Simplifying algebraic expressions

A core Edexcel skill spanning Foundation (collecting terms, expanding single brackets) and Higher (expanding double brackets, factorising quadratics).

Collecting like terms

Like terms have identical letter parts. Add/subtract their coefficients.

3x + 5x = 8x. 3x + 2y − x + 4y = 2x + 6y. 4a²b − a²b = 3a²b.

Expanding single brackets (distributive)

a(b + c) = ab + ac.

3(2x − 5) = 6x − 15. −2(x − 4) = −2x + 8 (sign change!).

Expanding double brackets (FOIL)

(a + b)(c + d) = ac + ad + bc + bd.

(x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15. (x − 2)(x + 7) = x² + 7x − 2x − 14 = x² + 5x − 14.

Special expansions

(a + b)² = a² + 2ab + b². (a − b)² = a² − 2ab + b². (a + b)(a − b) = a² − b² (difference of two squares).

Factorising — single bracket

Identify common factor (number and/or letter) and bring outside.

6x + 9 = 3(2x + 3). 4x² − 6x = 2x(2x − 3). 9a²b − 6ab² = 3ab(3a − 2b).

Factorising quadratics — leading coefficient 1

For x² + bx + c: find two numbers that multiply to c, add to b.

x² + 7x + 12: numbers 3 and 4 (product 12, sum 7) ⇒ (x + 3)(x + 4). x² − 5x + 6: numbers −2 and −3 ⇒ (x − 2)(x − 3). x² + x − 12: numbers 4 and −3 ⇒ (x + 4)(x − 3).

Factorising quadratics — leading coefficient ≠ 1 (Higher)

For ax² + bx + c: find two numbers that multiply to ac and add to b. Split the middle term.

2x² + 7x + 3: ac = 6, sum 7 ⇒ 1 and 6. 2x² + x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (2x + 1)(x + 3).

Edexcel exam tip

For "factorise fully", check that nothing further can come out. e.g. 4x² + 8x = 4x(x + 2), not 2(2x² + 4x).