**NCERT Class 9 Maths Lab Manual – Verify the Algebraic Identity a³-b³ = (a-b) (a²+ab+b²)**

**OBJECTIVE**

To verify the algebraic identity a³-b³ = (a-b) (a²+ab+b²).

**Materials Required**

- Acrylic sheet
- Geometry box
- Scissors
- Adhesive/Adhesive tape
- Cutter

**Prerequisite Knowledge**

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

**Theory**

- For concept of cuboid and its volume refer to Activity 7.
- For concept of cube and its volume refer to Activity 7.

**Procedure**

- Using acrylic sheet, make a cuboid of dimensions (a – b) x a x a, where b < a. (see Fig. 10.1)

- Using acrylic sheet, make another cuboid of dimensions (a-b) x a x b, where b < a. (see Fig. 10.2).

- Now, make one more cuboid of dimensions (a-b) x b x b. (see Fig. 10.3)

- Now, make a cube of dimensions b x b x b. (see Fig. 10.4)

- Arrange the cube and cuboids obtained in Fig. 10.1 to 10.4 to form a solid as shown in Fig. 10.5, which is a cube of side a units.

- Now, remove a cube of side b units from the solid obtained in Fig. 10.5, thus we obtain solid as shown in Fig. 10.6.

**Demonstration**

For Fig. 10.1, volume of cuboid = (a-b) x a x a = (a-b)a²

For Fig. 10.2, volume of cuboid = (a-b) x a x b = (a-b)ab

For Fig. 10.3, volume of cuboid = (a – b) x b x b = (a – b)b²

For Fig. 10.4, volume of cube =b³

For Fig. 10.5, volume of cube = Sum of volume of all cubes and cuboids

= (a – b)a² + (a – b)ab + (a – b)b² + b³ …..(i)

The cube obtained in Fig. 10.5 has its each side a.

Its volume = (side)³ = a³ …..(ii)

From Eqs. (i) and (ii), we get

a³ = (a – b)a² + (a – b)ab + (a – b)b² + b³ …..(iii)

For Fig. 10.6, volume of solid obtained = a³ – b³

= (a – b)a² + (a – b)ab + (a – b)b² + b³ – b³ [from Eq.(iii)]

= (a – b)a² + (a – b)ab + (a – b)b² = (a-b) (a² +ab + b²)

Therefore, a³-b³ = (a-b) (a²+ab+b²)

Here, volume is in cubic units.

**Observation**

On actual measurement, we get

a = ……. , b = ……. ,

So, a² =…….. , b² = ……. ,

(a- b) = ……. , ab = ……. ,

a³ =…….. , b³ = ……. ,

Hence, a³-b³ = (a-b) (a²+ab+b²).

**Result**

The algebraic identity a³-b³ = (a-b) (a²+ab+b²) has been verified.

**Application**

The identity can be used in simplification and factorisation of algebraic expressions.

**Viva Voce
**

**Question 1:**

What is the expanded form of a³ – b³?

**Answer:**

Expanded form of a³-b³ = (a-b) (a²+ab+b²).

**Question 2:**

If (a² + ab + b²) = 0, then what will be the value of a³ – b³?

**Answer:**

a³-b³ = (a-b) (a²+ab+b²)

a³-b³=0

**Question 3:**

If x = y, then what will be the value of x³ – y³?

**Answer:**

Now, x³ – y³ = x³ – x³ = 0

**Question 4:**

What is degree of the expression y³ – x³ ?

**Answer:**

The degree of given expression is 3.

**Question 5:**

If we replace a by -a and b by -b, then what is the expansion of a³ – b³ ?

**Answer:**

a³ + b³ = (-a³)-(-b³)

=-a³+b³

b³ -a³ =(b-a)(b² +a² +ab)

**Suggested Activity**

Verify the algebraic identity x³ – y³ = (x – y) (x² + xy+ y²) by using x =7, y = 5.

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