Polynomials and Curve Intersections
Use factor and remainder information to factorise or solve higher polynomials and form intersection equations.
How to study Higher Mathematics
Practise a skill without a calculator first, connect it to neighbouring techniques, then apply it in unfamiliar contexts with complete working and interpretation.
Core concepts
Concept 1
factor theorem
Exam cue: What does substituting a candidate root reveal?
Concept 2
remainder theorem
Exam cue: Can the polynomial be reduced to a quadratic factor?
Concept 3
cubic factorisation
Exam cue: Which equation results when the two y-values are equal?
Concept 4
quartic equations
Concept 5
simultaneous intersections
Risk pitfalls and guardrails
Stopping after finding one factor
Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.
Losing roots during division or factorisation
Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.
Solving each curve separately instead of equating them
Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.
Memory anchors
Root gives zero
If f(a) = 0, then x − a is a factor.
Remainder is f(a)
Division by x − a leaves remainder f(a).
Reduce the degree
Remove a known linear factor before solving the remaining polynomial.
Solve every factor
A complete solution includes roots from all factors.
Intersections share coordinates
Set the two function expressions equal, then recover y.
Check multiplicity and context
Repeated or inadmissible roots may change the final answer.
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
Factorise x³−4x²−x+4 fully.
Factorise 2x³+x²−8x−4 fully.
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
What is Pass Harbor?
Completely free exam prep for 247 UK exams.
- Practice questions
- Flashcards
- Study guides
- Mock exams
- No registration
- No paywall
- Start instantly
“No more expensive exam prep. Quality study tools should be accessible to everyone.”