Class 09 To verify the algebraic identity :a3 – b3 = (a – b)(a2 + ab + b2)

 

Activity 10

OBJECTIVE

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

 METHOD OF CONSTRUCTION

MATERIAL REQUIRED

Acrylic sheet, sketch pen, glazed papers, scissors, adhesive, cello-tape, coloured papers, cutter.

1.   Make a cuboid of dimensions (a–b) × a × a (b < a), using acrylic sheet and cellotape/adhesive as shown in Fig. 1.

 2.   Make another cuboid of dimensions (a–b) × a × b, using acrylic sheet and cellotape/adhesive as shown in Fig. 2.

 3.   Make one more cuboid of dimensions (a–b) × b × b as shown in Fig. 3.

 4.    Make a cube of dimensions b × b × b using acrylic sheet as shown in Fig. 4.

5.   Arrange the cubes and cuboids made above in Steps (1), (2), (3) and (4) to obtain a solid as shown in Fig. 5, which is a cube of volume a³ cubic units.

Fig. 5

 Fig. 6

 DEMONSTRATION

 Volume of cuboid in Fig. 1 = (a–b) × a × a cubic units.

 Volume of cuboid in Fig. 2 = (a–b) × a × b cubic units.

 Volume of cuboid in Fig. 3 = (a–b) × b × b cubic units.

 Volume of cube in Fig. 4 = b³ cubic units.

 Volume of solid in Fig. 5 = a³ cubic units.

 Removing a cube of size b³ cubic units from the solid in Fig. 5, we obtain a solid as shown in Fig. 6.

 Volume of solid in Fig. 6 = (a–b) a² + (a–b) ab + (a–b) b²

 =  (a–b) (a² + ab + b²)

 Therefore, a³ – b³ = (a – b)(a² + ab + b²)

OBSERVATION

 On actual measurement:

 a = ..............,        b = ..............,

 So, a³ = ..............,       b³ = .............., (a–b) = ..............,    ab = ..............,

 a²  = ..............,      b² = ..............,

 Therefore, a³ – b³ = (a – b) (a² + ab + b²).

 APPLICATION

 The identity may be used in simplification/factorisation of algebraic expressions.