Extend Polynomial Identities to Complex Numbers

Lesson 8 of 9: Complex Number System

In this lesson:

• Factor sums of squares using complex numbers
• Write factored forms of quadratics using complex roots
• Verify polynomial factorizations by expanding

What You Will Learn Today

1. Factor over the complex numbers:
2. Write the factored form of any real-coefficient quadratic using its complex roots
3. Verify polynomial factorizations by expanding complex factors
4. Recognize that every real-coefficient polynomial factors completely over

Recall: Difference of Squares Identity

• Identity:
• Example: — check by expanding
• Key move: middle terms cancel; only perfect squares remain

Can you factor in one step?

Can You Factor ?

Over the reals: No. The discriminant is . No real factors exist.

Over the complex numbers: Yes.

Check: ✓

Difference vs. Sum of Squares: The Parallel

Difference of squares (real):

Sum of squares (complex):

Replace with : , turning into .

Derivation: Why the Identity Works

Key step: flips the sign from to .

Middle terms cancel; only the i² step differs from difference of squares.

Worked Example: The Canonical Case

Identify: ,

Apply:

Verify: ✓

Three More Sum-of-Squares Factoring Examples

Quick Check: Factor Using Sum of Squares

Factor each expression over the complex numbers:

Root-to-Factor: The Connection to CN.C.7

Factor Theorem: If is a root, then is a factor.

This works for complex , not just real .

From Roots to Factored Form

For roots and :

Cross-terms cancel; product is real-coefficient.

Worked Example:

Roots: ,

Factored form:

Verify: ✓

Worked Example:

Roots:

Factored form:

Quick Check: Roots to Factored Form

1. The roots of are . Write the factored form and verify.

2. The roots of are . Write the factored form.

Error Hunt: Two Mistakes to Find

Student 1 factors :

Student 2 factors :

Identify each error. Write the correct factorization.

Every Polynomial Factors Completely Over

For any polynomial of degree with real coefficients:

• Exactly roots exist over (counting multiplicity)
• Complex roots come in conjugate pairs
• Odd-degree polynomials must have at least one real root

This is the preview of the Fundamental Theorem of Algebra (CN.C.9).

Factor Completely Over

Step 1:

Step 2: Roots of :

Full factorization:

Three linear factors for degree 3.

Three Problems With No Scaffolding

1. Factor over the complex numbers.

2. The roots of are . Write the factored form.

3. Factor completely over .

Watch for These Three Common Errors

1. Sign error: — sum of squares uses , not real numbers

2. Same-sign error: — conjugate pair needed:

3. Verification skip: always expand to confirm — complex factors must produce real coefficients

What You Now Know: Complex Polynomial Identities

✓ Sum of squares:
✓ Root-to-factor: if is a root, is a factor
✓ Conjugate pairs always multiply to real coefficients

Sum of squares needs — not
Use , not

Next: Fundamental Theorem of Algebra

In HSN.CN.C.9, you'll see the theorem made explicit:

Every non-constant polynomial with complex coefficients has at least one complex root.

This lesson showed the pattern. CN.C.9 proves it applies to every polynomial — including higher-degree cases beyond quadratics and cubics.