**CBSE Class 9 Maths Lab Manual – Algebraic Identity (a**^{3} – b^{3}) = (a – b) (a^{2} + ab + b^{2})

^{3}– b

^{3}) = (a – b) (a

^{2}+ ab + b

^{2})

**Objective**

To verify the identity a^{3} – b^{3} = (a – b)(a^{2} + ab + b^{2}) geometrically by using sets of unit cubes.

**Prerequisite Knowledge**

Volume of a cube = (Edge)^{3}

Volume of a cuboid = l x b x h

a^{3} – b^{3} = (a – b)(a^{2} + ab + b^{2})

**Materials Required**

A set of 53 plastic or wooden cubes each of dimensions (1 x 1 x 1 unit)

**Procedure**

To verify a^{3} – b^{3} = (a – b)(a^{2} + ab + b^{2}). Let a = 3 and b =1.

- Take 27 cubes and place them to form a stack consisting of a 9 columns, each column consisting 3 cubes [fig. (i)].
- Remove one cube from this stack get a stack of 26 cubes (Arrangement I)

- Make arrangement II of 26 cubes. This arrangement consists of three stacks.
- The first stack consists of 18 cubes such as 9 columns of two cubes each. .
- The second stack consists of 6 cubes such as two rows of three cubes each.
- Third stack consist of 1 row of 2 cubes.

**Observation**

Since the two arrangements have equal number of cubes (each arrangement has 26 cubes), the total volume in both the arrangements must be equal.

- Volume of arrangement I

Volume of stack in fig. 1(i) = a^{3}

Volume of stack in fig. 1(ii) = b^{3}

∴Volume of arrangement I = Volume of stack in fig. 1(i) – Volume of stack in fig. 1(ii) = a^{3}– b^{3} - Volume of arrangement II

Volume of the stack in fig. 2 (i) = (a – b) a^{2}

Volume of the stack in fig. 2(ii) = (a – b)ab

Volume of the stack in fig. 2 (iii) = (a – b)b^{2}

Total volume of arrangement II = (a – b)a^{2}+ (a – b)ab + (a – b)b^{2}= (a – b)(a^{2}+ ab + b^{2}).

Since number of cubes in arrangement I and II are equal.

∴a^{3}– b^{3}= (a – b)(a^{2}+ ab + b^{2}).

**Result**

The identity a^{3} – b^{3} = (a – b)(a^{2} + ab + b^{2}) is verified geometrically by using cubes and cuboids.

**Learning Outcome**

In this way, students can learn the concept of verifying the identity geometrically by adding different cubes and cuboids.

**Activity Time**

Students can perform this activity for any other values of a and b e.g. a = 6, b = 2 and find volumes of different cubes and cuboids through this activity.

**Viva Voce**

**Question 1.**

What is the value of 3^{3} – 2^{3} ?

**Answer:**

(3 – 2) (9 + 4 + 6) = 19.

**Question 2.**

What is the simplification of 12^{3} – 2^{3}?

**Answer:**

(12 – 2) (144 + 4 + 24) = 1720.

**Question 3.**

Factorize: 27x^{3} – 64y^{3}.

**Answer:**

(3x – 4y) (9x^{2} + 16y^{2} + 12xy).

**Question 4.**

Find 8^{3} – 3^{3}

**Answer:**

(5) (8^{2} + 9 + 24) = 485.

**Question 5.**

Factorize: (2x + y)^{3} – (x + 2y)^{3}.

**Answer:**

(2x + y – x – 2y) [(2x + y)^{2} + (x + 2y)^{2} + (2x + y) (x + 2y)]

= (x – y) (7x^{2} + 7y^{2} + 13xy)

**Question 6.**

How many factors are possible of x^{3} – 27?

**Answer:**

2 factors.

**Question 7.**

Write the real zero of 8x^{3} – 1.

**Answer:**

\(\frac { 1 }{ 2 }\)

**Question 8.**

Is x – 2, a factor of x^{3} – 8?

**Answer:**

Yes.

**Question 9.**

Flow many real zeroes are possible of x^{3} + 1 and x^{3} – 1 ?

**Answer:**

Only one real zero is possible.

**Multiple Choice Questions**

**Question 1.**

Write the factors of a^{3} – b^{3}:

(i) (a – b)(a^{2} + b^{2} + ab)

(ii) (a – b)(a^{2} + b^{2} – ab)

(iii) (a – b)(a + b)

(iv) none of these

**Question 2.**

Write the factors of 27 – x^{3}:

(i) (3 – x)(9 + x^{2} + 3x)

(ii) (3 – x)(9 + x^{2} – 3x)

(iii) (x – 3)(9 + x^{2} + 3x)

(iv) none of these

**Question 3.**

Find the product of (3x – y) (9x^{2} + y^{2} + 3xy):

(i) 27x^{3} + y^{3}

(ii) 27x – y^{3}

(iii) y^{3} – 27x^{3}

(iv) none of these

**Question 4.**

Write the degree of x^{3} – y^{3}:

(i) 3

(ii) 2

(iii) 6

(iv) none of these

**Question 5.**

What is the coefficient of y in 343x^{3} – 27y^{3}:

(i) -27

(ii) 343

(iii) 27

(iv) none of these

**Question 6.**

Write the coefficient of a^{3} in (125a^{3} – 8y):

(i) 125

(ii) -8

(iii) 8

(iv) none of these

**Question 7.**

If the area of rectangle is (125x^{3} -y^{3}), find the dimensions of rectangle :

(i) (5x – y) and (5x + y)

(ii) (5x – y) and (25x^{2} + y^{2} – 5xy)

(iii) (5x – y) and (25x^{2} + y^{2} + 5xy)

(iv) none of these

**Question 8.**

Write the coefficient of a^{2} in (343a^{3} – 125b^{3}):

(i) (7a – 5b)

(ii) (7a + 5b

(iii) (343 – 15b^{3})

(iv) none of these

**Question 9.**

64m^{3} – 343x^{3} is a

(i) binomial

(ii) monomial

(iii) trinomial

(iv) none of these

**Question 10.**

Evaluate: \(\frac { { 369 }^{ 3 }-{ 139 }^{ 3 } }{ { 369 }^{ 2 }+\left( { 369 }\times { 139 } \right) +{ 139 }^{ 2 } }\)

(i) 508

(ii) 230

(iii) 24508

(iv) none of these

**Answers**

- (i)
- (i)
- (ii)
- (i)
- (iv)
- (i)
- (iii)
- (iv)
- (i)
- (ii)

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