Polynomial Roots Finder
Understanding Polynomial Roots
What are Polynomial Roots?
Polynomial roots (or zeros) are values of x where P(x) = 0. For a polynomial of degree n:
P(x) = anx^n + an-1x^(n-1) + ... + a1x + a0
where an ≠ 0, and all coefficients are real numbers
Fundamental Theorem of Algebra
- Every polynomial of degree n has exactly n complex roots (counting multiplicity)
- Real polynomials have:
- All real roots, or
- Complex conjugate pairs
- Number of real roots can be:
- Equal to degree
- Less than degree by even number
Root Finding Methods
Newton's Method
Iterative method using derivatives
Bisection Method
Divides interval containing root
Synthetic Division
For finding remaining roots
Numerical Methods
For higher degree polynomials
Properties of Roots
- Vieta's Formulas
- Sum of roots = -an-1/an
- Product of roots = (-1)^n * a0/an
- Rational Root Theorem
Helps find potential rational roots
- Descartes' Rule of Signs
Bounds number of positive/negative real roots
Applications
Engineering
System stability analysis
Physics
Wave equations
Control Theory
System response analysis
Computer Graphics
Curve intersection