Grade 10 Algebraic Expressions & Factorising

Grade 10 algebraic expressions: products, factorising trinomials, difference of squares, sum and difference of cubes, and simplifying fractions, with worked…

Algebraic expressions are the toolkit for the whole of Grade 10 Paper 1. Together with equations and inequalities they carry about 30 of the 100 marks, and every functions, patterns and finance question leans on the same skills: multiplying out brackets, factorising, and simplifying algebraic fractions.

Grade 10 extends the Grade 9 work in three ways. You factorise trinomials where the coefficient of x² is not 1, you meet the sum and difference of two cubes, and you simplify algebraic fractions by factorising first and cancelling common factors. You also learn to tell rational and irrational numbers apart and to round irrational numbers sensibly.

Key concepts

  • Products of a binomial and a trinomial, e.g. (x + 2)(x² − 3x + 4)
  • Factorising by common factor, grouping, difference of two squares and trinomials
  • Sum and difference of two cubes: a³ ± b³ = (a ± b)(a² ∓ ab + b²)
  • Simplifying algebraic fractions by factorising and cancelling
  • Adding and subtracting algebraic fractions with the lowest common denominator
  • Rational and irrational numbers; rounding off and estimating surds

Worked examples

Factorise fully: 3x² − 12

Take out the common factor first: 3(x² − 4). Then x² − 4 is a difference of two squares, so 3x² − 12 = 3(x − 2)(x + 2).

Factorise: 6x² + x − 2

Find two numbers that multiply to 6 × (−2) = −12 and add to 1: 4 and −3. Split the middle term: 6x² + 4x − 3x − 2 = 2x(3x + 2) − 1(3x + 2) = (3x + 2)(2x − 1).

Simplify: (x² − 9) ÷ (x² + x − 6)

Factorise the top and bottom: (x − 3)(x + 3) ÷ [(x + 3)(x − 2)]. Cancel (x + 3) to get (x − 3)/(x − 2), with x ≠ −3 and x ≠ 2.

Common mistakes

  • Forgetting to take out a common factor first, so the factorisation is not complete.
  • Cancelling terms instead of factors, e.g. crossing out x² in (x² + 3)/x².
  • Getting the signs wrong in the sum and difference of cubes.
  • Leaving out the restrictions that make a denominator zero.

Exam tips

  • Check every factorisation by multiplying it out again; it takes ten seconds.
  • The word 'fully' in a question is a hint that there is a common factor to take out first.
  • With algebraic fractions, factorise everything before you cancel or find a common denominator.

Frequently asked questions

What is the difference between Grade 9 and Grade 10 factorising?

Grade 9 covers common factors, the difference of squares and trinomials like x² + 5x + 6. Grade 10 adds trinomials with a leading coefficient other than 1 (such as 6x² + x − 2), grouping in pairs, and the sum and difference of two cubes.

How do I remember the sum and difference of cubes?

The bracket keeps the sign of the question: a³ − b³ = (a − b)(…). In the second bracket the middle term has the opposite sign and the last term is always positive: a² + ab + b². So a³ − b³ = (a − b)(a² + ab + b²) and a³ + b³ = (a + b)(a² − ab + b²).

How many marks is algebra worth in Grade 10?

Algebraic expressions, equations and inequalities together are about 30 marks of the 100-mark Paper 1, according to the CAPS examination guidelines.

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