Know the Fundamental Theorem of Algebra

Lesson 9 of 9: Complex Number System

In this lesson:

• State the Fundamental Theorem of Algebra
• Demonstrate it for quadratic polynomials (all three cases)
• Apply it to count roots and factor polynomials

What You Will Learn Today

1. State the FTA: every degree- polynomial has exactly complex roots (counting multiplicity)
2. Demonstrate the FTA for quadratics using all three discriminant cases
3. Explain why real-coefficient polynomials factor into linear complex factors
4. Explain why odd-degree real-coefficient polynomials must have at least one real root

The Motivation:

Over the real numbers: No solution. .

Over the complex numbers: Two solutions — and .

The Fundamental Theorem of Algebra confirms: every polynomial has exactly roots in .

The Pattern You Have Been Observing

Degree gives exactly roots in .

The Fundamental Theorem of Algebra

Theorem: Every polynomial of degree has exactly roots in , counted with multiplicity.

Guarantees existence — not a formula for finding roots.

Multiplicity: What "Counting Multiplicity" Means

Multiplicity = how many times a root appears as a factor.

Degree 2 → 2 roots counting multiplicity. FTA ✓

The Theorem Guarantees Existence, Not Method

FTA says: Roots exist in .

FTA does NOT say: How to find them.

Demonstrating the FTA for Quadratics

All three cases give exactly 2 roots. The FTA holds for all quadratics.

Case 1: Two Real Roots (Positive Discriminant)

Roots: and

Factored form:

Count: 2 real roots. FTA ✓

Case 2: One Repeated Root (Zero Discriminant)

Root: (multiplicity 2)

Factored form:

Count: 2 roots counting multiplicity. FTA ✓

Case 3: Two Complex Roots (Negative Discriminant)

Roots: and

Factored form:

Count: 2 complex roots. FTA ✓

The Quadratic Proof Is Complete

Every quadratic has exactly 2 roots in . QED.

Two Common Errors About the FTA

A student says:

1. "The FTA says every polynomial has real roots."

2. " has 2 roots: and ."

What is wrong in each? State the correct version.

Complex Roots Come in Conjugate Pairs

For real-coefficient polynomials:

If is a root, then is also a root.

Why: Conjugating gives when coefficients are real.

Complex roots always come in conjugate pairs.

Odd-Degree Polynomials Must Have a Real Root

• Complex roots pair up → even count
• Degree is odd → at least one root unpaired
• That unpaired root must be real

Example: — one real root .

Odd-degree real-coefficient polynomial: real root guaranteed.

Two Cubics: Counting All Three Roots

Both cubics have exactly 3 roots in . FTA holds for degree 3.

Check-In: State and Apply the FTA

1. State the Fundamental Theorem of Algebra in one sentence.

2. Explain in one sentence why a degree-7 polynomial with real coefficients must have at least one real root.

Factor Completely Over

Roots: — two real, two complex.

Count: 4 roots for degree 4. FTA ✓

The theorem predicts the answer before you compute it.

The Complex Number System: The Full Story

The complex numbers are sufficient for all polynomial algebra.

You Have Completed the Complex Number System

The Fundamental Theorem of Algebra:

Every polynomial of degree has exactly roots in , counted with multiplicity.

This is why complex numbers were worth building.

Carl Friedrich Gauss first proved this in 1799. It took new mathematics to do it.