The Substitution Move and Multi-Step Strategy

Lesson 2 of 2: Revealing Hidden Structure

In this lesson:

• Use substitution to expose hidden algebraic patterns
• Factor expressions with compound and high-power terms
• Apply multi-step strategy and verify at every stage

What You Will Learn Today

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

1. Use substitution (e.g., ) to expose hidden algebraic structure
2. Apply patterns after substituting, then substitute back
3. Explain why a rewriting strategy was chosen
4. Verify equivalence at each step by expanding or testing values

x⁴ − y⁴: Does a Pattern Apply Here?

• Two terms, subtraction — looks like difference of squares
• But requires both terms to be perfect squares

Question: Is a perfect square? What is ?

What expression, when squared, equals ?

The Substitution Insight: x⁴ = (x²)²

The expression becomes — a familiar difference of squares.

Six Steps for Any Hidden Structure

1. Examine — what does it look like?
2. Substitute — let $u = $ (inner expression)
3. Recognize — which pattern fits?
4. Factor — apply the pattern
5. Back-substitute — restore the original variable
6. Check further — any factor factorable again?

x⁴ − y⁴: Full Worked Example

Sum of Squares: a Factoring Dead End

After back-substituting, you may encounter — a dead end.

The sign makes all the difference.

Second Type: Quadratic in After Substituting

Factor:

Let : becomes

• Factor:
• Back-substitute:
• Check further: both sums of squares — done

Check-In: How Many Factors After Substituting?

Factor:

1. What substitution is useful?
2. What pattern appears after substituting?
3. How many factors does the complete factorization have?

Count carefully — check further after back-substituting.

Check-In Answer: Three Factors, Not Two

Factor:

• Let :
• Back-substitute:
• : sum of squares — no further factoring
• : difference of squares →

Compound Expressions as Values of a and b

Now: and in a pattern can be compound expressions.

Example:

• Two terms, subtraction, both squares → difference of squares
• ,
• Factor:

Simplify each bracket before advancing.

Always Factor Out the GCF First

Before any pattern: factor out the GCF.

Example:

• GCF = 8:
• Recognize sum of cubes ():
• Verify at : ✓

Worked Example: Compound Difference of Squares

Factor:

Step 1: DoS with ,

Step 2:

Step 3: Check further — both linear, done

Step 4: Verify at : ; ✓

Worked Example: Two Rounds of Difference of Squares

Factor:

• Round 1: , so ,

• Round 2:
• Round 3:

Why does stay unfactored over the reals?

Guided Practice: Identify and Start

Factor:

Your turn — fill in:

• Pattern: _______________
• (hint: )
• (hint: )
• Middle-term check: , middle = ?

Identify the pattern and verify before writing the factored form.

Guided Practice: Confirm Your Work

Factor:

• Pattern: Perfect square trinomial
• , ; middle check: ✓

Independent Practice: Apply and Verify

Factor:

Steps:

1. Identify the pattern
2. Identify and (remember: , so )
3. Apply the formula and simplify
4. Verify by expanding

Work all four steps independently.

Independent Practice: Confirm Your Work

Factor:

• ,
• ; verify: both sides give ✓

What If the Sign Were Plus Instead?

Consider:

• Let : becomes — a sum of squares
• Sum of squares does not factor over the reals

Conclusion: does not factor over the reals.

Substitution is powerful — but it can't bypass dead ends.

Full Procedure: No Prompts This Time

Factor and verify at .

Run the six steps independently:

1. Examine; 2. Identify pattern; 3. Identify and
2. Apply formula and simplify
3. Check further; 6. Verify at

Full Procedure: Answer and Verification

Factor:

• DoS: ,
• ; both linear — done
• Verify at : ; ✓

Complete Strategy: The Full Picture

✓ Examine first — structure reveals the strategy

✓ GCF before patterns — always

✓ Four tools: DoS, PST, SoC/DoC, -substitution

✓ Check every factor — done when no degree-≥2 factor remains

Sum of squares = dead end over the reals

and can be compound expressions

What These Two Lessons Equip You to Do

SSE.B.3: factoring reveals polynomial zeros — the factored form shows where the graph crosses the x-axis.

REI.B.4: substitution converts higher-degree equations to quadratic form — solves .

The structural analysis habit is the foundation for both.