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
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If α, β are roots of x² - 6x + 9 = 0, then (α - β)² = ?
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Answer: 0
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For x² = 4x - 3, evaluate x⁴ in terms of x without higher powers.
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Answer: 12x² - 16x + 9 = 48x - 48 - 16x + 9 = ...
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The number of roots of xⁿ = 1 in ℂ is:
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Answer: n
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The factor theorem states that (x - a) is a factor of p(x) iff:
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Answer: p(a) = 0
Practice Questions
Practise PRMO questions on Polynomials. Answers are revealed after each question.
Start Practice →Frequently Asked Questions
What are the key concepts in Polynomials for PRMO?
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.
Which formulae are important for Polynomials in PRMO?
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).
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