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Mathematics9Updated 7 Sept 20262 pages

Mastering Factorisation Techniques

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Louis Glancy@louisglancy
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Factorising is like breaking down maths expressions into smaller, simpler parts - think of it as finding what pieces multiply together to make the original expression. You'll use these skills constantly in your Junior Cert maths, so mastering these four main types will make algebra much easier!
Factorising notes – page 1

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The Four Types of Factorising

There are four main types of factorising you need to know, and each one has a pattern you can spot. Most expressions with 2 terms use either HCF or DOTS, 3 terms usually need the quadratic method, and 4 terms typically use grouping.

Highest Common Factor (HCF) is your starting point - find the biggest number and letter that appears in all terms. For example, with 3x² + 6x, both terms contain 3x, so you can factor out 3xx+2x + 2. It's like reverse multiplication!

The grouping method works brilliantly with 4 terms. You split them into two pairs, factor each pair separately, then spot the common bracket that appears twice. When you see 3ac + 6c + ad + 2d, you get 3ca+2a + 2 + da+2a + 2, which becomes 3c + d$$a + 2.

Quick Tip: Always check your factorising by multiplying the brackets back out - you should get your original expression!

Factorising notes – page 2

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Quadratics and DOTS Methods

Quadratic factorising uses trial and error, but there's a system to it. Start with x $$x since you need x² as your first term. Then hunt for two numbers that multiply to give your last term - if it's negative, your signs will be different in the brackets.

Check your work by using FOIL (First, Outer, Inner, Last) to multiply back out. The outer and inner terms should add up to your middle term. Don't worry if your first guess is wrong - just try different number pairs until it works!

DOTS (Difference of Two Squares) is actually the easiest once you spot the pattern. You need something squared minus something else squared, like x² - 36. Just rewrite it as xx² - (6)², then use the magic formula: (first + second)(first - second).

Remember: With DOTS, you always get one bracket with addition and one with subtraction - it's the same numbers, just different signs!

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