O-Level E-Math and SEC G3 Mathematics K310
N5: Algebraic expressions and formulae
Manipulate expressions, factorise, change the subject, and substitute with brackets and units kept visible.
Core notes
Algebra records a general relationship. Reliable manipulation preserves equality, keeps restrictions visible, and chooses expansion, factorisation or a common denominator according to the required form.
Notation, substitution and modelling
Letters may represent unknown or variable quantities. Juxtaposition means multiplication, powers show repeated factors, and brackets group operations. Substitute with brackets, especially for negative values, and translate a real situation by defining each letter before writing the expression.
Patterns and linear expressions
For a sequence with constant first difference, the coefficient of is that difference. Combine like terms only when their variable parts match. Expand brackets by multiplying every enclosed term, and factorise by extracting the greatest common factor or a shared binomial.
Products, identities and quadratics
Expand products systematically, then collect like terms. The three standard identities allow rapid expansion or factorisation. For , seek two factors whose product and sum reproduce the required coefficients.
Formulae and algebraic fractions
Changing the subject uses inverse operations while preserving equality. For algebraic fractions, factor before cancelling, multiply numerators and denominators, invert the second fraction when dividing, and use a lowest common denominator for addition or subtraction.
Formulae and methods
| Relationship | Use |
|---|---|
| Expand a squared sum. | |
| Expand a squared difference. | |
| Factorise a difference of two squares. |
Worked examples
Example 1: Simplify .
- Expand the bracket to obtain .
- Add and combine like terms.
- Write the result as .
Answer: The simplified expression is .
Example 2: Simplify .
- Use common denominator .
- Rewrite the first numerator as .
- Combine to get , with .
Answer: , where .
Chapter checkpoint
- Translate every symbol and bracket before manipulating the expression.
- Choose whether the target form calls for expansion, factorisation, substitution or rearrangement.
- State excluded values when an algebraic denominator could be zero.
Exam traps and retrieval check
Avoid these traps
- Cancelling terms across addition instead of cancelling common factors.
- Losing a negative sign when expanding or substituting.
- Giving a simplified algebraic fraction without its excluded values.
Check from memory
When can two algebraic terms be combined?
When their variable parts, including powers, are identical.
What should be done before cancelling an algebraic fraction?
Factorise the numerator and denominator.
Why use brackets when substituting a negative value?
They preserve the sign when powers or products are evaluated.
Official K310 coverage
This chapter maps to 16 official outcomes: N5.1, N5.2, N5.3, N5.4, N5.5, N5.6, N5.7, N5.8, N5.9, N5.10, N5.11, N5.12, N5.13, N5.14, N5.15, N5.16. Check the official 2027 K310 syllabus for exact assessable wording and paper details.