Factoring Quadratics & Polynomials: Complete Guide with 8 Worked Examples

Master SAT factoring with this comprehensive guide. Learn GCF, difference of squares, perfect square trinomials, AC method, and grouping with 8 fully worked examples and expert strategies.

SAT Math – Advanced Math

Factoring Quadratic and Polynomial Expressions

Breaking down expressions into products of simpler factors

Factoring is the process of rewriting an expression as a product of simpler factors. On the SAT, you'll factor to solve equations, simplify expressions, find zeros of functions, and reveal important characteristics of parabolas and polynomials—skills fundamental to advanced algebra and calculus.

Success requires recognizing factoring patterns (greatest common factor, difference of squares, perfect square trinomials, sum/difference of cubes), applying systematic approaches to trinomials, and verifying results through multiplication. These techniques aren't just algebraic manipulations—they're the foundation for solving real-world optimization problems, analyzing projectile motion, and understanding parabolic relationships in physics, engineering, and economics.

Understanding Factoring

What is Factoring?

Factoring reverses multiplication, expressing a sum as a product.

Expanded form: \(x^2 + 5x + 6\)
Factored form: \((x + 2)(x + 3)\)
Purpose: Simplify expressions, solve equations, find zeros
Verification: Always multiply out to check your factoring

Greatest Common Factor (GCF)

Always factor out the GCF first—the largest expression that divides all terms.

Example: \(6x^3 + 9x^2 = 3x^2(2x + 3)\)
GCF: 3x² divides both terms
Strategy: Always check for GCF before other methods
Tip: Don't forget to factor out negative signs when helpful

Special Factoring Patterns

Recognize these patterns for instant factoring.

Difference of squares: \(a^2 - b^2 = (a+b)(a-b)\)
Perfect square trinomial: \(a^2 + 2ab + b^2 = (a+b)^2\)
Difference of cubes: \(a^3 - b^3 = (a-b)(a^2+ab+b^2)\)
Sum of cubes: \(a^3 + b^3 = (a+b)(a^2-ab+b^2)\)

Factoring Trinomials

For \(ax^2 + bx + c\), find factors that multiply to \(ac\) and add to \(b\).

When a = 1: Find two numbers that multiply to \(c\) and add to \(b\)
When a ≠ 1: Use AC method or trial and error
Example: \(x^2 + 7x + 12 = (x+3)(x+4)\)
Check: 3 × 4 = 12, 3 + 4 = 7 ✓

Essential Factoring Formulas

Difference of Squares

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

Recognizable: Two perfect squares separated by minus

Example: \(x^2 - 25 = (x + 5)(x - 5)\)

Perfect Square Trinomials

\(a^2 + 2ab + b^2 = (a + b)^2\)

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

Check: First and last terms are perfect squares, middle = \(2 \times \sqrt{\text{first}} \times \sqrt{\text{last}}\)

Simple Trinomial (a = 1)

\(x^2 + bx + c = (x + m)(x + n)\)

Where \(m \times n = c\) and \(m + n = b\)

Example: \(x^2 + 8x + 15 = (x + 3)(x + 5)\) because 3 × 5 = 15, 3 + 5 = 8

Grouping Method

For four terms, group pairs and factor each:

1. Group into two pairs

2. Factor GCF from each pair

3. Factor out common binomial

Common Pitfalls & Expert Tips

❌ Forgetting to factor out GCF first

Always check for common factors before other methods. Factoring \(2x^2 + 10x + 12\) without removing GCF = 2 makes it harder!

❌ Sign errors in factoring

For \(x^2 - 5x + 6\), factors must multiply to +6 and add to -5, so both must be negative: \((x-2)(x-3)\), not \((x+2)(x+3)\)!

❌ Missing difference of squares

\(x^2 - 16\) factors to \((x+4)(x-4)\). Don't leave it unfactored! Also check if factors can be factored further.

❌ Not verifying by multiplying back

Always multiply your factors to verify you get the original expression. This catches sign errors and missing terms!

✓ Expert Tip: Know your perfect squares

Memorize: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144... Instant recognition speeds up factoring dramatically!

✓ Expert Tip: Factor completely

After factoring, check if any factor can be factored further. \(x^4 - 16 = (x^2+4)(x^2-4) = (x^2+4)(x+2)(x-2)\)

✓ Expert Tip: Use factoring to solve equations

Once factored, set each factor equal to zero. \((x-3)(x+5) = 0\) means \(x = 3\) or \(x = -5\). Zero Product Property!

