6.4 Factoring Polynomials & Special Products
Key Takeaways
- Factoring is the algebraic process of decomposing a polynomial into a product of simpler polynomials; it is the exact reverse operation of polynomial multiplication.
- The universal first step in any factoring problem is extracting the Greatest Common Factor (GCF), which consists of the largest common integer divisor and the lowest common exponent of shared variables.
- Quadratic trinomials of the form $x^2 + bx + c$ ($a = 1$) factor into $(x + p)(x + q)$, where $p \cdot q = c$ and $p + q = b$.
- Trinomials with leading coefficient $a > 1$ ($ax^2 + bx + c$) are factored using the $ac$-method (grouping): find two integers whose product is $a \cdot c$ and sum is $b$, split the middle term, and factor by grouping.
- Special factoring formulas include the Difference of Two Squares ($a^2 - b^2 = (a - b)(a + b)$) and Perfect Square Trinomials ($a^2 \pm 2ab + b^2 = (a \pm b)^2$); the sum of squares ($a^2 + b^2$) cannot be factored over real numbers.
Factoring Polynomials & Special Products
Quick Summary: Factoring breaks an algebraic sum down into a product of simpler linear or polynomial factors. Factoring allows us to solve quadratic equations, simplify complex rational expressions, and find graph intercepts ($x$-intercepts). Factoring follows a strict procedural hierarchy: always check for a Greatest Common Factor (GCF) first, then inspect the number of terms to apply trinomial factoring ($a = 1$ or $ac$-method), Difference of Two Squares ($a^2 - b^2$), or Perfect Square Trinomial rules.
Mastering factoring is essential for the HiSET exam, where factoring questions appear in pure algebraic simplification, root-finding, and geometric area decomposition contexts.
Step 0: Extracting the Greatest Common Factor (GCF)
Before applying any advanced factoring patterns, you must always factor out the Greatest Common Factor (GCF) of all terms.
How to Identify the GCF:
- Numerical GCF: Find the greatest integer that divides evenly into all numerical coefficients.
- Variable GCF: For each variable shared by all terms, take the variable with the lowest exponent present.
- Divide: Divide every term of the polynomial by the GCF and place the quotient inside parentheses.
Finding the GCF of: 12x³y² - 18x²y³ + 6x²y²
┌────────────────────────────────────────────────────────┐
│ Coefficients: GCF of 12, 18, 6 ──► 6 │
│ x-variable: x³, x², x² ──► x² (Lowest power) │
│ y-variable: y², y³, y² ──► y² (Lowest power) │
│ Complete GCF: 6x²y² │
└────────────────────────────────────────────────────────┘
Factored Result: 6x²y²(2x - 3y + 1)
Caution — The Missing "1" Trap: When an entire term is identical to the GCF (such as $+6x^2 y^2$ divided by $6x^2 y^2$), you must write $+1$ as a placeholder. Omitting the $+1$ destroys the polynomial's original value upon expansion.
Factoring Quadratic Trinomials $x^2 + bx + c$ ($a = 1$)
When the leading coefficient is $1$, factoring $x^2 + bx + c$ requires finding two numbers $p$ and $q$ that satisfy two simultaneous conditions:
Trinomial Sign Decision Rules
| Signs of $b$ and $c$ | Factor Structure | Condition on Numbers $p$ and $q$ | Example |
|---|---|---|---|
| $c > 0$ and $b > 0$ | $(x + p)(x + q)$ | Both $p$ and $q$ are positive | $x^2 + 7x + 12 = (x + 3)(x + 4)$ |
| $c > 0$ and $b < 0$ | $(x - p)(x - q)$ | Both $p$ and $q$ are negative | $x^2 - 8x + 15 = (x - 3)(x - 5)$ |
| $c < 0$ and $b > 0$ | $(x + p)(x - q)$ | Opposite signs; positive number has larger absolute value | $x^2 + 3x - 10 = (x + 5)(x - 2)$ |
| $c < 0$ and $b < 0$ | $(x + p)(x - q)$ | Opposite signs; negative number has larger absolute value | $x^2 - 7x - 18 = (x - 9)(x + 2)$ |
Factoring Trinomials $ax^2 + bx + c$ ($a > 1$) via the $ac$-Method
When the leading coefficient $a$ is not $1$ and cannot be factored out via GCF, use the systematic $ac$-method (factoring by grouping).
The 5-Step ac-Method Protocol
┌─────────────────────────────────────────────────────────────┐
│ Step 1: Multiply a · c. │
│ Step 2: Find two factors of (a·c) that ADD up to b. │
│ Step 3: SPLIT the middle term bx into two terms: px + qx. │
│ Step 4: GROUP into two pairs: (ax² + px) + (qx + c). │
│ Step 5: Factor out GCFs to reveal a common binomial factor. │
└─────────────────────────────────────────────────────────────┘
Worked Example: Factoring $6x^2 + 11x - 10$
- Calculate $a \cdot c$:
- Find factors of $-60$ that sum to $+11$:
- Pairs of $-60$: $(-1, 60), (-2, 30), (-3, 20), (-4, 15)$
- Test $-4$ and $15$: $(-4)(15) = -60$ and $-4 + 15 = 11$ (Match!)
- Rewrite the middle term $11x$ as $-4x + 15x$:
- Factor by grouping (two pairs):
- Extract the GCF from each pair:
- Factor out the shared binomial $(3x - 2)$:
- Verify via FOIL:
Special Factoring Patterns
Recognizing algebraic patterns allows instant factoring without testing factor pairs.
1. Difference of Two Squares
A binomial consisting of two perfect squares separated by a minus sign factors into conjugate pairs:
- $x^2 - 49 = (x - 7)(x + 7)$
- $9x^2 - 25 = (3x - 5)(3x + 5)$
- $16x^4 - 81 = (4x^2 - 9)(4x^2 + 9) = (2x - 3)(2x + 3)(4x^2 + 9)$
Crucial HiSET Rule — Sum of Squares is Prime: The sum of two squares, $a^2 + b^2$, CANNOT be factored over real numbers. For example, $x^2 + 25$ is prime.
2. Perfect Square Trinomials
When the first and last terms are positive perfect squares and the middle term equals twice the product of their square roots:
- $x^2 + 10x + 25 = (x + 5)^2$
- $4x^2 - 12x + 9 = (2x - 3)^2$
Multi-Step Factoring: "Factor Completely"
On the HiSET, a question stating "Factor completely" signals that multiple factoring operations must be applied sequentially.
Multi-Step Factoring Protocol:
- Always factor out the GCF first.
- Inspect the remaining polynomial inside parentheses.
- Factor the remaining polynomial using trinomial methods or difference of squares.
- Confirm that no factor can be broken down further.
Worked Example: Complete Factoring
Factor completely: $3x^3 - 75x$
- Step 1: Extract the GCF ($3x$):
- Step 2: Recognize the difference of two squares inside parentheses ($x^2 - 25$):
- Step 3: Combine all factors:
Which of the following represents the completely factored form of 12x^3 y^2 - 18x^2 y^3 + 6x^2 y^2?
What is the factored form of the quadratic trinomial x^2 - 7x - 18?
Factor the quadratic expression: 2x^2 + 7x - 15
Which of the following is the complete factorization of 3x^3 - 75x over the real numbers?