What is Factorise in Maths? | Handsome Youth Of Classical Mythology

Factorising is a mathematical process of finding factors of a given number, polynomial or algebraic expression. It is a technique used in algebra to simplify an expression by breaking it down into a product of its factors. This technique is widely used in mathematics, science and engineering to solve complex problems. In this article, we will discuss what factorising is and how it can be used to solve mathematical problems.

What is a Factor?

A factor is a number that divides another number without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6 and 12. When we factorise a number, we write it as a product of its factors. For example, we can factorise 12 as 2 x 2 x 3 or 2^2 x 3.

What is Factorisation?

Factorisation is the process of finding the factors of a given number or algebraic expression. In algebra, factorisation involves breaking down an expression into simpler factors. The resulting factors are then multiplied together to get the original expression. Factorisation can be used to solve equations, simplify expressions, and find roots of polynomials.

How to Factorise?

There are different techniques used to factorise numbers and expressions. The most common ones are:

Factorisation by Common Factor: In this technique, we look for a common factor that can be factored out of all the terms in the expression. For example, to factorise 6x + 9, we can factor out 3 to get 3(2x + 3).
Factorisation by Grouping: This technique involves grouping the terms in an expression in a way that allows us to factor out a common factor. For example, to factorise x^2 + 3x + 2, we can group the first two terms as x(x + 3) and the last two terms as 1(x + 2). This gives us (x + 2)(x + 1).
Factorisation by Quadratic Formula: This technique is used to factorise quadratic expressions of the form ax^2 + bx + c. The quadratic formula is given by x = (-b ?? ???(b^2 - 4ac))/2a. For example, to factorise 2x^2 + 7x + 3, we can use the quadratic formula to get x = -1/2 or x = -3/2. This gives us (2x + 1)(x + 3).
Factorisation by Completing the Square: This technique is used to factorise quadratic expressions of the form ax^2 + bx + c. It involves adding and subtracting a constant to make the expression a perfect square. For example, to factorise x^2 + 4x + 4, we add and subtract 4 to get (x + 2)^2.

Examples of Factorisation

Let's look at some examples of factorisation:

Factorise 12: 12 = 2 x 2 x 3 = 2^2 x 3
Factorise x^2 + 5x + 6: x^2 + 5x + 6 = (x + 2)(x + 3)
Factorise 3x^2 - 10x - 8: 3x^2 - 10x - 8 = (3x + 2)(x - 4)
Factorise 2x^2 + 5x + 2: 2x^2 + 5x + 2 = (2x + 1)(x + 2)

Applications of Factorisation

Factorisation is an important technique used in various fields of mathematics, science and engineering. Some of its applications are:

Algebraic Equations: Factorisation is used to solve algebraic equations by simplifying expressions and finding roots of polynomials.
Trigonometry: Factorisation is used to simplify trigonometric expressions and solve trigonometric equations.
Statistics: Factorisation is used in statistical analysis to find factors that affect a given phenomenon.
Cryptography: Factorisation is used in cryptography to break down complex codes and find the prime factors of large numbers.

Conclusion

Factorisation is a powerful technique used in mathematics, science and engineering to simplify expressions, solve equations, and find roots of polynomials. It involves breaking down an expression into simpler factors and multiplying them together to get the original expression. Factorisation is an important tool for solving complex problems in various fields and is widely used in research and development.