Algebraic manipulation; factorisation

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.

Here's how to factorise by finding the Highest Common Factor (HCF), which is the biggest number or letter that divides into all parts of an expression.

1. Look at all the terms (parts of the expression separated by + or - signs). For example, in 6x + 9.
2. Find the HCF of the numbers. For 6 and 9, the biggest number that divides both is 3.
3. Find the HCF of the letters (variables). In 6x + 9, only '6x' has an 'x', so there's no common letter.
4. Write the HCF outside a bracket. So far, we have 3( ).
5. Divide each original term by the HCF. 6x divided by 3 is 2x. 9 divided by 3 is 3.
6. Write the results inside the bracket. So, 3(2x + 3). You've factorised it! To check, multiply it back out: 3 × 2x = 6x and 3 × 3 = 9, so 6x + 9.

Not all expressions are the same, so we have different ways to factorise them, like having different tools for different jobs!

• Common Factor: This is what we just did. Look for a number or letter that divides into every term. E.g., 5y - 10 = 5(y - 2).
• Difference of Two Squares (DOTS): This is for expressions like a² - b². It's always factorised into (a - b)(a + b). Think of it like a special pattern! For example, x² - 9 = (x - 3)(x + 3) because x² is x times x, and 9 is 3 times 3.
• Quadratic Trinomials: These are expressions with three terms, usually in the form ax² + bx + c (like x² + 5x + 6). You need to find two numbers that multiply to 'c' and add up to 'b'. For x² + 5x + 6, the numbers are 2 and 3 (because 2 × 3 = 6 and 2 + 3 = 5). So it factorises to (x + 2)(x + 3). This is like finding two secret codes that work together!

Even superheroes make mistakes sometimes, but knowing them helps you avoid them!

• Forgetting the '1' when everything factors out: ❌ 3x + 3 = 3(x) ✅ 3x + 3 = 3(x + 1) (Remember, 3 divided by 3 is 1, not nothing!)
• Not finding the Highest Common Factor: ❌ 4x + 8 = 2(2x + 4) (You can still take out another 2 from 2x+4!) ✅ 4x + 8 = 4(x + 2) (Always look for the biggest number or letter that divides all terms.)
• Incorrect signs in Difference of Two Squares: ❌ x² - 16 = (x - 4)(x - 4) or (x + 4)(x + 4) ✅ x² - 16 = (x - 4)(x + 4) (One must be minus, one must be plus to cancel out the middle term.)
• Mixing up addition and multiplication for quadratic trinomials: ❌ For x² + 7x + 10, thinking the numbers are 1 and 10 (multiply to 10, but add to 11, not 7). ✅ For x² + 7x + 10, the numbers are 2 and 5 (multiply to 10, and add to 7). So, (x + 2)(x + 5).