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Polynomials & Inequalities

Polynomial roots, factorisation, AM-GM, Cauchy-Schwarz, and algebraic inequalities

Key Concepts
  • Vieta's formulas and polynomial root structure
  • AM–GM, Cauchy–Schwarz and the rearrangement inequality
  • Sum-of-squares (SOS) arguments prove non-negativity
  • Equality cases identify the extremal configuration
Important Formulae
AM–GM (a₁+…+aₙ)/n ≥ ⁿ√(a₁…aₙ)
Cauchy–Schwarz (Σaᵢbᵢ)² ≤ (Σaᵢ²)(Σbᵢ²)
Rearrangement Σ aᵢb_{σ(i)} is max when both are sorted the same way
Quick Tips
  • Try to write an expression as a sum of squares to show it is ≥ 0.
  • Always state and check the equality case in an olympiad inequality proof.
Sample Practice Questions

  1. The product of the roots of x² − 5x + 6 = 0 is:

    • 5
    • 6
    • −6
    • 1
    Show answer

    Answer: 6

  2. Simplify (a + b)² − (a − b)².

    • 2ab
    • 4ab
    • a² + b²
    • 2a²
    Show answer

    Answer: 4ab

  3. If a + b = 5 and ab = 6, then a² + b² equals:

    • 11
    • 13
    • 25
    • 19
    Show answer

    Answer: 13

  4. How many real roots does x² + 1 = 0 have?

    • 0
    • 1
    • 2
    • Infinite
    Show answer

    Answer: 0

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Practice Questions

Practise RMO questions on Polynomials & Inequalities. Answers are revealed after each question.

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Algebra