equation which is only true for specific values).

Expansion vs. Factorisation:
While expansion multiplies factors into a sum, factorisation resolves a polynomial sum back into a product of simpler linear expressions.

⭐ Key Properties & Visual Proofs

• Geometric Area Preservation: Algebraic identities can be visualised using geometrical models where squares and rectangles represent terms. The total partitioned area equals the area of the whole.
• Trinomial Partitioning: A square of side (a + b + c) partitions into exactly 9 pieces: 3 squares (a², b², c²) and 6 rectangles (2ab, 2bc, 2ca).
• Volume Decomposition: A cube of edge (a + b) can be split into 8 parts: two smaller cubes (a³, b³) and six cuboids (three of a²b and three of ab²).

📐 Essential Formulas & Geometric Identities

Square of a Binomial

(a + b)² = a² + 2ab + b²

Square of a Trinomial

(a+b+c)² = a²+b²+c²+2ab+2bc+2ca

Difference of Squares

a² − b² = (a + b)(a − b)

🧩 Visual Factorisation Algorithm (Using Algebra Tiles)

Goal: Factorise the quadratic expression x² + 7x + 12.

1

Step 1: Select Your Tiles

To factorise x² + 7x + 12, pick 1 square tile (x² area), 7 rectangular strips (x area), and 12 unit tiles (1 area).

2

Step 2: Split the Middle Term

The 7x in x² + 7x + 12 has to be split as 3x + 4x so that a solid rectangle can be formed.

3

Step 3: Form a Rectangle

Arrange the tiles: place 3 x-tiles on the right side of the x²-tile and 4 x-tiles below it. Place the 12 unit tiles in a 3 by 4 array.

4

Step 4: Measure the Boundaries

Once the rectangular arrangement is formed, we observe that the dimensions are (x + 3) and (x + 4). Thus, the factored form is (x + 3)(x + 4).

⚠️ Common Pitfalls & Exam Traps

❌ Trap 1: The Exponent Distribution Error

Avoid writing (a + b)² as a² + b². Doing so ignores the outer rectangular cross-terms (like 2ab) of the geometric model.

⚠️ Trap 2: Incorrect Signs in Cubes

When expanding (a − b)³, remember that the signs must alternate. Replacing b with −b makes odd powers negative: a³ − 3a²b + 3ab² − b³.

Designed by Akash Srivastva | Mathematics Hub • Visual Learning Series

🚀 Loved this? Get more visual study materials at: www.akashmaths.online