**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**

- Hardboard
- Coloured papers
- Adhesive
- White paper
- Scissors
- Geometry Box

**Prerequisite Knowledge**

- Square and its area.
- Rectangle and its area.

**Theory**

- For square and its area refer to Activity 3.
- For rectangle and its area refer to Activity 3.

**Procedure**

- Take a hardboard of suitable size and paste a white paper on it.
- From a coloured paper, cut out a square of side a units, (see Fig. 6.1)

- Further, cut out a square of sided units (b < a)from another coloured paper, (see Fig. 6.2)

- Also, cut out a square of sidec units (c < b)from different coloured paper.(see Fig. 6.3)

- Cut out two rectangles of dimensions b x a from different coloured paper, (see Fig. 6.4)

- Also, cut out two rectangles of dimensions c x b from different coloured paper, (see Fig. 6.5)

- Now further, cut out two rectangles of dimensions c x a from another coloured paper, (see Fig. 6.6)

- 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

- calculating the square of a number which can be expressed as a sum of three convenient numbers.
- 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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