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Verify the Algebraic Identity (a+b)³ = a³+b³+ 3a²b + 3ab² Experiment Class 9 Maths Practical NCERT

The experiment to determine Verify the Algebraic Identity (a+b)³ = a³+b³+ 3a²b + 3ab² are part of the Class 9 Maths Lab Manual provides practical activities and experiments to help students understand mathematical concepts effectively. It encourages interactive learning by linking theoretical knowledge to real-life applications, making mathematics enjoyable and meaningful.

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

Determine Verify the Algebraic Identity (a+b)³ = a³+b³+ 3a²b + 3ab² Class 9 Practical

OBJECTIVE

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

Materials Required

  1. Acrylic sheets
  2. Adhesive/Adhesive tape
  3. Scissors
  4. Geometry Box
  5. Cutter

Prerequisite Knowledge

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

Theory

  1. 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)
    NCERT Class 9 Maths Lab Manual - Verify the Algebraic Identity (a+b)³ = a³+b³+ 3a²b + 3ab² 1
    Properties of cuboid are

    1. In a cuboid, there are 6 faces, 12 edges and 8 corners (four at bottom and four at top face) which are called vertices.
    2. Opposite faces of a cuboid are equal and parallel.
    3. The line segment joining the opposite vertices of cuboid is called the diagonal of a cuboid.
    4. There are four diagonals in a cuboid which are equal in length.
      Volume of cuboid = lbh
      where, l = length, b = breadth and h = height
  2. Cube A cuboid whose length, breadth and height are same, is called a cube, (see Fig. 7.2)
    NCERT Class 9 Maths Lab Manual - Verify the Algebraic Identity (a+b)³ = a³+b³+ 3a²b + 3ab² 2
    Properties of cube are

    1. In a cube, there are 6 faces, 12 edges and 8 corners (four at bottom and four at top face) which are called vertices.
    2. All the six faces of a cube are congruent square faces.
    3. Each edge of a cube have same length.
      Volume of cube = a³
      where, a is side of cube.

Procedure

  1. 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)
    NCERT Class 9 Maths Lab Manual - Verify the Algebraic Identity (a+b)³ = a³+b³+ 3a²b + 3ab² 3
  2. 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)
    NCERT Class 9 Maths Lab Manual - Verify the Algebraic Identity (a+b)³ = a³+b³+ 3a²b + 3ab² 4
  3. 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)
    NCERT Class 9 Maths Lab Manual - Verify the Algebraic Identity (a+b)³ = a³+b³+ 3a²b + 3ab² 5
  4. 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)
    NCERT Class 9 Maths Lab Manual - Verify the Algebraic Identity (a+b)³ = a³+b³+ 3a²b + 3ab² 6
  5. 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
    NCERT Class 9 Maths Lab Manual - Verify the Algebraic Identity (a+b)³ = a³+b³+ 3a²b + 3ab² 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

  1. calculating the cube of a number which can be expressed as the sum of two convenient numbers.
  2. 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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NCERT Class 9 Maths Lab Manual