Expanding and Factorising

Expanding brackets

Expanding means multiplying out brackets. Multiply the term outside the bracket by each term inside.

Single brackets: 3(x + 4) = 3x + 12. Watch signs: −2(x − 5) = −2x + 10.

Expanding double brackets

Multiply every term in the first bracket by every term in the second. A helpful order is FOIL (First, Outside, Inside, Last):

(x + 3)(x + 5)

• First: x × x = x²
• Outside: x × 5 = 5x
• Inside: 3 × x = 3x
• Last: 3 × 5 = 15

Combine: x² + 5x + 3x + 15 = x² + 8x + 15.

Difference of two squares: (x + a)(x − a) = x² − a², e.g. (x + 4)(x − 4) = x² − 16.

Factorising

Factorising is the reverse of expanding — writing an expression as a product of factors (putting brackets back in).

Common factor

Take out the highest common factor of every term:

• 6x + 9 = 3(2x + 3) (HCF is 3).
• 4x² + 8x = 4x(x + 2) (HCF is 4x).

Factorising quadratics (x² + bx + c)

Find two numbers that multiply to c and add to b.

Example: factorise x² + 7x + 12.

• Two numbers that multiply to 12 and add to 7 → 3 and 4.
• So x² + 7x + 12 = (x + 3)(x + 4).

With a negative: x² − 2x − 15 → two numbers multiplying to −15, adding to −2 → −5 and +3 → (x − 5)(x + 3).

Difference of two squares (factorising)

x² − 25 = (x + 5)(x − 5) (both terms are perfect squares, separated by a minus).

Worked example

Factorise x² + 2x − 8.

• Two numbers multiplying to −8, adding to +2 → +4 and −2.
• Answer: (x + 4)(x − 2). Check by expanding: x² − 2x + 4x − 8 = x² + 2x − 8 ✓

Common mistakes

• Only multiplying the first term inside a bracket — multiply every term.
• Sign errors when expanding a negative outside the bracket.
• For quadratics, picking numbers that multiply correctly but don't add to b (or vice versa).

Exam tips

• Use FOIL to keep double-bracket expansions organised.
• Always check factorising by expanding back.
• Spot the difference of two squares (a² − b²) — it factorises instantly.

Key facts to remember

• Expanding: multiply the outside term by each inside term; use FOIL for double brackets.
• Factorising (reverse): take out the common factor, or for x² + bx + c find two numbers that multiply to c, add to b.
• Difference of two squares: a² − b² = (a + b)(a − b).