Chapter 5 of 12
Revision Notes
Theory of Equations — Revision Notes
Tamil Nadu Board · Class 12 · Mathematics
Theory of Equations revision notes for Tamil Nadu Board Class 12 Mathematics: 4 topics in quick points. Includes Basics of Polynomial Equations, Vieta's.
45 questions25 flashcards8 formulas & key relations5 concepts
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Key Topics to Revise
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1. Basics of Polynomial Equations
- A polynomial of degree n is P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where aₙ ≠ 0.
- The equation P(x) = 0 is called a polynomial equation.
- A value c ∈ ℂ satisfying P(c) = 0 is called a root or zero of the polynomial.
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2. Vieta's Formulae – Relations Between Roots and Coefficients
- Vieta's formulae connect the COEFFICIENTS of a polynomial to SUMS and PRODUCTS of its roots.
- For a quadratic ax² + bx + c = 0 with roots α, β: α + β = -b/a and αβ = c/a.
- A quadratic with roots α and β is: x² - (α + β)x + αβ = 0.
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3. Fundamental Theorem of Algebra
- FUNDAMENTAL THEOREM: Every polynomial equation of degree n ≥ 1 has at least one root in ℂ (complex numbers).
- Consequence: A polynomial equation of degree n has EXACTLY n roots in ℂ when counted with multiplicity.
- A polynomial equation of degree n cannot have MORE than n roots, even counting multiplicity.
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4. Nature of Roots – Complex Conjugate and Irrational (Surd) Roots
- COMPLEX CONJUGATE ROOT THEOREM: If a + ib (b ≠ 0) is a root of a polynomial with REAL coefficients, then a - ib is also a root.
- Non-real complex (imaginary) roots always occur in CONJUGATE PAIRS for polynomials with real coefficients.
- Since imaginary roots come in pairs, an odd-degree polynomial with real coefficients always has at least one real root.
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Full NotesKey Concepts
A polynomial equation of degree nVieta's formulae establish algebraic relationships betweenWhen roots of a polynomialEvery polynomial equation of degree nIf a polynomial equation has real
Frequently Asked Questions
What are the important topics in Theory of Equations for Tamil Nadu Board Class 12 Mathematics?
Key topics in Theory of Equations include Basics of Polynomial Equations, Vieta's Formulae – Relations Between Roots and Coefficients, Fundamental Theorem of Algebra, Nature of Roots – Complex Conjugate and Irrational (Surd) Roots. Study these first, then practise questions on each for the Tamil Nadu Board Class 12 board exam.
How should I revise Theory of Equations for the Tamil Nadu Board Class 12 board exam?
Learn the core ideas first, then work through the 45 practice questions on Theory of Equations. Revise definitions regularly and use flashcards for quick recall before the exam.
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