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

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

1. Acrylic sheet
2. Geometry box
3. Scissors
4. Adhesive/Adhesive tape
5. Cutter

Prerequisite Knowledge

1. Concept cuboid and its volume.
2. Concept of cube and its volume.

Theory

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

Procedure

1. Using acrylic sheet, make a cuboid of dimensions (a – b) x a x a, where b < a. (see Fig. 10.1)
   
2. Using acrylic sheet, make another cuboid of dimensions (a-b) x a x b, where b < a. (see Fig. 10.2).
   
3. Now, make one more cuboid of dimensions (a-b) x b x b. (see Fig. 10.3)
   
4. Now, make a cube of dimensions b x b x b. (see Fig. 10.4)
   
5. 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.
   
6. 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.

Math LabsMath Labs with ActivityMath Lab ManualScience LabsScience Practical Skills