Notation, vocabulary and manipulation: simplify and manipulate algebraic expressions (including those involving surds) by: •• expanding products of two binomials •• factorising quadratic expressions of the form x² + bx + c, including the difference of two squares

The idea

This skill set covers expanding and factorising. Expanding two binomials uses the distributive law four times, often remembered as FOIL: (x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15. Factorising x² + bx + c reverses this by finding two numbers that multiply to c and add to b, so x² + 7x + 12 = (x + 3)(x + 4). The difference of two squares, a² − b², factorises as (a + b)(a − b), for example x² − 49 = (x + 7)(x − 7). For Higher tier candidates, surds may appear in these expressions, for example (2 + √3)(2 − √3) = 4 − 3 = 1, using the difference of two squares. Always expand fully, collect like terms and check by substituting a value.

What earns the marks

• Expand the product of two binomials by multiplying every term in the first bracket by every term in the second, for example (x + 2)(x + 6) = x² + 8x + 12.
• Collect like terms after expansion, for example (2x + 1)(x + 4) = 2x² + 9x + 4.
• Factorise x² + bx + c by finding two numbers with product c and sum b, for example x² + 9x + 20 = (x + 4)(x + 5).
• Recognise and factorise the difference of two squares, for example x² − 36 = (x + 6)(x − 6).
• Expand and simplify products involving surds (Higher tier only), for example (3 + √2)(3 − √2) = 9 − 2 = 7.
• Check a factorisation by expanding the brackets and confirming the original expression is recovered.

Common mistakes and how to put them right

• Expanding (x + 3)(x + 5) as x² + 15 by multiplying only the first and last terms.

  Put it right: Multiply all four pairs, giving x² + 8x + 15.

• Factorising x² + 7x + 12 as (x + 7)(x + 12).

  Put it right: Find numbers with product 12 and sum 7, namely 3 and 4, giving (x + 3)(x + 4).

• Treating x² − 25 as (x − 5)².

  Put it right: The difference of two squares factorises as (x + 5)(x − 5).

• Incorrectly evaluating the product of surds, such as √3 × (−√3) = −9.

  Put it right: √3 × √3 = 3, so the product is −3, giving 4 − 3 = 1 for (2 + √3)(2 − √3) (Higher tier only).

Exam tips

• After expanding, count the terms before collecting to ensure no product has been missed.
• For factorising, list factor pairs of the constant term and test which pair adds to the coefficient of x.
• Check a factorisation by expanding it mentally or on paper and comparing with the original expression.
• In multiple-choice questions, substitute a simple value such as x = 2 into the original and each option to identify the equivalent expression.