Pattern Recognition and Factoring

Lesson 1 of 2: Seeing Before Acting

In this lesson:

• Classify expressions by algebraic pattern
• Factor using difference of squares and perfect square trinomials
• Apply the sum and difference of cubes formulas

What You Will Learn Today

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

1. Identify algebraic patterns — difference of squares, perfect square trinomials, sum/difference of cubes
2. Factor expressions by recognizing and applying known patterns
3. Explain why a rewriting strategy was chosen
4. Verify equivalence by expanding or testing values

Recall: Expression Vocabulary and Exponent Rules

• Term: piece separated by or ; coefficient: the numeric factor in a term
• Factor: a quantity being multiplied — has factors and
• Exponent rule: , so

Is a perfect square? How many terms in ?

An Area You Can Compute Instantly

A fountain has outer radius and inner radius .

The ring area is .

• Compute , then ... or...
• Factor first:

What Do You Notice? No Factoring Yet

Describe each expression: count terms, check signs, look for perfect squares.

Write one observation for each. What features stand out?

Pattern 1: Difference of Squares

Form:

Diagnostic criteria — all three must hold:

• Exactly two terms
• Subtracted from each other
• Both terms are perfect squares

Examples: , , (with a closer look)

When Difference of Squares Fails

Try 1: ?

• ✗

Try 2: ?

• — wrong sign ✗

Sum of squares does not factor over the reals.

Pattern 2: Perfect Square Trinomial

Form:

Middle-term check — the signature:

Example:

• , , ✓

Pattern 3: Sum and Difference of Cubes

Forms:

SOAP sign rule: Same · Opposite · Always Positive

• Binomial sign = Same as original
• Trinomial middle sign = Opposite of binomial
• Trinomial last sign = Always positive

Classify: Which Pattern Fits Each?

Which pattern fits each one? Don't factor yet — just identify.

Think about each one, then advance to check.

Answers: All Four Patterns Identified

1. → DoS (, )
2. → PST ( ✓)
3. → Sum of cubes (, )
4. → DoS (, )

Factoring as Running Expansion Backwards

You know how to expand:

Now reverse the arrow:

• Recognize the structure ( with , )
• Write
• Done — verified by running expansion forward

Worked Example: Difference of Squares

Factor:

• Identify: two terms, subtraction, both perfect squares
• , (since )
• Apply:
• Verify: ✓

Middle-Term Check: Factor

• , ; middle: ✓
• Factor: ; Verify: ✓

Non-1 Leading Coefficient: Find the Root

Key move: , so

Apply to :

• ,
• Middle check: ✓

Worked Example: Difference of Cubes

Factor:

• , (since )
• SOAP: difference → binomial sign "−"

• Verify at : ; ✓

Guided Practice: Factor Step by Step

Factor:

1. Pattern: _______________
2. ___, ___
3. Factored form:
4. Verify by expanding

Work it, then advance.

Guided Practice: Check Your Work

Factor:

1. Pattern: Difference of squares
2. , (since )
3. Factored form:
4. Verify: ✓

Independent Practice: Factor and Verify

Factor:

Steps:

1. Identify the pattern (which one is it?)
2. Run the middle-term check to confirm
3. Write the factored form
4. Verify by expanding

Work through all four steps independently.

Independent Practice: Check Your Work

Factor:

Check: , , ✓

Verify: ✓

Mixed Example: Non-Monomial Values of a and b

Factor:

• Two terms, subtraction: check for perfect squares
• , so ; , so
• Apply:
• Check further: both linear — done

Find the Error: Sum of Squares

Claim:

1. What is the error in this reasoning?
2. What is the correct conclusion about ?
3. Verify at : does ?

Identify the error before advancing.

Error Resolved: The Missing Middle Term

Error:

Squaring a binomial always produces three terms — the middle term is missing.

Conclusion: is a sum of squares — does not factor.

Verify: at : ✗

Full Procedure: Three Expressions Independently

For each expression:

1. Identify the pattern
2. Factor completely
3. Check for further factoring
4. Verify at a value you choose

Expressions: , ,

Work all three before advancing.

Full Procedure: Answers and Verifications

• — DoS; at : ✓
• — PST ( ✓); at : ✓
• — SoC (SOAP); at : ✓

Key Takeaways: Structure Before Action

✓ Look first — structure reveals the strategy

✓ Three templates: difference of squares, PST, sum/difference of cubes

✓ Factoring = reversing expansion

Sum of squares does NOT factor over the reals

Always check further factoring after each step

PST: confirm equals the middle term

Lesson 2 Preview: Hidden Algebraic Structure

In Lesson 2, we tackle .

• Two terms, subtraction — looks like difference of squares
• Is a perfect square? What if ?

Substitution reveals patterns hiding in plain sight.