Theory of Equations

Introduction to Theory of Equations

• This lecture discusses the relations between roots and coefficients of polynomial equations.
• Focusing initially on quadratic equations and extending to cubic, bi-quadratic, and n-degree polynomial equations.

Quadratic Equation

• General form: ax² + bx + c = 0
  • Sum of roots (α+β): -b/a
  • Product of roots (αβ): c/a

Cubic Equation

• General form: ax³ + bx² + cx + d = 0
  • After division by a: x³ + (b/a)x² + (c/a)x + (d/a) = 0
  • It has 3 roots: α, β, γ
  • Expressed as: (x - α)(x - β)(x - γ) = 0

Coefficients Relationship

• Coefficient of x² (sum of roots): -b/a
• Coefficient of x (sum of products of roots taken two at a time): c/a
• Constant term (product of roots): -d/a

Bi-quadratic Equation

• General form: ax^4 + bx^3 + cx^2 + dx + e = 0
  • Sum of roots: -b/a
  • Product of roots taken two at a time: c/a
  • Product of roots: -e/a

Generalizing the Pattern

• For n-degree polynomial: a₀xⁿ + a₁xⁿ⁻¹ + ... + aₙ = 0
  • Roots: α₁, α₂, ..., αₙ

Sum of Roots and Products

• Sum: Σαᵢ = -a₁/a₀
• Sum of products of two distinct roots: Σ(αᵢαⱼ) for i ≠ j = a₂/a₀
• Products of roots: (-1)ⁿ aₙ/a₀ (for n roots)

Finding Roots Products

• Example: For polynomial of degree 17:
  • Coefficient of xⁱ contributes to roots sum and product decisions.
• Given polynomial: 2x¹⁷ + 19x³ - 11, product of its roots = (-1)¹⁷ * -11/2 = value calculation.

Applications of Theoretic Concepts

• Use of concepts in transformation of equations.
• When roots are altered, apply transformation to derive new polynomial equations.

Examples of Root Transformations

1. Cubic equation
   • Given roots: α, β, γ, derive a polynomial for roots α+β, β+γ, γ+α by substituting x with -x.
2. Reciprocal Roots
   • If given relations involve reciprocals, derive equations based on substitutions.
3. Transformations for Squared Roots
   • Replace x with √x to adapt to specified transformations.

Relation in Arithmetic Progression (AP)

• If roots are in AP, establish relation through average definitions.
• Form equation based on given p, q, and r related to cubic equations in AP context. Transfer findings into designed format for ease of derivability.

Descartes' Rule of Signs

• A pivotal conclusion on determining positive and negative roots based on the sign of coefficients.
  • This factor provides insights into the number of possible real roots.

Conclusion

• Understanding the relationships between roots and coefficients across various polynomial equations is essential for theoretical aspects of Algebra and practical applications.