Numbers and Algebraic Expressions
What this domain tests
Short-answer items on signed arithmetic, order of operations, expansion, factorisation and square-root simplification. Marks are lost on the intermediate lines rather than on the final answer, so the exam rewards clean, justified steps.
Core rules
- Brackets first, innermost outward; then powers, then multiplication/division, then addition/subtraction.
- In −3² the exponent binds before the minus, giving −9; (−3)² = 9. Keep the distinction.
- Distributive law works both ways: a(b+c) = ab+ac, and factorising is the reverse. Check a factorisation by expanding mentally.
- Three identities carry most of the marks: (a+b)² = a²+2ab+b², (a−b)² = a²−2ab+b², (a+b)(a−b) = a²−b².
- √(a²) = |a|, never automatically a. Like radicals combine: 3√2+5√2 = 8√2, but √2+√3 stays as it is.
- Rationalise a denominator by multiplying numerator and denominator by the conjugate.
Worked example
Simplify (2x−3)² − (x+1)(x−5). First term: 4x²−12x+9. Second: x²−5x+x−5 = x²−4x−5. Subtracting the entire second bracket: 4x²−12x+9 − x² + 4x + 5 = 3x²−8x+14. Check with x=1: original gives 1 − (−4) = 5; the answer gives 3−8+14 = 9. The mismatch signals an error in the expansion, so recalculate rather than move on.
Traps that cost marks
Squaring each term separately and writing (2x−3)² as 4x²+9; letting the minus sign strike only the first term of the subtracted bracket; combining unlike radicals; cancelling terms inside a square root.