4.1 Factoring & Equivalent Algebraic Expressions

4.1 Factoring & Equivalent Algebraic Expressions

Quick Summary: Factoring transforms sums and differences of algebraic terms into products of simpler factors. On the Digital SAT, factoring is essential not only for solving quadratic and higher-degree polynomial equations, but also for identifying structurally equivalent algebraic expressions. Mastering the universal Factoring Hierarchy allows you to break down complex expressions efficiently without getting lost in algebraic manipulation.

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The Universal Factoring Hierarchy

Whenever you encounter an expression to factor or simplify, always follow this strict four-step hierarchy:

                  +-----------------------------------------+
                  |       STEP 1: Greatest Common Factor    |
                  |  Always factor out the GCF first!       |
                  +--------------------+--------------------+
                                       |
         +-----------------------------+-----------------------------+
         |                             |                             |
         v                             v                             v
+------------------+          +------------------+          +------------------+
|   2 TERMS        |          |   3 TERMS        |          |   4 TERMS        |
| - Difference of  |          | - Perfect Square |          | - Factoring by   |
|   Squares        |          |   Trinomial      |          |   Grouping       |
| - Sum/Diff of    |          | - Standard       |          | - 2x2 or 3x1     |
|   Cubes          |          |   Trinomial (ac) |          |   Grouping       |
+------------------+          +------------------+          +------------------+

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1. Greatest Common Factor (GCF)

The Greatest Common Factor is the largest monomial (coefficient and variable powers) that divides evenly into every term of the polynomial. Factoring out the GCF reduces the coefficients and degrees of remaining terms before applying any further factoring techniques.

$$\text{Expression: } 12x^4y^2 - 18x^3y^3 + 30x^2y^4$$

• Greatest numerical factor of $12, -18, 30$: $6$
• Lowest power of $x$: $x^2$
• Lowest power of $y$: $y^2$
• GCF: $6x^2y^2$

$$\implies 6x^2y^2(2x^2 - 3xy + 5y^2)$$

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2. Factoring Two-Term Expressions (Binomials)

A. Difference of Two Squares (DOTS)

Any binomial of the form $a^2 - b^2$ factors into the product of conjugate binomials:

$$a^2 - b^2 = (a - b)(a + b)$$

Extended Multi-Layer Difference of Squares

High-difficulty SAT questions often require applying the difference of squares rule repeatedly:

$$x^4 - 81 = (x^2 - 9)(x^2 + 9) = (x - 3)(x + 3)(x^2 + 9)$$ $$16a^4 - 625b^4 = (4a^2 - 25b^2)(4a^2 + 25b^2) = (2a - 5b)(2a + 5b)(4a^2 + 25b^2)$$

B. Sum and Difference of Cubes

While less frequent than quadratics, sum and difference of cubes appear on harder module questions:

$$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$ $$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$

Mnemonic (SOAP):

• Same sign: $(a - b)$ or $(a + b)$
• Opposite sign: $+ab$ or $-ab$
• Always Positive: $+b^2$

Example: $8x^3 - 27 = (2x)^3 - 3^3 = (2x - 3)(4x^2 + 6x + 9)$

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3. Factoring Three-Term Expressions (Trinomials)

A. Perfect Square Trinomials

Recognizing perfect square trinomials eliminates tedious trial-and-error:

$$(a + b)^2 = a^2 + 2ab + b^2$$ $$(a - b)^2 = a^2 - 2ab + b^2$$

To identify a perfect square trinomial:

1. The first term must be a perfect square: $a^2$.
2. The last term must be a positive perfect square: $b^2$.
3. The middle term must equal $\pm 2ab$.

Example: $25x^2 - 70x + 49$

• $a = \sqrt{25x^2} = 5x$
• $b = \sqrt{49} = 7$
• Middle term check: $2(5x)(7) = 70x$ (Matches!)
• Factored form: $(5x - 7)^2$

B. Monic Trinomials ($x^2 + bx + c$)

Find two numbers $p$ and $q$ that multiply to $c$ and add to $b$:

$$p \cdot q = c \quad \text{and} \quad p + q = b \implies (x + p)(x + q)$$

Example: Factor $x^2 - 5x - 24$

• Find factors of $-24$ that sum to $-5$: $(-8) \times (3) = -24$ and $-8 + 3 = -5$.
• Factored form: $(x - 8)(x + 3)$.

