Lesson 5.5

Complex Zeros & FTA

The Fundamental Theorem of Algebra guarantees that every equation has a solution—if you are willing to look beyond the real number line.

Introduction

Prerequisite Connection: We know that has roots . But has no real roots because the graph never touches the x-axis.

Today's Increment: We introduce the Complex Number System again. By allowing , we ensure that an -th degree polynomial ALWAYS has exactly zeros.

Why This Matters: This theorem provides completeness. It assures engineers and physicists that their equations aren't "broken"—the solutions exist, they just might be complex numbers (which model rotations and oscillations).

Explanation of Key Concepts

The Fundamental Theorem of Algebra (FTA)

Every polynomial of degree has exactly zeros in the complex number system.

Note: This counts multiplicity! A double root counts as 2.

Conjugate Pairs Theorem

Complex roots love company. They never travel alone.

is a root...
Then

MUST also be a root.

*This applies only when the polynomial coefficients are real numbers (which is almost always true in this class).

Worked Examples

Level: Basic

Example 1: Find the Missing Zeros

A degree 4 polynomial has roots and . What are the other roots?

Root 1:

(Real).

Root 2:

(Complex).

Root 3:

(Conjugate Pair).

Wait... degree 4 means 4 roots.

If no other information is given, we cannot determine the 4th specific root. But usually questions like this imply minimal conditions or give multiplicity.
If the problem said "Double root at 5", then the roots are

Level: Intermediate

Example 2: Construct the Equation

Find a polynomial of degree 3 with zeros and .

Step 1: Identify all zeros

Given: 4, 2i. Missing Conjugate:

Total 3 zeros. Matches degree 3.

Step 2: Write factors

Step 3: Multiply conjugates first

Final Expansion

Level: Advanced

Example 3: Complex Factoring

Factor completely over the complex numbers.

Phase 1: Real Factoring (Difference of Squares)

Phase 2: Complex Factoring
Typically we stop above. But if asked for "linear factors over complex numbers":

Answer:

Common Pitfalls

Conjugating Real Numbers:
If a root is , the conjugate is NOT . The conjugate of is (still 5). Only the imaginary part flips sign.
Multiplying Error:
When multiplying , students often get .
Remember: . So it becomes .

Real-World Application

Control Systems: Stability

In engineering, we model systems (like cruise control or a thermostat) with polynomials. The roots of these polynomials determine stability.

If a root has a positive real part (e.g., ), the system is unstable and will oscillate out of control (the amplitude grows geometrically). If all roots have negative real parts, the system is stable and will settle down. We need the FTA to ensure we've found ALL the roots to guarantee safety!