Turn expressions into factors, roots and useful identities. Begin at the first unfamiliar idea; the prerequisite links help you find a shorter route. You can open any lesson.

Unsure where to start? Try the starting-point check.

  1. A01

    Polynomial functions

    Recognise a polynomial and explain how its degree can change under addition.

    Inside: Expressions versus polynomial functions · Degree, coefficients and evaluation · Zeros and the zero polynomial

  2. A02

    Polynomial division

    Write and check P = DQ + R with the correct remainder degree.

    Inside: Polynomial division · Quotient-remainder identity and degree bounds

  3. A03

    Remainder and factor theorems

    Choose a substitution or a new polynomial that turns given values into factors.

    Inside: Remainders by substitution · Roots and linear factors · Constructing an auxiliary polynomial · Integer-polynomial divisibility bridge

  4. A04

    Polynomial roots and multiplicity

    Use the theorem with the correct coefficient field and multiplicities.

    Inside: Complex roots and multiplicity · Root-counting consequences and polynomial identities

  5. A05

    Solving polynomial equations

    Solve structured equations without losing or inventing roots.

    Inside: Factorisation and substitutions · Real-root restrictions · Finding all roots and checking completeness

  6. A06

    Vieta’s formulas

    Find root expressions without calculating each root separately.

    Inside: Roots and coefficients of a quadratic · Cubic and higher relations · Transformed-root equations

  7. A07

    Symmetric polynomials

    Rewrite symmetric expressions in a smaller set of quantities.

    Inside: Elementary symmetric expressions · Power sums and reductions · Symmetric systems

  8. A08

    Common roots of polynomials

    Reduce the degree of an equation satisfied by a shared root.

    Inside: Elimination using a common root · Polynomial gcd and parameter cases

  9. A09

    Irreducible polynomials

    Prove irreducibility with a valid criterion over a specified field.

    Inside: Coefficient fields and reducibility · Rational-root tests and their limits · Eisenstein and modular tests · Integer-value arguments

These original lessons introduce the methods and give practice with solutions. A single short session is not a full assessment of Olympiad readiness. Build depth through the written challenges and your country’s official past papers.