Expansion and Factorisation
Expand bracket structures systematically and reverse the process through complete factorisation.
How to study National 5 Mathematics
Secure one operational skill at a time, connect it to neighbouring techniques, then apply it in unfamiliar contexts with complete working and a checked conclusion.
Core concepts
Concept 1
Expansion distributes every term and requires careful signs when brackets or squared binomials are involved.
Exam cue: Mark each multiplication when expanding more than two terms.
Concept 2
Factorisation can use a common factor, difference of squares, unitary or non-unitary trinomial structure, or combinations.
Exam cue: Check for a common factor before using another pattern.
Concept 3
A fully factorised result has every common factor extracted and every required polynomial factor completed.
Exam cue: Multiply factors back mentally to verify coefficients and signs.
Risk pitfalls and guardrails
Losing the middle term when squaring a binomial.
Guardrail: Check signs, brackets, restrictions, units, accuracy and whether the answer needs a reason.
Stopping after extracting only part of a common factor.
Guardrail: Check signs, coefficients and both sides of every algebraic transformation.
Using difference of squares when the terms are added.
Guardrail: Check signs, brackets, restrictions, units, accuracy and whether the answer needs a reason.
Memory anchors
Distribution
Every term outside a bracket multiplies every term inside it.
Common factor
Extract the greatest expression shared by all terms first.
Difference of squares
a² − b² = (a − b)(a + b).
Unitary trinomial
Find two numbers whose product is c and sum is b.
Non-unitary trinomial
Use factors whose cross-products combine to the middle coefficient.
Factor check
Expand the answer to recover the original expression.
Checkpoint rule
Do the check-up only after you can summarize each concept in one sentence and identify one dangerous pitfall from memory.
Knowledge Check (after reading)
Short check-up to confirm understanding of this module.
Check-up Questions
Expand 3(x + 4).
Expand −2(3x − 5).
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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