Algebraic identities are equations that are always true, regardless of the value of the variables involved. While the prompt refers to "algebraic identity class 9," it's likely referring to a specific algebraic identity commonly taught around the 9th grade. The most probable identity being referenced is:
(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
Explanation
This identity shows the expansion of the square of a trinomial (an expression with three terms). Here's a breakdown:
- (a + b + c)²: This means (a + b + c) multiplied by itself: (a + b + c) * (a + b + c).
- a² + b² + c²: These are the squares of each individual term in the trinomial.
- 2ab + 2bc + 2ca: These are twice the product of each pair of terms in the trinomial.
Why is it Important?
This identity is important for several reasons:
- Simplifying Expressions: It allows you to quickly expand and simplify expressions involving the square of a trinomial.
- Factoring: It can sometimes be used in reverse to factor complex algebraic expressions.
- Problem Solving: It's helpful in solving various algebraic problems and proving other mathematical relationships.
Example
Let's say you have the expression (x + 2y + 3)². Using the identity, you can expand it as follows:
(x + 2y + 3)² = x² + (2y)² + 3² + 2(x)(2y) + 2(2y)(3) + 2(x)(3)
= x² + 4y² + 9 + 4xy + 12y + 6x
Other Important Identities often taught in Class 9
While the previous identity is the most likely one, it's worth noting some other common algebraic identities taught at this level:
- (a + b)² = a² + 2ab + b²
- (a - b)² = a² - 2ab + b²
- (a + b)(a - b) = a² - b²
- (x + a)(x + b) = x² + (a + b)x + ab
- (a + b)³ = a³ + b³ + 3ab(a + b) = a³ + 3a²b + 3ab² + b³
- (a - b)³ = a³ - b³ - 3ab(a - b) = a³ - 3a²b + 3ab² - b³
- (a³ + b³) = (a + b)(a² - ab + b²)
- (a³ - b³) = (a - b)(a² + ab + b²)
- a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)
It is essential to know these identities and their applications to solve various algebraic problems efficiently.
