Simple Formulae and their Application

In the article Simple Formula and their Applications I we dealt with algebraic formulas in the second degree, i.e., formulas related to perfect squares and the sum and difference of two squares. In this article we will be covering the algebraic formulas in the third degree, i.e., formulas related to perfect cubes and the sum and difference of two cubes.

Formula:

Again, [Taking common from ]

Proof:

Corollary:

or,

Example: Find the cube of

Solution:

Example: Find the cube of

Solution:

Example: If , show that

Solution:

[ ] (Proved)

Example: Simplify

Solution:

Putting for and for we have,

The given expression

Example: Show that when

Solution:

from the given condition we have,

(Proved)

Formula:

Again, [Taking common from ]

Proof:

Corollary:

or,

Example: Find the cube of

Solution:

Example: Find the cube of

Solution:

Example: Find the cube of 297

Solution:

[ ]

Example: If , find the value of

Solution:

…(1)

[ ]

[Using (1)]

Exercise 1:

1. Find the cube of:

a)

b)

c)

d)

2. Simplify:

a)

b)

c)

d)

3. Find the value of when and

4. Find the value of when and

5. If , show that,

6. If show that

7. If show that

8. If find the value of

Formula:

Proof:

Conversely, . Hence, any expression of the form can be resolved into factors.

Example: Multiply by

Solution:

Putting for and for we have,

Hence,

Example: Resolve into factors

Solution:

Example: Resolve into factors

Solution:

Formula:

Proof:

Conversely, . Hence, any expression of the form can be resolved into factors.

Example: Multiply by

Solution:

Putting for and for we have,

Example: Resolve into factors

Solution:

Example: Resolve into factors

Solution:

Exercise 2:

1. Multiply:

a) by

b) by

c) by

d) by

2. Resolve into factors:

a)

b)

c)

d)