NCERT Class 9 Maths Lab Manual – Verify the Algebraic Identity (a+b+c)² = a²+b²+c²+2ab+2bc+2ca

NCERT Class 9 Maths Lab Manual – Verify the Algebraic Identity (a+b+c)² = a²+b²+c²+2ab+2bc+2ca

OBJECTIVE

To verify the algebraic identity (a+b+c)² = a²+b²+c²+2ab+2bc+2ca .

Materials Required

1. Hardboard
2. Coloured papers
3. Adhesive
4. White paper
5. Scissors
6. Geometry Box

Prerequisite Knowledge

1. Square and its area.
2. Rectangle and its area.

Theory

1. For square and its area refer to Activity 3.
2. For rectangle and its area refer to Activity 3.

Procedure

1. Take a hardboard of suitable size and paste a white paper on it.
2. From a coloured paper, cut out a square of side a units, (see Fig. 6.1)
   
3. Further, cut out a square of sided units (b < a)from another coloured paper, (see Fig. 6.2)
   
4. Also, cut out a square of sidec units (c < b)from different coloured paper.(see Fig. 6.3)
   
5. Cut out two rectangles of dimensions b x a from different coloured paper, (see Fig. 6.4)
   
6. Also, cut out two rectangles of dimensions c x b from different coloured paper, (see Fig. 6.5)
   
7. Now further, cut out two rectangles of dimensions c x a from another coloured paper, (see Fig. 6.6)
   
8. Paste the squares and rectangles on the hardboard as shown in Fig. 6.7.
   

Demonstration
From Fig. 6.7, it is clear that from the arrangement of sqaures and rectangle, square PQRS of side (a + b+c) units is obtained.
Area of square PQRS = (a + b +c)² [∴ area of square = (side)²] … (i)
Also, area of square PQRS = Sum of the areas of all the squares and rectangles, which are
used to make the square PQRS = a² + b² + c² + ab + ab + bc +bc +ca+ca = (a² + b² + c² + 2ab + 2bc + 2ca) …(ii)
From Eqs. (i) and (ii), we have
(a + b+c)² =(a² +b² +c² + 2ab + 2bc + 2ca) Here, area is in square units.

Observation
On actual measurement, we get
a = …….. , b = …….. ,  c = …….. ,
So, a² = …….. , b² = …….. , c² = …….. ,
ab = …….. , bc = …….. , ca = …….. ,
2ab = …….. , 2bc = …….. , 2ca = …….. ,
a + b+c = …….. ,
and (a + b+c)² = …….. ,
Hence, (a + b+c)² = a² + b² + c² + 2ab + 2bc +2ca

Result
Identity (a + b+c)²= a² +b² +c² +2ab + 2bc + 2ca has been verified.

Application
This identity may be used for

1. calculating the square of a number which can be expressed as a sum of three convenient numbers.
2. simplification and factorisation of algebraic expressions.

Viva Voce
Question 1:
Is the identity (a+b+c)² = a²+b²+c²+2ab+2bc+2ca, holds for all real values of a, b and c?
Answer:
Yes

Question 2:
Which identity should be used to expand (3x -√y + z)² ?
Answer: 
(a+b+c)² = a²+b²+c² + 2ab + 2bc + 2ca .

Question 3:
What do you mean by a polynomial?
Answer: 
An algebraic expression, in which the variables involved have only non-negative integral power, is called a polynomial.

Question 4:
What do you mean by zeroes of a polynomial?
Answer: 
If p(x) is a polynomial, then a real number k is said to be zero of the polynomial p(x), If p(k) = 0.

Question 5:
What do you mean by a trinomial?
Answer: 
A polynomial with three terms is called a trinomial.

Question 6:
Is the expansion of (x + y + z)², trinomial?
Answer: 
No, because on expanding (x + y + z)², we get six terms.

Question 7:
Write the condition that the three variables in identity
(a+b+c)² = a²+b²+c²+2ab+2bc+2ca should be written in two variables form (x + y)² =x² + y² + 2xy.
Answer: 
Anyone of the variable in the identity (a+b+c)² = a²+b²+c²+2ab+2bc+2ca should be zero.

Question 8:
If we take all the variables are equal in the identity
(a+b+c)² = a²+b²+c²+2ab+2bc+2ca, then it will give us a cube of one variable. What is the name of the solid figure in which it gives out a cube of one variable?
Answer: 
Cubic solid figure

Suggested Activity
Verify the identity (a+b+c)² = a²+b²+c²+2ab+2bc+2ca by taking different values of a,b,c.

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