Theory of Equations — Formula Sheet

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.

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.

FUNDAMENTAL THEOREM

Every polynomial equation of degree n ≥ 1 has at least one root in ℂ (complex numbers).

A polynomial equation of degree n in one variable x is expressed as aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ = 0, where aₙ ≠ 0.

Every polynomial equation of degree n ≥ 1 has at least one root in the complex number system ℂ.