Factorisation is a mathematical concept that involves finding the factors of a given number or expression. It is an essential component of algebraic identities and plays a crucial role in solving mathematical problems. In this response, we will discuss factorisation and the process of finding common factors on algebraic identities.

Understanding Factorisation

Factorisation is the process of breaking down a number or expression into its factors. A factor is a number or expression that can be multiplied together to give the original number or expression. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because these numbers can be multiplied together to give 12.

The process of factorisation is useful in many mathematical applications, including simplifying algebraic expressions, solving equations, and finding the prime factorisation of a number.

Finding Common Factors on Algebraic Identities

An algebraic identity is an equation that is true for all values of the variables involved. For example, (a + b)^2 = a^2 + 2ab + b^2 is an algebraic identity.

To find the common factors on an algebraic identity, we need to first expand it using the distributive property of multiplication. For example, (a + b)^2 can be expanded as:

(a + b)^2 = (a + b)(a + b)
= a(a + b) + b(a + b)
= a^2 + ab + ba + b^2
= a^2 + 2ab + b^2

Once the identity is expanded, we can look for common factors between the terms. In this case, we can see that a^2, 2ab, and b^2 all have a common factor of a. We can factor out the common factor to get:

a^2 + 2ab + b^2 = a(a + 2b) + b^2

This is an example of factorisation, where we have broken down the expression into its factors.

Conclusion

Factorisation is an essential skill in mathematics that plays a crucial role in solving problems involving algebraic expressions and equations. Understanding the process of factorisation is important for students in Class 8 and can help them develop a strong foundation in mathematics.