🎯 What You're About to Master
Expanding brackets means multiplying everything outside by everything inside. Factorising is the reverse—finding what's common and putting things back into brackets. Together, these are the foundations of algebraic manipulation. Whether you're simplifying expressions, solving equations, or working with quadratics, you'll need these skills constantly.
Why this matters
Expanding and factorising appear in almost every algebra question. Mastering them now saves time in exams and makes harder topics (like solving quadratics) feel much easier. Plus, the methods for quadratics extend to calculus later.
📖 Expanding Brackets
Single Brackets (Multiplying One Term)
a(b + c) = ab + ac
Multiply the term outside by each term inside. It's distributive multiplication.
Example: 3x(x + 2) = 3x·x + 3x·2 = 3x² + 6x
Watch Out: Negative Signs
Critical rule
Negative × Positive = Negative
Negative × Negative = Positive
Example: -7x(4 - 5y) = -7x(4) + (-7x)(-5y) = -28x + 35xy
Multiple Single Brackets Added Together
Expand each bracket separately, then collect like terms
Example: Expand and simplify 2(x + 5) + 3x(x - 8)
Step 1: Expand first bracket: 2(x + 5) = 2x + 10
Step 2: Expand second bracket: 3x(x - 8) = 3x² - 24x
Step 3: Combine: 2x + 10 + 3x² - 24x
Step 4: Collect like terms (the x terms): 2x - 24x = -22x
Final: 3x² - 22x + 10
1
Expand and simplify: 4(x + 7) + 5x(3 - x)
Double Brackets (Two Brackets Multiplied)
Every term in bracket 1 × every term in bracket 2
(a + b)(c + d) = ac + ad + bc + bd (four multiplications total)
Method 1: FOIL
First, Outer, Inner, Last
Example: Expand (2x - 3)(x + 4)
First: 2x × x = 2x²
Outer: 2x × 4 = 8x
Inner: -3 × x = -3x
Last: -3 × 4 = -12
Combine: 2x² + 8x - 3x - 12
Collect like terms: 2x² + 5x - 12
Method 2: Grid Method (For Complex Expressions)
Write brackets as row and column headings, multiply cells, then add all terms.
2
Expand: (x - 3)(3x - 5)
Squared Brackets (Special Case)
(x + a)² = (x + a)(x + a) — treat as two separate brackets
Common mistake: (x + 3)² ≠ x² + 9. You must expand as (x + 3)(x + 3).
Example: Expand (2x + 3)²
Rewrite: (2x + 3)(2x + 3)
FOIL: 4x² + 6x + 6x + 9
Collect: 4x² + 12x + 9
Triple Brackets (Three Brackets Multiplied)
Expand two brackets first, then multiply the result by the third
(a + b)(c + d)(e + f) → First do (a + b)(c + d), then multiply by (e + f)
3
Expand: (x + 1)(x + 2)(x + 3) — just expand the first two brackets
🔄 Factorising Basics
What Is Factorisation?
Factorisation is the reverse of expanding
Instead of removing brackets, you put things into brackets by finding common factors.
3x + 6 is not factorised (two separate terms)
3(x + 2) is factorised (multiplication of factors)
Factorising Out Common Factors
Find the highest common factor (HCF) of all terms, write it outside brackets
Example: Factorise 12x² + 18x
Final: