Algebra and Functions
Use rational indices, surds, quadratics, inequalities, polynomials, factor and remainder theorems and function mappings.
How to prepare for the current TMUA
Secure the shared mathematical specification, practise application in unfamiliar settings, then make every Paper 2 implication and proof step explicit.
Core concepts
Concept 1
Manipulate exact algebraic forms while preserving domains and equivalence.
Exam cue: Factor or simplify before selecting a solving route.
Concept 2
Use discriminants, factorisation, division and theorems to analyse polynomial roots.
Exam cue: Check restrictions before cancelling or applying an inverse.
Concept 3
Interpret functions as mappings and distinguish one-to-one from many-to-one behaviour.
Exam cue: Use a value such as f(a) to test a claimed factor or remainder.
Risk pitfalls and guardrails
Cancelling terms across addition.
Guardrail: Do not treat examples as proof, reverse an implication, cancel a possible zero or rely on calculator-only intuition.
Ignoring extraneous or excluded values.
Guardrail: Do not treat examples as proof, reverse an implication, cancel a possible zero or rely on calculator-only intuition.
Assuming every function is one-to-one.
Guardrail: Do not treat examples as proof, reverse an implication, cancel a possible zero or rely on calculator-only intuition.
Memory anchors
Discriminant
b² − 4ac controls the number of real quadratic roots.
Complete the square
Expose a quadratic's turning point and range.
Factor theorem
x − a is a factor exactly when f(a) = 0.
Remainder theorem
Division by x − a leaves remainder f(a).
Surd discipline
Simplify and rationalise while keeping values exact.
Function domain
Check permitted inputs before composing or inverting.
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
A positive number x satisfies x^(3/2)=8. What is x+1?
The expression √48−√12 is written as a√3. What is a²?
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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