Theory of Equations — Chapter Summary
Tamil Nadu Board · Class 12 · Mathematics
Summary of Theory of Equations for Tamil Nadu Board Class 12 Mathematics. Part of the Tamil Nadu Board Class 12 Mathematics syllabus.
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Overview
The Theory of Equations is a fundamental branch of algebra that deals with finding solutions (roots) of polynomial equations. This chapter explores the relationships between the coefficients and roots of polynomial equations, methods to form equations with given roots, and techniques to solve higher
Key Concepts
A polynomial equation of degree n
A polynomial equation of degree n in one variable x is expressed as aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ = 0, where aₙ ≠ 0. The value c is called a root o
Vieta's formulae establish algebraic relationships between
Vieta's formulae establish algebraic relationships between the roots and coefficients of a polynomial without explicitly finding the roots. For a quad
When roots of a polynomial
When roots of a polynomial are known, we can construct the equation using: (1) Direct multiplication: (x - α)(x - β)(x - γ)... = 0, or (2) Using Vieta
Every polynomial equation of degree n
Every polynomial equation of degree n ≥ 1 has at least one root in the complex number system ℂ. Using this theorem repeatedly, we conclude that a poly
If a polynomial equation has real
If a polynomial equation has real coefficients and a complex number z₀ = a + bi (where b ≠ 0) is a root, then its complex conjugate z̄₀ = a - bi must
Learning Objectives
- Form polynomial equations satisfying given conditions on roots using Vieta's formulae and relationships between roots and coefficients
- Demonstrate and apply techniques to solve polynomial equations of higher degree (cubic, quartic, and beyond) using factorization and substitution methods
- Solve equations of higher degree when some roots are known to be complex numbers, irrational numbers (surds), or rational numbers
- Identify and solve reciprocal equations by recognizing their special structure and applying appropriate transformation techniques
- Determine the number of positive and negative roots of polynomial equations using Descartes Rule of Signs
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Sources & Official References
Content is aligned to the official syllabus. Refer to the board website for the latest curriculum.
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