Complex Zeros & Fundamental Theorem Flashcards

Complex Variables and Functions

• A complex variable is a variable that exists within the complex number system, featuring both real and imaginary components.

Complex Polynomial Functions

• The general form of a complex polynomial function of degree n is expressed as:
  f(x) = aₙxⁿ + aₙ-₁ xⁿ-¹ + ... + ax + a₀
• In this polynomial, the coefficients (aₙ, aₙ-₁, a, a₀) are complex numbers, n is a nonnegative integer, and x is the complex variable.

Fundamental Theorem of Algebra

• Asserts that every complex polynomial function f(x) of degree n (where n is at least 1) possesses at least one complex zero.

Factoring Complex Polynomials

• A complex polynomial function f(x) of degree n can be factored into n linear factors of the form:
  f(x) = aₙ (x - r₁)(x - r₂)...(x - rₙ)
• In this factorization, aₙ, r₁ through rₙ are complex numbers, indicating that each polynomial of degree n has exactly n complex zeros, which may include repeated values.

Conjugate Pairs Theorem

• For polynomial functions with real number coefficients, if r = a + bi (where i is the imaginary unit) is a zero of f(x), then its complex conjugate r = a - bi is also a zero of f(x).

Real Zeros in Odd-Degree Polynomials

• A polynomial of odd degree guarantees at least one real zero, illustrating a distinct trait of odd-degree polynomials compared to even-degree ones.