Expanding, Factorising and Algebraic Manipulation
This is the toolkit the rest of algebra is built on. Almost every error recorded in lessons came from one of three things: signs, not multiplying everything, and not factorising fully.
1. Like terms
Like terms have exactly the same letters with the same powers. Only like terms can be added or subtracted.
- 3x + 5x = 8x ✓
- 3x + 5y — cannot be simplified ✓
- 3x + 5x² — cannot be simplified (different powers)
- 4xy + 3xy = 7xy ✓
x and x² are not like terms. Combining them was a recorded error — 9x² and 9x³ are different things entirely, and so are 8x² and 9x².
The sign belongs to the term in front of it. In 7x − 3x + 2x, the terms are +7x, −3x, +2x, giving 6x.
2. Expanding brackets
Single brackets
Multiply everything inside by the term outside.
- 3(x + 4) = 3x + 12
- 5(2x − 3) = 10x − 15
- x(x + 7) = x² + 7x
- 2x(3x − 4) = 6x² − 8x
Multiply the letters as well as the numbers. x × x = x², not 2x or x.
With a negative outside — the sign of every term flips:
- −2(x − 5) = −2x + 10
- −(x + 3) = −x − 3
A minus in front changes every sign inside. −2(x − 5) gives +10, because −2 × −5 = +10. Getting “−18x + 72” instead of “+18x − 72” (or the reverse) was a documented error, and it comes from dropping this rule halfway through.
Negative × negative = positive. And −3 × −3 = 9, not 6 — a real error from a lesson. Multiplying, not adding.
Double brackets
Multiply every term in the first by every term in the second — four products (FOIL, or a grid).
(x + 3)(x + 5)
- x × x = x²
- x × 5 = 5x
- 3 × x = 3x
- 3 × 5 = 15
- = x² + 8x + 15
(2x − 3)(x + 4)
- 2x² + 8x − 3x − 12 = 2x² + 5x − 12
Four products, every time. Adding the brackets instead of multiplying them, or missing the cross terms, was recorded repeatedly. A grid makes it hard to miss one.
Squared brackets:
(x + a)² means (x + a)(x + a) — expand it properly.
- (x + 5)² = x² + 10x + 25 — not x² + 25
- (2x − 4)² = 4x² − 16x + 16 — note the 4x²
Squaring a bracket is not squaring each term. This is one of the highest-frequency errors across the whole syllabus.
Triple brackets
Expand two brackets first, simplify, then multiply that result by the third.
(x + 1)(x + 2)(x + 3)
- (x + 1)(x + 2) = x² + 3x + 2
- (x² + 3x + 2)(x + 3) = x³ + 3x² + 3x² + 9x + 2x + 6
- = x³ + 6x² + 11x + 6
Simplify before the third bracket. Skipping that step multiplies the number of chances to slip, and was flagged as a cause of wrong coefficients.
3. Factorising — the reverse
Factorising puts an expression back into brackets by taking out what is common.
Common factors
Find the HCF of the numbers and the lowest power of each common letter.
- 6x + 9 = 3(2x + 3)
- 4x² + 8x = 4x(x + 2)
- 16x² + 20xy = 4x(4x + 5y)
“Factorise fully” means take out EVERYTHING common. For 16x² + 20xy the common factor is 4x, not 2 and not 4 — a specific error recorded in lessons. A partly factorised answer does not get full marks.
Check by expanding. 4x(4x + 5y) = 16x² + 20xy ✓
Difference of two squares
a² − b² = (a + b)(a − b)
- x² − 36 = (x + 6)(x − 6)
- z² − 36 = (z + 6)(z − 6)
- 9x² − 25 = (3x + 5)(3x − 5)
Take the square root of each part. For x² − 36 the roots are x and 6 — the number whose square is 36. Confusing “the square root of 36” with “36 squared” was a documented slip.
Only works for a subtraction of two squares. x² + 36 does not factorise.
Quadratic expressions
For x² + bx + c, find two numbers that multiply to c and add to b:
- x² + 7x + 12 = (x + 3)(x + 4)
- x² − 5x + 6 = (x − 2)(x − 3)
Factorising by grouping
For four terms, group them in pairs:
ax + ay + bx + by
- a(x + y) + b(x + y)
- = (a + b)(x + y)
The bracket must be identical in both halves. If it isn’t, regroup or check your signs.
4. Index laws in algebra
The same rules as with numbers, applied to letters:
- x⁵ × x³ = x⁸ (add)
- x⁷ ÷ x³ = x⁴ (subtract)
- (x⁵)³ = x¹⁵ (multiply)
- (3x²)³ = 27x⁶ — the power applies to both the 3 and the x²
Add powers when multiplying; multiply them for a power of a power. Producing p⁸ where p¹⁵ was needed came from mixing these two rules.
A power outside a bracket hits everything inside. (3x²)³ = 3³ × (x²)³ = 27x⁶ — the coefficient must be cubed too.
5. Simplifying algebraic fractions
Factorise the top and bottom first, then cancel common factors:
(x² − 9)/(x + 3) = (x + 3)(x − 3)/(x + 3) = x − 3
You can only cancel FACTORS, never individual terms. In (x + 3)/(x + 5) nothing cancels — the 3 and 5 are not factors of anything. “Cancelling” the x is a serious and common error.
6. Mistakes that cost marks
Combining unlike terms such as x and x².
Losing a sign when a bracket has a minus in front.
Multiplying only the first term inside a bracket.
Adding the brackets instead of multiplying them.
Missing one of the four products in a double expansion.
Expanding (x + a)² as x² + a².
Not simplifying before the third bracket.
Factorising incompletely when the question said “fully”.
Taking the wrong root in a difference of two squares.
Multiplying indices when you should add them.
Forgetting to raise the coefficient to the outside power.
Cancelling terms instead of factors in a fraction.
Frequently asked questions
What are like terms? Terms with exactly the same letters and powers — only these can be combined.
How do I expand a single bracket? Multiply everything inside by the term outside, including the letters.
What happens with a negative outside the bracket? Every sign inside changes.
How do I expand double brackets? Multiply every term in the first by every term in the second — four products — then collect like terms.
What is (x + 5)²? x² + 10x + 25. It is (x + 5)(x + 5), not x² + 25.
How do I expand three brackets? Expand two, simplify, then multiply by the third.
What does “factorise fully” mean? Take out the highest common factor — numbers and letters — so nothing common remains.
How do I factorise a difference of two squares? a² − b² = (a + b)(a − b), taking the square root of each part.
How do I factorise four terms? Group them in pairs, factorise each pair, and take out the common bracket.
Can I cancel the x in (x + 3)/(x + 5)? No. You can only cancel factors, and here there are none.
Quick revision checklist
- I only combine genuinely like terms
- I keep the sign attached to each term
- I expand single brackets, multiplying letters as well as numbers
- I flip every sign when there’s a minus outside
- I know negative × negative = positive
- I do all four products in a double expansion
- I expand (x + a)² properly
- I simplify before multiplying by a third bracket
- I factorise fully, taking out the HCF including letters
- I check factorising by expanding back
- I recognise and factorise a difference of two squares
- I can factorise quadratics and by grouping
- I apply index laws correctly, including to coefficients in brackets
- I cancel only factors in algebraic fractions
These notes cover expanding, factorising and algebraic manipulation in the Cambridge IGCSE Mathematics (0580) syllabus, and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Maths lessons, with particular attention to the errors students make most often and the wording examiners reward. Always check the current syllabus and formula list for your own exam series.
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