Notation, vocabulary and manipulation: simplify and manipulate algebraic expressions (including those involving surds and algebraic fractions) by: •• expanding products of two or more binomials •• factorising quadratic expressions of the form ax² + bx + c (Higher content only)

The idea

This Higher-only skill covers simplifying and manipulating algebraic expressions, including surds and algebraic fractions. Expanding products of two or more binomials means multiplying every term in one bracket by every term in the other, then collecting like terms; for example (x + 3)(x − 5) = x² − 2x − 15. Factorising quadratics of the form ax² + bx + c reverses expansion: find two numbers that multiply to ac and sum to b, split the middle term, then factorise in pairs. Surds such as √8 simplify to 2√2, and algebraic fractions are simplified by factorising numerators and denominators and cancelling common factors. Always check by expanding or substituting a value.

What earns the marks

• Expanding two binomials correctly: multiply each term in the first bracket by each term in the second, then collect like terms, e.g. (x + 4)(x − 2) = x² + 2x − 8.
• Expanding three or more binomials by multiplying two brackets first and then multiplying the resulting expression by the remaining binomial.
• Factorising ax² + bx + c by finding two numbers with product ac and sum b, splitting the middle term, and factorising in pairs, e.g. 6x² + 11x + 4 = (3x + 4)(2x + 1).
• Simplifying surds by extracting the largest square factor, e.g. √50 = 5√2, and rationalising denominators where required.
• Simplifying algebraic fractions by factorising numerator and denominator fully and cancelling common factors, stating any excluded values where relevant.

Common mistakes and how to put them right

• Sign errors when expanding brackets with negative terms.

  Put it right by checking each product carefully.

• Forgetting to multiply all terms when expanding three binomials, such as omitting the product of the first and third brackets.

  Put it right by expanding two brackets first and then multiplying every term.

• Incorrect factorisation of ax² + bx + c when a is not 1, often by assuming the factors are simply (x + p)(x + q).

  Put it right by using the ac method or checking by expansion.

• Simplifying algebraic fractions by cancelling terms rather than factors, e.g. cancelling x from (x + 2)/x incorrectly.

  Put it right by factorising first and cancelling only common factors.

Exam tips

• Always show your expansion or factorisation steps clearly so method marks can be awarded even if the final answer is wrong.
• Check your factorisation by expanding it back to the original expression; this catches sign and coefficient errors quickly.
• When simplifying surds, look for the largest square factor to reach the simplest form in one step, and rationalise denominators if the question requires it.