Algebra, Number and Structure
What you must understand
This domain covers manipulation of algebraic expressions and the structure of number systems. Candidates must be confident with index laws for integer and rational indices, expansion and factorisation (including difference of squares, perfect squares, grouping, and the factor theorem for cubics), simplification of algebraic fractions, solving linear and quadratic equations and inequalities, simultaneous equations (substitution and elimination), and the discriminant Δ = b² − 4ac as a classifier of roots. You also need surds and rationalising denominators, logarithmic and exponential laws, and the ability to recognise equivalent forms of an expression.
Traps that appear
- Sign errors when expanding brackets with a negative leading coefficient.
- Forgetting to reverse an inequality when multiplying or dividing by a negative number.
- Assuming a quadratic has two real roots without checking the discriminant; 'no real solutions' means Δ < 0.
- Cancelling terms rather than factors in algebraic fractions, e.g. writing (x+3)/x as 3.
- Mishandling rational indices: x^(2/3) means the cube root squared, and the base must be non-negative for even roots.
- Losing the negative root when solving x² = k.
Worked example
Solve 2x² − 5x − 3 = 0. Factorise: (2x + 1)(x − 3) = 0, so x = −1/2 or x = 3. Check with the discriminant: Δ = 25 + 24 = 49 > 0, confirming two distinct real roots. Rule of thumb: for any quadratic, compute Δ first - it tells you how many roots to expect before you factorise.