Prove and Use Polynomial Identities

Lesson 4 of 7: Arithmetic with Polynomials

In this lesson:

• Distinguish polynomial identities from equations
• Prove identities by algebraic expansion
• Use the Pythagorean triple identity to generate integer triples

Learning Objectives for This Lesson

By the end of this lesson, you should be able to:

1. Distinguish a polynomial identity (true for all values) from an equation (true for specific values)
2. Prove polynomial identities by expanding algebraically
3. Apply standard identities (difference of squares, sum/difference of cubes)
4. Use the Pythagorean-triple identity to generate Pythagorean triples
5. Explain how one identity generates an infinite family of numerical relationships

Recall: How to Multiply Polynomials

• Distributing:
• FOIL (binomials): First, Outer, Inner, Last terms
• Expand means: multiply out all factors, then collect like terms

Expand — what do you get?

Always, Sometimes, or Never True?

Statement:

• : ✓
• : and ✓

Both check out — but there are infinitely many values. Is it always true?

Polynomial Multiplication Was Already Identity-Proving

In APR.A.1, you expanded:

That step-by-step algebra didn't depend on any specific value of .

That's exactly what proving an identity means.

The multiplication you already know is the proof technique.

Identity vs. Equation: The Key Distinction

Equation: — true only for or

Identity: — true for every

• Equation: restricts → specific solutions
• Identity: universal → true for all substitutions

Why Substitution Can't Prove Universality

Claim: is "true."

• : ✓ — : ✓
• : ✗ — the claim fails

Substitution can disprove, but cannot prove. We need algebra valid for ALL values.

Probe: Is This an Identity?

Is a polynomial identity?

Test at , :

Left side: . Right side: .

Not equal — so this is NOT an identity.

The counterexample disproves it. One failure is enough.

Classify Each as Equation or Identity

For each: does it restrict values, or claim universal equality?

Proof Methodology: One Side to the Other

Do NOT Work on Both Sides at Once

Wrong approach:

"Add and subtract..." (manipulates both sides)

Why this is logically circular: You're assuming the equality to prove the equality.

Proof 1: Perfect Square Trinomial

Prove:

Each step: distributive property, commutativity, combining like terms.

Proof 2: Difference of Cubes

Prove:

Expand RHS:

Middle terms cancel in pairs.

Spot the Error in This Proof

A student "proves" :

Line 1:
Line 2: ✓

What is wrong? The student wrote the conclusion first — that's circular, not a proof.

Check-In: Your First Independent Proof

Prove:

Start from the LHS. Show all steps.

Remember: start from , expand, and arrive at .

Same Proof Technique Applied to a Bigger Claim

You've proved identities of degree 2.

Now: an identity that generates infinitely many Pythagorean triples.

The same expansion technique — applied carefully to a degree-4 identity — turns every pair of positive integers into a Pythagorean triple.

This is why algebraic universality matters.

Pythagorean Triples: What Are They?

A Pythagorean triple is three positive integers with .

Classic example: — verify: ✓

Other examples: , ,

How do we generate them systematically?

The Pythagorean Triple Identity: Statement

This means: if , , , then .

Every pair with gives a Pythagorean triple!

Proof: Expand Both Sides and Compare

Expand both sides:

LHS:

RHS:

LHS = RHS

Generate Pythagorean Triples: Three Examples

Check-In: Generate and Verify a Triple

Use the formula to find the Pythagorean triple for , :

1. Compute
2. Compute
3. Compute
4. Verify

Identities as Replacement Rules Both Ways

Identity works in both directions:

Forward (expand):

Backward (factor):

An identity is a two-way substitution: use it in whichever direction helps.

Mental Arithmetic Using Polynomial Identities

Difference of squares:

Perfect square:

Key step: recognize the pattern, then substitute into the identity.

Applying the Sum of Cubes Identity

Sum of cubes:

Compute with , :

Your Turn: Prove This Identity

Prove:

Start from the RHS. Expand completely. Show all steps. Arrive at .

No template. Use the same technique as the difference of cubes proof.

Three Errors to Catch Before They Cost You

Substitution ≠ proof: checking values shows consistency, not universality

Both sides at once: starting with LHS = RHS assumes the conclusion

Cube sign trap: sum: — diff:

Polynomial Identities Are Algebraically Universal

✓ Identity: true for all values — proved by algebraic expansion

✓ Equation: true for specific values — solved, not proved

✓ Proof method: one side → transform → arrive at the other

✓ Pythagorean identity: one proof → infinitely many triples

Next: Generalizing to Any Power

You proved — degree 2.

The Binomial Theorem (APR.C.5) generalizes this to for any positive integer :

The coefficients are the numbers in Pascal's Triangle.