**NCERT Class 9 Maths Lab Manual – Verify the Algebraic Identity (a+b)³ = a³+b³+ 3a²b + 3ab²**

**OBJECTIVE**

To verify the algebraic identity (a+b)³ = a³+b³+ 3a²b + 3ab².

**Materials Required**

- Acrylic sheets
- Adhesive/Adhesive tape
- Scissors
- Geometry Box
- Cutter

**Prerequisite Knowledge**

- Concept of cuboid and its volume.
- Concept of cube and its volume.

**Theory**

- Cuboid A cuboid is a solid bounded by six rectangular plane surfaces, e.g. Match box, brick, box, etc., are cuboid, (see Fig. 7.1)

**Properties of cuboid are**- In a cuboid, there are 6 faces, 12 edges and 8 corners (four at bottom and four at top face) which are called vertices.
- Opposite faces of a cuboid are equal and parallel.
- The line segment joining the opposite vertices of cuboid is called the diagonal of a cuboid.
- There are four diagonals in a cuboid which are equal in length.

Volume of cuboid = lbh

where, l = length, b = breadth and h = height

- Cube A cuboid whose length, breadth and height are same, is called a cube, (see Fig. 7.2)

**Properties of cube are**- In a cube, there are 6 faces, 12 edges and 8 corners (four at bottom and four at top face) which are called vertices.
- All the six faces of a cube are congruent square faces.
- Each edge of a cube have same length.

Volume of cube = a³

where, a is side of cube.

**Procedure**

- Cut six squares of equal side a units from acrylic sheet. Paste all of them to form a cube by using adhesive tape/adhesive, (see Fig. 7.3)

- Cut six squares of equal side b units (b < a) from acrylic sheet. Paste all of them to form a cube by using adhesive tape/adhesive, (see Fig. 7.4)

- Also, cut 12 rectangles of length b units and breadth a units and 6 squares of side a units. Paste all of them to form a cuboid, (see Fig. 7.5)

- Cut 12 rectangles of length a units and breadth b units and 6 squares of side b units. Paste all of them to form a cuboid, (see Fig. 7.6)

- Arrange the cubes obtained in Fig 7.3 and Fig 7.4 and the cuboids obtained in Fig 7.5 and Fig 7.6 as shown in Fig 7.7

**Demonstration**

For Fig. 7.3, volume of cube of side a units = a³

For Fig. 7.4, volume of cube of side b units = b³

For Fig. 7.5, volume of a cuboid of dimensions a x a x b units = a²b

So, volume of all three such cuboids = a²b + a²b + a²b = 3a²b

For Fig. 7.6, volume of a cuboid of dimensions a x b x b units = ab²

So, volume of all three such cuboids = ab² + ab² + ab² = 3ab²

In Fig. 7.7, we have obtained the cube of side (a + b) units.

So, volume of cube = (a + b)³

As, volume of cube of Fig. 7.7 = (Volume of cube of Fig. 7.3) + (Volume of cube of Fig. 7.4) + (Volume of three cuboids of Fig. 7.5) + (Volume of three cuboids of Fig. 7.6)

=> (a+b)³ = a³+b³+ 3a²b + 3ab²

Here, volume is in cubic units.

**Observation**

On actual measurement, we get

a =…….. , b = …….. ,

So, a³ =…….. , b³ = …….. ,

a2b = …….. , 3a²b = …….. ,

ab² = …….. , 3ab² = …….. ,

(a + b)³ = …….. ,

Hence, (a+b)³ = a³+b³+ 3a²b + 3ab²

**Result**

The algebraic identity (a+b)³ = a³+b³+ 3a²b + 3ab² has been verified.

**Application**

The identity is useful for

- calculating the cube of a number which can be expressed as the sum of two convenient numbers.
- simplification and factorization of algebraic expressions.

**Viva Voce**

**Question 1:**

Is (a + b)³ a trinomial?

**Answer:**

No, because (a + b)³ has four terms.

**Question 2:**

What is the degree of polynomial (x + 2y)³ ?

**Answer:**

3, because the highest power of variable in the expansion of (x + 2 y)³ will be 3.

**Question 3:**

In the identity of (a + b)³, what do you mean by a³ and 3a²b?

**Answer:**

a³ means volume of cube of side a and 3a²b means volume of three cuboids of dimensions a, a and b.

**Question 4:**

What is the maximum number of zeroes that a cubic polynomial can have?

**Answer:**

Three

**Question 5:**

What is the expanded form of (a + b)³ ?

**Answer:**

(a+b)³ = a³+b³+ 3a²b + 3ab²

**Question 6:**

For evaluating (101)³, which formula we should use?

**Answer:**

We should use (a+b)³ = a³+b³+ 3a²b + 3ab² by taking a = 100 and b = 1

**Suggested Activity**

Verify that (a+b)³ = a³+b³+ 3a²b + 3ab² by taking x = 11 units, y- 3 units.

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