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
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How many real roots does x² + 1 = 0 have?
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Answer: 0
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The product of the roots of x² − 5x + 6 = 0 is:
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Answer: 6
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Simplify (a + b)² − (a − b)².
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Answer: 4ab
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If a + b = 5 and ab = 6, then a² + b² equals:
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Answer: 13
Practice Questions
Practise RMO questions on Polynomials & Inequalities. Answers are revealed after each question.
Start Practice →Frequently Asked Questions
What are the key concepts in Polynomials & Inequalities for RMO?
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.
Which formulae are important for Polynomials & Inequalities in RMO?
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.
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