Factorising is a mathematical process where we break down a mathematical expression into smaller parts that can be multiplied together to give the original expression. It is also called "factoring" in some countries.
Why do we need to factorise?
Factorising is an important skill in mathematics that is used in many areas, including algebra, calculus, and trigonometry. It is used to simplify complex mathematical expressions and solve equations.
Factorising also helps us find common factors and simplify fractions. It is also useful in solving problems related to areas and volumes, as we can factorise the expressions to find the factors that are common to both.
How do we factorise?
Factorising involves breaking down an expression into its factors. We look for common factors or patterns in the expression and use these to simplify the expression.
For example, if we have the expression 2x + 4, we can factorise it by finding the common factor of 2. We write it as 2(x + 2).
Another example is the expression x^2 + 4x + 4. We can factorise it by finding two numbers that multiply to give 4 and add to give 4. These numbers are 2 and 2. We write it as (x + 2)^2.
Types of Factorisation
There are different types of factorisation techniques that are used in mathematics. Some of the most common ones are:
Common Factor Factorisation
This involves finding the common factor in an expression and factoring it out. For example, in the expression 6x + 9, the common factor is 3. We can write it as 3(2x + 3).
Difference of Two Squares Factorisation
This involves factoring an expression that is in the form a^2 - b^2. For example, in the expression x^2 - 4, we can write it as (x + 2)(x - 2).
Trinomial Factorisation
This involves factoring an expression that is in the form ax^2 + bx + c. For example, in the expression 2x^2 + 5x + 3, we can write it as (2x + 3)(x + 1).
Grouping Factorisation
This involves grouping terms in an expression and factoring them out. For example, in the expression 3x^3 + 6x^2 + 2x + 4, we can group the first two terms and the last two terms together. We can then factor out the common factors to get 3x^2(x + 2) + 2(x + 2). We can then factor out the common factor of (x + 2) to get (x + 2)(3x^2 + 2).
Applications of Factorisation
Factorisation has many applications in mathematics and real-life situations. Some of the most common applications are:
Simplifying Algebraic Expressions
Factorisation is used to simplify algebraic expressions and make them easier to solve. This is especially useful in calculus and trigonometry, where complex expressions can be simplified using factorisation.
Solving Equations
Factorisation is used to solve algebraic equations. By factoring out the common factors, we can simplify the expressions and solve for the variables.
Area and Volume Problems
Factorisation is used to solve problems related to areas and volumes. By factorising the expressions, we can find the common factors and simplify the expressions to find the solutions.
Cryptography
Factorisation is used in cryptography to encode and decode messages. It is used to factorise large numbers into their prime factors, which is a difficult task that requires a lot of computing power.
Conclusion
Factorisation is an important skill in mathematics that is used in many areas, including algebra, calculus, and trigonometry. It involves breaking down an expression into its factors and simplifying it. There are different types of factorisation techniques that are used depending on the form of the expression. Factorisation has many applications in mathematics and real-life situations, including simplifying algebraic expressions, solving equations, solving problems related to areas and volumes, and cryptography.
