Algebraic manipulation; factorisation - Mathematics IGCSE Study Notes

Algebraic manipulation is like being a detective for numbers and letters (called variables). You're given a jumbled-up clue (an algebraic expression), and your job is to rearrange it, simplify it, or rewrite it in a different form to make it clearer or easier to use.

Factorisation is a special trick within algebraic manipulation. Think of it like this: Imagine you have a delicious cake. Factorisation is like figuring out all the ingredients that went into making that cake – the flour, sugar, eggs, etc. In maths, when we factorise an expression, we're finding the 'ingredients' (smaller expressions or numbers) that multiply together to make the original, bigger expression.

For example, if you have the number 6, its factors are 2 and 3 because 2 × 3 = 6. In algebra, if you have an expression like 2x + 4, we can see that both '2x' and '4' can be divided by 2. So, we can 'take out' the 2, and write it as 2(x + 2). Here, '2' and '(x + 2)' are the factors. It's like putting common items into a shopping bag!

Let's say you're planning a party, and you want to buy snacks. You need 3 bags of chips and 3 cans of soda for each of your 5 friends. How many items do you need in total?

Without factorisation (the 'long' way): You could calculate for each friend: (3 bags of chips + 3 cans of soda) = 6 items per friend. Then, for 5 friends: 6 items/friend × 5 friends = 30 items.

With factorisation (the 'smart' way): Let 'C' be chips and 'S' be soda. For one friend, you need 3C + 3S. Notice that '3' is common to both! So you can 'factor out' the 3: 3(C + S). This means 3 times (chips + soda).

Now, for 5 friends, you need 5 × [3(C + S)]. This is the same as 5 × 3 × (C + S) = 15(C + S). If C=1 bag and S=1 can, then 15(1+1) = 15(2) = 30 items. See how finding the common '3' first made it easier to think about the total items for each friend before multiplying by the number of friends? It groups things logically.