\(P(a)=0\)
Use a known factor to produce an equation, verify a candidate root, or reduce the degree.
IB MATHEMATICS AA · HIGHER LEVEL · PREMIUM
Polynomials
A focused Higher Level chapter built around the factor theorem, remainder theorem, polynomial division, repeated roots, and the two root relationships students actually need: the total sum and total product of the roots. The chapter then moves into parameters, transformed roots, GDC investigations and long connected exam-style synthesis.
CORE IDEA
Turn every sentence into an equation.
“\(x-a\) is a factor” becomes \(P(a)=0\). “The remainder is \(R\)” becomes \(P(a)=R\). “\(a\) is a repeated root” becomes \(P(a)=0\) and \(P'(a)=0\). Once that translation is automatic, most polynomial questions become a structured system rather than guesswork.
\(P(a)=R\)
Find remainders by substitution instead of doing unnecessary long division.
\(P(a)=P'(a)=0\)
Two conditions create the extra equation needed for parameter problems.
Sum + product
Use only the total sum and total product formulas to extract global information without solving every root.
EXAM TECHNIQUE
Use the representation that matches the task.
Evaluation form is fastest for remainders. Factor form is best for roots and multiplicity. Expanded form is useful for coefficient comparison. Substitution is the clean route for transformed roots. The notes repeatedly show why one representation is more efficient than another.
\[P(a)=R\qquad P(a)=0\qquad P(a)=P'(a)=0\]
The final examples combine several of these ideas, including a GDC-plus-calculus investigation of when a cubic has three distinct real roots.
INSIDE THE PACK
Detailed teaching, selective practice.
30 pages
Narrative explanations, worked examples, theorem translation, long division, sum/product of roots, GDC reasoning, reconstructed past-paper structures and original synthesis.
20 questions · 7 pages
Deliberately varied rather than repetitive: theorem problems, parameter recovery, non-linear remainders, repeated roots, root totals, GDC and multi-part HL finishers.
10 pages
Full algebraic reasoning for every worksheet question, including GDC interpretation, long division, transformed roots and the final synthesis problem.
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Mathematical Induction
Continue with the separate AA HL induction chapter for proof, divisibility, recursive sequences, calculus and trigonometric identities.