Algebraic Expressions and Identities
This is the chapter where algebra stops being 'move the letters around' and starts being a modelling tool — an engineer's load ratio, a technician's mass-per-portion, a signal processor's transmission rate are all just rational expressions in disguise. Every rule below exists for one reason: to let you rewrite a formula without changing its value, so you can read off what it actually means.
Overview — The Shape of This Chapter
Six skills, one underlying discipline
Algebraic fractions, exponent laws, identities, expansion, substitution and factorisation are not six separate topics — they're six applications of the same discipline: an expression is only 'simplified' when every legal step has been applied and no further legal step remains. Examiners test whether you know which moves are legal (cancel a common FACTOR) and which aren't (cancel a common TERM).
- Algebraic fractions — multiply, divide, add and simplify expressions with polynomial numerators/denominators, always tracking restricted values.
- Exponent laws — the mechanical rules (product, quotient, power, zero, negative, fractional) that every other skill depends on.
- Special identities — and as instant shortcuts that replace slow FOIL expansion.
- Simplification and expansion — distributive law, collecting like terms, expanding double brackets fully.
- Substitution and evaluation — turning a general formula into a specific number, correctly ordered.
- Factorisation — running expansion in reverse: GCF, grouping, trinomials, difference of squares.
The shape of the chapter
Command terms you'll actually see on this topic
| Command term | What it demands | AO | Mark-earning move |
|---|---|---|---|
| Show that | Prove a given result using valid algebraic steps — the answer is already known, so the working is what's assessed. | AO2 | Every line must follow logically; jumping straight to the given answer without showing the identity/expansion scores 0, even if it matches. |
| Simplify | Reduce an expression to its lowest, fully-cancelled equivalent form. | AO1 | The accuracy mark is only awarded for the FULLY reduced form — partial cancelling loses it. |
| Factorise fully | Write as a product of factors that cannot be broken down any further. | AO1 | Stopping after removing just the GCF when a further difference-of-squares or trinomial split is still possible loses the final mark. |
| Hence | Use the previous part's result directly in your method. | AO2 | Restarting from scratch instead of quoting the earlier result can cost the method mark even if the final answer is correct. |
| Deduce | State a conclusion supported by reasoning already established, without re-deriving everything. | AO2 | A bare, unexplained answer loses the reasoning mark; over-working from first principles wastes time but isn't penalised. |
| Justify | Give a reasoned argument, typically comparing a computed value against a stated condition. | AO3 | A correct number with no concluding sentence answering the actual question ('yes, because…') loses the final justification mark. |
| Evaluate | Substitute given values and compute a numerical answer. | AO1 | Dropping the context units (kN, cups, hours) can cost a mark in modelling questions. |
| State | Give a rule or fact with no working required. | AO1 | One clean line is enough — writing a full derivation wastes time but isn't marked down. |
Key point
Overview