2.3 Factoring & Expanding Polynomials

2.3 Factoring & Expanding Polynomials

Polynomial arithmetic and factoring form a core computational pillar of the CLEP College Algebra examination. Expanding polynomial products converts factored expressions into expanded standard form, while factoring reverses this process by decomposing polynomial sums into products of simpler algebraic factors.

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Polynomial Vocabulary, Taxonomy & Standard Form

A polynomial in one variable $x$ is an algebraic expression of the form: $P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$ where $n$ is a non-negative integer ($n \in \{0, 1, 2, \dots\}$), $a_n \neq 0$, and coefficients $a_n, a_{n-1}, \dots, a_0 \in \mathbb{R}$.

• Standard Form: Terms ordered by degree in strict descending order ($x^n \to x^{n-1} \to \dots \to x^0$).
• Degree of Polynomial ($n$): The highest exponent power of the variable.
• Leading Coefficient ($a_n$): The coefficient attached to the highest-degree term.
• Constant Term ($a_0$): The term with degree $0$ (no variable).

Classification Table

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Polynomial Multiplication & Expansion Methods

Expanding polynomial products requires multiplying every term in the first factor by every term in the second factor using the distributive law.

1. The FOIL Method (Binomial $\times$ Binomial)

For multiplying two binomials $(a + b)(c + d)$:

• First terms: $a \cdot c$
• Outer terms: $a \cdot d$
• Inner terms: $b \cdot c$
• Last terms: $b \cdot d$ $(a + b)(c + d) = ac + ad + bc + bd$

2. General Expansion (Binomial $\times$ Trinomial)

Worked Example: Fully expand $(3x - 4)(2x^2 + 5x - 6)$.

1. Distribute $3x$ to all three terms:
   $3x(2x^2) + 3x(5x) + 3x(-6) = 6x^3 + 15x^2 - 18x$.
2. Distribute $-4$ to all three terms:
   $-4(2x^2) - 4(5x) - 4(-6) = -8x^2 - 20x + 24$.
3. Combine like terms:
   $6x^3 + (15x^2 - 8x^2) + (-18x - 20x) + 24 = 6x^3 + 7x^2 - 38x + 24$.

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Systematic Factoring Hierarchy Roadmap

To factor any polynomial efficiently on the CLEP exam, follow this decision tree:

Step 1: Always extract the Greatest Common Factor (GCF) first.
        │
Step 2: Count the terms inside the remaining polynomial:
        ├─► 2 Terms: Check Special Binomial Formulas
        │    ├─ Difference of Squares: a² - b² = (a - b)(a + b)
        │    ├─ Sum of Cubes: a³ + b³ = (a + b)(a² - ab + b²)
        │    └─ Difference of Cubes: a³ - b³ = (a - b)(a² + ab + b²)
        │    (Note: Sum of Squares a² + b² is PRIME over Reals!)
        │
        ├─► 3 Terms: Trinomial Factoring
        │    ├─ Simple (x² + bx + c): Find factors of c that sum to b.
        │    ├─ General (ax² + bx + c): Use ac-method or Trial & Error.
        │    └─ Perfect Square Trinomial: a² ± 2ab + b² = (a ± b)²
        │
        └─► 4 Terms: Factoring by Grouping
             └─ Group into pairs, extract GCF from each pair.

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Advanced Factoring Techniques & Worked Examples

1. Extraction of Greatest Common Factor (GCF)

Always factor out the highest common coefficient and variable powers before applying other rules.

Example: Factor $12x^4y^2 - 18x^3y^3 + 6x^2y^4$.

• $\text{GCF} = 6x^2y^2$.
• Factored form: $6x^2y^2 (2x^2 - 3xy + y^2)$.

2. Difference of Squares Identity

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

Worked Example (Multi-Step): Factor $81x^4 - 16$ completely.

1. Recognize difference of squares: $(9x^2)^2 - (4)^2 = (9x^2 - 4)(9x^2 + 4)$.
2. Factor the first binomial again: $(3x - 2)(3x + 2)(9x^2 + 4)$. (Note: $9x^2 + 4$ is a sum of squares and cannot be factored over real numbers).

3. Sum and Difference of Cubes (SOAP Mnemonic)

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

• Same sign as binomial operator.
• Opposite sign for middle trinomial term.
• Always Positive for last trinomial term.

Worked Example: Factor $27x^3 - 64y^3$.

• Identify $a = 3x$ and $b = 4y$.
• Apply formula: $(3x - 4y)((3x)^2 + (3x)(4y) + (4y)^2) = (3x - 4y)(9x^2 + 12xy + 16y^2)$.

4. General Trinomial Factoring ($ax^2 + bx + c$) using $ac$-Method

To factor $ax^2 + bx + c$ ($a \neq 1$):

1. Calculate the product $a \cdot c$.
2. Find two integers $p$ and $q$ such that $p \cdot q = ac$ and $p + q = b$.
3. Split middle term $bx$ into $px + qx$.
4. Factor by grouping.

Worked Example: Factor $8x^2 - 14x - 15$ completely.

1. Calculate $a \cdot c = 8 \cdot (-15) = -120$.
2. Find factors of $-120$ summing to $-14$: $-20$ and $+6$ ($(-20)(6) = -120$ and $-20 + 6 = -14$).
3. Split middle term: $8x^2 - 20x + 6x - 15$.
4. Group in pairs: $(8x^2 - 20x) + (6x - 15)$.
5. Factor GCF from pairs: $4x(2x - 5) + 3(2x - 5)$.
6. Factor common binomial $(2x - 5)$:
   $(4x + 3)(2x - 5)$

5. Factoring Four Terms by Grouping

Worked Example: Factor $2x^3 - 3x^2 - 8x + 12$ completely.

1. Group first two and last two terms: $(2x^3 - 3x^2) - (8x - 12)$.
2. Extract GCF from each group: $x^2(2x - 3) - 4(2x - 3)$.
3. Factor common binomial: $(x^2 - 4)(2x - 3)$.
4. Factor difference of squares: $(x - 2)(x + 2)(2x - 3)$.