Section 24.2
Linear Homogeneous Equations
Finding roots of higher degree polynomials to solve .
1
Introduction
For constant coefficients, we guess .
This leads to an -th degree polynomial equation.
2
Characteristic Polynomials
Fundamental Theorem of Algebra
An -th degree polynomial has exactly roots (counting multiplicity).
We might need synthetic division or grouping to factor them.
3
Roots & Solutions
- Real Distinct:
- Real Repeated (k times):
- Complex Conjugate Pairs:
4
Worked Examples
Example 1: 3rd Order
Solve .
.
.
Roots: .
.
Example 2: 4th Order
Solve .
.
.
Roots: .
.
Example 3: Repeated Complex
.
Roots with multiplicity 2.
.
This is a resonance system.
5
Practice Quiz
Loading...