C. Non-Monic Trinomials ($ax^2 + bx + c$, $a \neq 1$) via the $ac$-Method

1. Multiply $a$ and $c$ to get the key product $ac$.
2. Find two integers $p$ and $q$ such that $p \cdot q = ac$ and $p + q = b$.
3. Split the middle term $bx$ into $px + qx$.
4. Factor the resulting four-term expression by grouping.

Worked Example: Factoring $6x^2 + 11x - 10$

• Step 1: $ac = 6 \times (-10) = -60$.
• Step 2: Find factors of $-60$ that sum to $11$: $+15$ and $-4$ ($15 \times (-4) = -60$, $15 + (-4) = 11$).
• Step 3: Split middle term: $6x^2 + 15x - 4x - 10$.
• Step 4: Group in pairs: $$3x(2x + 5) - 2(2x + 5) = (3x - 2)(2x + 5)$$

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4. Factoring Four-Term Polynomials: Factoring by Grouping

A. Standard 2x2 Grouping

When given four terms, group them into pairs that share common factors:

$$ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y)$$

Worked Example: Cubic Grouping

Factor completely: $x^3 - 4x^2 - 9x + 36$

• Step 1: Group the first two and last two terms: $$(x^3 - 4x^2) - (9x - 36)$$
• Step 2: Extract the GCF from each pair: $$x^2(x - 4) - 9(x - 4)$$
• Step 3: Factor out the common binomial $(x - 4)$: $$(x^2 - 9)(x - 4)$$
• Step 4: Check for further factoring (Difference of Squares): $$(x - 3)(x + 3)(x - 4)$$

B. 3x1 Grouping (Trinomial Square Minus a Square)

Occasionally, four terms consist of a perfect square trinomial paired with a subtracted square:

$$x^2 + 6x + 9 - y^2 = (x + 3)^2 - y^2 = (x + 3 - y)(x + 3 + y)$$

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5. Advanced Structure Recognition & Substitution

When an equation involves higher powers or nested expressions, substituting $u$ transforms the problem into a standard quadratic:

$$\text{Expression: } (x - 2)^2 - 5(x - 2) - 14$$

1. Let $u = (x - 2)$.
2. Rewrite in terms of $u$: $u^2 - 5u - 14$.
3. Factor the monic quadratic: $(u - 7)(u + 2)$.
4. Substitute back $u = (x - 2)$: $$((x - 2) - 7)((x - 2) + 2) = (x - 9)(x)$$

Similarly, for quartics of quadratic type:

$$x^4 - 13x^2 + 36 = (x^2 - 9)(x^2 - 4) = (x - 3)(x + 3)(x - 2)(x + 2)$$

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Summary of Key Factoring Patterns

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SAT Strategies: Equivalence Verification

On the Digital SAT, questions frequently ask: "Which of the following expressions is equivalent to..."

Strategy 1: Algebraic Manipulation

Factor or expand using the standard rules above.

Strategy 2: Test Number Substitution

If algebraic factoring feels complex or error-prone under time pressure:

1. Choose a simple test number (e.g., $x = 2$ or $x = 3$; avoid $0$ and $1$ if multiple choices evaluate to the same value).
2. Evaluate the target expression at $x = 2$.
3. Evaluate each answer option at $x = 2$.
4. The matching numerical result identifies the equivalent expression.

Strategy 3: Desmos Graphical Equivalence

1. Type the given expression as y_1 = 16x^4 - 81.
2. Type the answer option as y_2 = (2x - 3)(2x + 3)(4x^2 + 9).
3. If the graphs overlap perfectly along the entire domain, the expressions are mathematically equivalent.
4. Alternatively, use a Desmos table to compare outputs across multiple $x$-values ($x = -2, -1, 0, 1, 2$).

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Common Pitfalls & Exam Traps

• Stopping too early: In $x^4 - 16 = (x^2 - 4)(x^2 + 4)$, forgetting that $(x^2 - 4)$ factors further into $(x - 2)(x + 2)$.
• Sign errors when grouping with negative signs: In $x^3 - 3x^2 - 4x + 12$, factoring out $-4$ gives $-4(x - 3)$, not $-4(x + 3)$.
• Canceling non-factors across fractions: In $\frac{x^2 - 9}{x + 3}$, you must factor numerator to $(x-3)(x+3)$ before canceling $(x+3)$.
• Treating $a^2 + b^2$ as factorable over reals: $a^2 + b^2$ does not factor into real binomials.