**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 cube = (edge)
^{3} - Volume of cuboid = I x b x h

**Materials Required**

A set of 56 cubes each has dimensions (1 x 1 x 1) cubic unit. Cubes may be of wood, plastic, cardboard or thermocol.

**Procedure**

To verify the identity a^{3} + b^{3}, we shall take a = 3 units and b = 1 unit.

- Make an arrangement of 28 cubes such that we get a cube of 3 x 3 x 3 cubic units and a single unit cube of b = 1 unit as shown in fig. (i) and fig.(ii).

- Now we will use second set of 28 cubes. Consider one stack (24 cubes) as {a x (a+b)} cubic units i.e., (3 x 4 x 2) cu. units and arrange as shown in fig. (iii):

- Consider an other stack of 4 cubes such as (a + b) (b) (b) i.e., (4) (1) (1) cubic units. Arrange them as shown in fig. (iv).

**Observation and Calculation**

As the arrangements shown in [fig. (i),fig. (ii)] and [fig. (iii), fig. (iv)], have equal number of cubes therefore the total volume in both the cases must be same.

In fig(i) and fig.(ii)

Volume of cube in fig. (i) = volume of 27 unit cubes = a^{3}

Volume of cube of 1 unit = b^{3}

Total volume of 28 cubes = a^{3} +b^{3} … (i)

In fig. (iii) and (iv),

Volume of cuboid of 24 unit cubes = (a + b) (a – b) (a)

Volume of cuboid of 4 unit cubes = (a + b)b^{2}

Total volume of 28 cubes = (a+b) (a – b) (a)+(a+b) (b^{2})

= (a + b) (a^{2} – ab + b^{2}) … (ii)

From (i) and (ii), we have

∴ 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**

Algebraic identity a^{3} + b^{3} = (a+b) (a^{2} – ab + b^{2}) is verified geometrically. This activity can be performed by using different colours of cubes as shown in fig. (i), (ii), (iii) and (iv). We have learnt making of cuboids of various dimensions by using unit cubes and adding and subtracting cuboids.

**Activity Time**

By using different values of a and b, students can verify the identity a^{3} + b^{3}, e.g., a = 6, b = 2 and also find volume of different cubes and cuboids used for this activity.

**Viva Voce**

**Question 1.**

Find 5^{3} + 2^{3} by using formula a^{3} + b^{3} = (a+b) (a^{2} – ab + b^{2}).

**Answer:**

(7) (25 + 4 – 10) = 7 (19) = 133.

**Question 2.**

How many factors are formed by x^{3} +y^{3 }?

**Answer:**

Two factors are formed (x +y) (x^{2} + y^{2} – xy).

**Question 3.**

What are the possible expressions for the area of a circular ring, whose internal radius is rand external radius is R ?

**Answer:**

π (R – r) (R + r).

**Question 4.**

What are the possible expressions for a dimension of a cuboid whose volume is 3x^{3} – 12x?

**Answer:**

3x (x – 2) (x + 2).

**Question 5.**

Simplify: (3 + x) (9 + x^{2} – 3x).

**Answer:**

33 + x^{3} = 27 + x^{3}.

**Question 6.**

Write the degree of (x + y) (x^{2} + y^{2} – xy).

**Answer:**

3.

**Question 7.**

Write the real zero of x^{3} + 1.

**Answer:**

-1

**Question 8.**

Is (2 + x) (4 + x^{2} – 2x) a binomial or trinomial ?

**Answer:**

(2 + x) (4 + x^{2} – 2x) = (23 + x) = 8 + x^{3} which is a binomial.

**Multiple Choice Questions**

**Question 1.**

Simplify: (3x + 1) (9x^{2} + 1 — 3x) by using identity

(i) (27x^{3} + 1)

(ii) (3x^{3} + 1)

(iii) (9x^{2} + 1)

(iv) none of these

**Question 2.**

Find the factors of (27x^{3} + 125) :

(i) (3x + 5)^{3}

(ii) (3x + 5) (9x^{2} – 15x + 25)

(iii) (3x – 5)^{3}

(iv) none of these

**Question 3.**

Write the factors of (x^{3} + 8) :

(i) (x + 2) (x^{2} + 4 – 2x)

(ii) x^{2} + 4

(iii) (x^{2} – 4)

(iv) none of these

**Question 4.**

Write the factors of (64a^{3} + 16√2) :

(i) (8a + 8)^{3}

(ii) (4a + 2√2) (6a^{2} + 8 – 8a√2)

(iii) 16a^{2} + 8√a + 8

(iv) none of these

**Question 5.**

Write the zeroes of (125x^{4} + 1) :

(i) 0, 5

(ii) 0, \(-\frac { 1 }{ 5 }\)

(iii) 0, \(\frac { 1 }{ 5 }\)

(iv) none of these

**Question 6.**

Check weather x = 2 is the zero of (125x^{3} + 1) :

(i) no

(ii) yes

(iii) can’t say

(iv) none of these

**Question 7.**

Write the integral zero of (64m^{3} – 343):

(i) no integral zero

(ii) \(\frac { 7 }{ 4 }\)

(iii) \(-\frac { 7 }{ 4 }\)

(iv) none of these

**Question 8.**

Write the degree of (4 + x) (16 + x^{2} – 4x):

(i) 1

(ii) 3

(iii) 2

(iv) none of these

**Question 9.**

Write the factors of x^{3} + \(\frac { 1 }{ { x }^{ 3 } }\)

(i) (\(x+\frac { 1 }{ x }\)) (x^{2} + \(\frac { 1 }{ { x }^{ 2 } }\) – 1)

(ii) (\(x-\frac { 1 }{ x }\)) (x^{2} + \(\frac { 1 }{ { x }^{ 2 } }\) + 1)

(iii) (\(x+\frac { 1 }{ x }\)) (x^{2} + \(\frac { 1 }{ { x }^{ 2 } }\) + 2)

(iv) none of these

**Question 10.**

Write the factors of (x – 1)^{3} + (y – 1)^{3}:

(i) (x – 1)(y – 1)

(ii) (x – 1 + y – 2) (x^{2} + y^{2} – 1)

(iii) (x + y – 2) (x^{2} + y^{2} – x – y – xy + 1)

(iv) none of these

**Question 11.**

Evaluate \(\frac { { 361 }^{ 3 }+{ 139 }^{ 3 } }{ { 361 }^{ 2 }-\left( 361\times 139 \right) +{ 139 }^{ 2 } }\) :

(i) 500

(ii) 600

(iii) 700

(iv) none of these

**Answers**

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

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