Algebra, Functions, Sequences and Series
Manipulate polynomials and functions and use recurrences, arithmetic and geometric series and binomial expansions.
How to prepare for the current ESAT
Confirm your course combination, build no-calculator fluency from Mathematics 1 upward, and use official Pearson materials for authentic timing and interface rehearsal.
Core concepts
Concept 1
Use rational indices, surds, discriminants, polynomial division, factor and remainder theorems and function mappings.
Exam cue: Check domain restrictions before manipulating a function.
Concept 2
Analyse sequences defined explicitly or recursively.
Exam cue: Test a factor by substitution before division.
Concept 3
Apply finite and infinite geometric sums and positive-integer binomial expansions.
Exam cue: For an infinite sum, verify |r| < 1 first.
Risk pitfalls and guardrails
Assuming every function is one-to-one.
Guardrail: Do not use calculator-dependent shortcuts, import a fact outside the annual specification or overlook a unit, sign, scale or course-module boundary.
Using an infinite geometric sum when the series diverges.
Guardrail: Do not use calculator-dependent shortcuts, import a fact outside the annual specification or overlook a unit, sign, scale or course-module boundary.
Dropping binomial coefficients.
Guardrail: Do not use calculator-dependent shortcuts, import a fact outside the annual specification or overlook a unit, sign, scale or course-module boundary.
Memory anchors
Discriminant
b² − 4ac determines the number of real quadratic roots.
Factor theorem
x − a is a factor exactly when f(a) = 0.
Remainder theorem
Dividing by x − a leaves remainder f(a).
Recurrence
Generate each term from the preceding term and rule.
Geometric infinity
S∞ = a/(1 − r) only when |r| < 1.
Binomial
Use coefficients with descending and ascending powers.
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
Simplify x^(3/2)x^(−1/2).
Rationalise 3/(2+√5).
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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