Theory of Equations

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