12.2. What is Factorisation?

When we factorise an algebraic expression, we write it as a product of factors. These factors may be numbers, algebraic variables or algebraic expressions.

Factorisation involves rewriting an expression in the form of a product of its factors. For example, the expression 12 can be factorised into 3 × 4. In algebra, we can do this with variables too, such as writing xy as x × y. The key idea here is that factorisation expresses an expression in a simpler form.

Think of factorisation like breaking down a complex recipe into its individual ingredients. Just as a cake can be expressed as flour, eggs, and sugar, algebraic expressions can be expressed in simpler terms (factors).

Expressions like 3xy, 5x2y, 2x(y + 2), 5(y + 1)(x + 2) are already in factor form. Their factors can be just read off from them, as we already know.

Certain expressions are already presented as products of factors. For instance, 3xy is already in factorised form because it's expressed as the product of 3, x, and y. Recognising these expressions makes it easier to work with them because we can directly see the factors involved.

Imagine you have a box of toys. If the box contains three types of toys - cars, dolls, and blocks -, you can think of the box as being in 'factor form' by directly counting how many of each toy you have (the factors) instead of listing each toy one by one.

On the other hand consider expressions like 2x + 4, 3x + 3y, x2 + 5x, x2 + 5x + 6. It is not obvious what their factors are. We need to develop systematic methods to factorise these expressions, i.e., to find their factors.

Some expressions are not immediately clear in terms of their factors. For example, with 2x + 4, we need to identify common elements and break it down into its simplest factor form. This becomes necessary for the expressions where visualising or seeing factors is not straightforward, requiring techniques such as factoring by grouping or using identities.

Think about a puzzle that is all mixed up. At first glance, you see a jumble of pieces (the expression) and it is not clear what the final picture (the factors) will be. You need to sort pieces into groups (factorisation techniques) to successfully put the puzzle together again.

We begin with a simple example: Factorise 2x + 4. We shall write each term as a product of irreducible factors; 2x = 2 × x, 4 = 2 × 2.

To factor the expression 2x + 4, we break each term down. We note that both terms share a common factor: 2. By factoring out 2, we can rewrite the expression as 2(x + 2). This process helps in simplifying expressions and solving equations more easily.

Consider gathering all your red apples and green apples into different baskets. If you have 2 apples in one basket and 4 apples in another basket with the same color, you can say you have a common factor of apples that you can put together to show a group of 2 apples, making it visual and easier to manage.

Example 1: Factorise 12a2b + 15ab2

First, we find the irreducible factors of each term. 12a2b can be broken down to 2 × 2 × 3 × a × a × b and 15ab2 as 3 × 5 × a × b × b. The common factors are 3ab. We can then factor the expression as 3ab(4a + 5b). This clearly shows how we can write complex algebraic expressions in simpler factorised terms.

Imagine packing boxes for a party. If you have a total of 12 cupcakes and 15 cookies, you can group them in twos to make organizing and serving easier, simplifying a later task in the event just as factorisation simplifies mathematical expressions.

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