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Polynomials

Roots, Vieta's formulas, polynomial identities, and factorisation

Key Concepts
  • Vieta's formulas link the roots of a polynomial to its coefficients
  • The remainder theorem: P(a) is the remainder when P(x) is divided by (x − a)
  • The factor theorem: (x − a) is a factor iff P(a) = 0
  • Symmetric functions of roots can be evaluated without finding the roots
Important Formulae
Vieta (quadratic) α+β = −b/a; αβ = c/a
Useful identity α² + β² = (α+β)² − 2αβ
Remainder theorem Remainder of P(x) ÷ (x−a) = P(a)
Quick Tips
  • Use Vieta's formulas to avoid solving for the roots explicitly.
  • Spot integer roots quickly with the rational root test.
Sample Practice Questions

  1. Given roots α and β, the quadratic is x² - (α+β)x + αβ. For roots 2+√3 and 2-√3:

    • x² - 4x + 1
    • x² + 4x - 1
    • x² - 4x - 1
    • x² + 4x + 1
    Show answer

    Answer: x² - 4x + 1

  2. If r, s, t are roots of x³ + px + q = 0, then r + s + t = ?

    • -p
    • 0
    • p
    • q
    Show answer

    Answer: 0

  3. By Descartes' rule, a polynomial with all positive coefficients has:

    • Only positive roots
    • No positive roots
    • At most one positive root
    • Two positive roots
    Show answer

    Answer: No positive roots

  4. If a + b + c = 1, ab + bc + ca = -1, abc = -1, find a³ + b³ + c³.

    • -3
    • -1
    • 1
    • 4
    Show answer

    Answer: 1

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Practice Questions

Practise PRMO questions on Polynomials. Answers are revealed after each question.

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Algebra