Special Algebraic Identities Obtained through Expanding - Revision Notes

In these revision notes for Special Algebraic Identities Obtained through Expanding, we cover the following key points:

• What is a binomial?
• How to expand the square of a sum.
• How to expand the square of a difference.
• What are conjugate expressions?
• How to expand the product of two conjugates.
• How to expand the cube of a sum.
• How to expand the cube of a difference.
• How to expand expressions of type (a - b) × (a2 + ab + b2).
• How to expand expressions of type (a + b) × (a2 - ab + b2).
• How to expand the square of the sum of three variables.
• How to combine special identities in the same exercise.

Special Algebraic Identities Obtained through Expanding Revision Notes

If an algebraic expression contains two terms added to or subtracted from each other, it is called a binomial. By definition, a binomial is an algebraic expression of the sum or the difference of two terms.

In symbols, a binomial is written as a + b, where a and b may represent a single number, a single variable, a product of numbers and variables where the latter can be raised in a certain power, etc.

There are eight algebraic identities based on the binomial structure that are widely used in many exercises involving expressions. They are:

1. Square of a sum, otherwise known as the square of a binomial, is an algebraic expression of the type
   (a + b)²
   where a and b are the terms of the binomial.
   When expanded, this algebraic expression becomes
   (a + b)² = a² + 2ab + b²
2. Square of a difference involves algebraic expressions of the type
   (a - b)²
   When expanded, such an algebraic expression becomes
   (a - b)² = a² - 2ab + b²
3. Product of conjugates. Two expressions of the form a + b and a - b are called the conjugates of each other. Basically, both are binomials that have the same terms but opposite signs in-between. When two conjugate expressions multiply with each other yields
   (a - b)(a + b) = a² - b²
4. Cube of a Sum is expanded in the following form:
   (a + b)³ = a³ + 3a² b + 3ab² + b³
5. Cube of a Difference is expanded in the following form:
   (a - b)³ = a³ - 3a² b + 3ab² - b³
6. Algebraic expression of the form (a - b) · (a2 + ab + b2) are expanded as
   (a - b) ∙ (a² + ab + b² ) = a³ - b³
7. Algebraic expression of the form (a + b) · (a2 - ab + b2) are expanded as
   (a + b) ∙ (a² - ab + b² ) = a³ - b³
8. Expanding expression of the type (a + b + c)² involves two steps, in the first step we consider the first two variables as a whole. Then, we continue with the rest. In this algebraic expression we use the first identity (a + b)² = a² + 2ab + b² twice. The final expanding form is
   (a + b + c)² = a² + b² + c² + 2ab + 2ac + 2bc

The 8 special algebraic identities can be combined or found more than once in the same expression. Being aware of this fact help us obtain a much shorter expression than the original.