What does Descartes' rule of signs tell us?

The number of positive real roots equals the number of sign changes in f(x) (or fewer by an even number); negative roots use f(−x).

How do complex/irrational roots occur?

For polynomials with real (rational) coefficients, complex roots occur in conjugate pairs a ± ib, and surd roots in pairs a ± √b.

Tamil Nadu Class 12 Maths: Theory of Equations

Samacheer Kalvi Class 12 Maths Theory of Equations full guide - Vieta's formulas (relation between roots and coefficients), nature of roots, transformation of equations, and Descartes' rule with worked examples.

Updated 6 June 20263 min read

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Overview

This chapter studies polynomial equations through the relation between their roots and coefficients (Vieta's formulas), the nature of roots, and methods to transform equations.

1. Relation Between Roots and Coefficients

Quadratic ax² + bx + c = 0, roots α, β:
   α + β = −b/a ,   αβ = c/a

Cubic ax³ + bx² + cx + d = 0, roots α, β, γ:
   α+β+γ = −b/a , αβ+βγ+γα = c/a , αβγ = −d/a

2. Nature of Roots

Complex roots of a real-coefficient polynomial occur in conjugate pairs (a ± ib).
Surd (irrational) roots occur in pairs (a ± √b).
A polynomial of degree n has exactly n roots (counting multiplicity).

3. Descartes' Rule of Signs

Number of positive real roots = number of sign changes in f(x) (or less by an even number). For negative roots, examine f(−x).

4. Transformation of Equations

To form an equation whose roots are k times / reciprocal of / increased by the original roots, substitute accordingly (e.g. replace x by x/k for roots multiplied by k).

Worked Examples

Example 1 — Sum and product of roots

For 2x² − 5x + 3 = 0, find α + β and αβ.

α + β = −(−5)/2 = 5/2 ;  αβ = 3/2

Example 2 — Using a known root

One root of x³ − 6x² + 11x − 6 = 0 is 1. Find the others.

Sum of roots = 6, product = 6. With α = 1:
β + γ = 5, βγ = 6 → t² − 5t + 6 = 0 → t = 2, 3
Roots: 1, 2, 3

Example 3 — Form a new equation

Roots of x² − 3x + 2 = 0 are α, β. Form the equation whose roots are 2α, 2β.

New sum = 2(α+β) = 6 ; new product = 4αβ = 8
Equation: x² − 6x + 8 = 0

Example 4 — Descartes' rule

Discuss real roots of f(x) = x³ − 3x² + 4x − 5.

Signs of f(x): + − + −  → 3 sign changes → 3 or 1 positive roots
f(−x) = −x³ − 3x² − 4x − 5 : all negative → 0 sign changes → 0 negative roots

Common Mistakes

Forgetting the sign in α+β = −b/a.
Assuming all roots are real (complex roots come in pairs).
Miscounting sign changes in Descartes' rule.
Wrong substitution while transforming equations.

Exam Tips & Revision Checklist

Memorise Vieta's formulas for quadratic and cubic.
Use a known root to reduce the degree (factor it out).
Practised: sum/product of roots, reducing a cubic, transforming roots, Descartes' rule.

Frequently Asked Questions

For ax³ + bx² + cx + d = 0 with roots α, β, γ: α+β+γ = −b/a, αβ+βγ+γα = c/a, αβγ = −d/a.