Fully Worked SAT-Style Examples

Example 1: Factoring with GCF

Factor completely: \(6x^3 + 15x^2 - 9x\)

Solution:

Step 1: Find the GCF

Coefficients: GCF of 6, 15, 9 is 3

Variables: All terms have at least \(x^1\)

GCF = 3x

Step 2: Factor out GCF

\(6x^3 + 15x^2 - 9x = 3x(2x^2 + 5x - 3)\)

Step 3: Factor the trinomial

For \(2x^2 + 5x - 3\), find factors of (2)(-3) = -6 that add to 5

Factors: 6 and -1 (6 × -1 = -6, 6 + (-1) = 5) ✓

\(2x^2 + 6x - x - 3\)

\(= 2x(x + 3) - 1(x + 3) = (2x - 1)(x + 3)\)

Answer: \(3x(2x - 1)(x + 3)\)

Example 2: Difference of Squares

Factor completely: \(x^2 - 49\)

Solution:

Step 1: Recognize the pattern

\(x^2\) is a perfect square

49 = \(7^2\) is a perfect square

Separated by minus sign

This is difference of squares: \(a^2 - b^2\)

Step 2: Apply formula

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

Here: \(a = x\), \(b = 7\)

\(x^2 - 49 = (x + 7)(x - 7)\)

Verification:

\((x + 7)(x - 7) = x^2 - 7x + 7x - 49 = x^2 - 49\) ✓

Answer: \((x + 7)(x - 7)\)

Example 3: Perfect Square Trinomial

Factor: \(x^2 + 12x + 36\)

Solution:

Step 1: Check for perfect square trinomial

First term: \(x^2 = (x)^2\) ✓

Last term: \(36 = (6)^2\) ✓

Middle term: \(2 \times x \times 6 = 12x\) ✓

This is a perfect square trinomial!

Step 2: Apply pattern

\(a^2 + 2ab + b^2 = (a + b)^2\)

Here: \(a = x\), \(b = 6\)

\(x^2 + 12x + 36 = (x + 6)^2\)

Answer: \((x + 6)^2\) or \((x + 6)(x + 6)\)

Example 4: Simple Trinomial (a = 1)

Factor: \(x^2 - 5x - 24\)

Solution:

Step 1: Find two numbers

Numbers that multiply to -24 (constant term)

AND add to -5 (coefficient of x)

Step 2: List factor pairs of -24

1 and -24 (sum = -23) ✗

2 and -12 (sum = -10) ✗

3 and -8 (sum = -5) ✓

Step 3: Write factored form

\(x^2 - 5x - 24 = (x + 3)(x - 8)\)

Answer: \((x + 3)(x - 8)\)

Example 5: Solving with Factoring

Solve for x: \(x^2 + 6x = 16\)

Solution:

Step 1: Set equal to zero

\(x^2 + 6x - 16 = 0\)

Step 2: Factor the trinomial

Find numbers that multiply to -16 and add to 6

8 and -2 (8 × -2 = -16, 8 + (-2) = 6) ✓

\((x + 8)(x - 2) = 0\)

Step 3: Apply Zero Product Property

\(x + 8 = 0\) OR \(x - 2 = 0\)

\(x = -8\) OR \(x = 2\)

Answer: \(x = -8\) or \(x = 2\)

Example 6: Trinomial with Leading Coefficient

Factor: \(3x^2 + 11x + 6\)

Solution:

Step 1: Use AC method

\(a = 3\), \(c = 6\), so \(ac = 18\)

Find factors of 18 that add to 11

9 and 2 (9 × 2 = 18, 9 + 2 = 11) ✓

Step 2: Rewrite middle term

\(3x^2 + 9x + 2x + 6\)

Step 3: Factor by grouping

\(3x(x + 3) + 2(x + 3)\)

\(= (3x + 2)(x + 3)\)

Answer: \((3x + 2)(x + 3)\)

Example 7: Factor Completely

Factor completely: \(x^4 - 81\)

Solution:

Step 1: Recognize difference of squares

\(x^4 = (x^2)^2\) and \(81 = 9^2\)

\(x^4 - 81 = (x^2 + 9)(x^2 - 9)\)

Step 2: Check if factors can be factored further

\(x^2 + 9\) is sum of squares (cannot factor with real numbers)

\(x^2 - 9\) is difference of squares! Factor again:

\(x^2 - 9 = (x + 3)(x - 3)\)

Complete Factorization:

\(x^4 - 81 = (x^2 + 9)(x + 3)(x - 3)\)

Answer: \((x^2 + 9)(x + 3)(x - 3)\)

Example 8: Grouping Method

Factor: \(x^3 + 3x^2 + 2x + 6\)

Solution:

Step 1: Group terms in pairs

\((x^3 + 3x^2) + (2x + 6)\)

Step 2: Factor GCF from each group

First group: \(x^2(x + 3)\)

Second group: \(2(x + 3)\)

\(x^2(x + 3) + 2(x + 3)\)

Step 3: Factor out common binomial

Both groups contain \((x + 3)\)

\(= (x + 3)(x^2 + 2)\)

Answer: \((x + 3)(x^2 + 2)\)

Factoring Strategy Flowchart

1. Always check for GCF first

2. Count terms:

• Two terms → Check for difference of squares or sum/difference of cubes

• Three terms → Check for perfect square trinomial, then factor as trinomial

• Four+ terms → Try grouping method

3. Check if any factor can be factored further (factor completely!)

SAT Factoring Checklist

Before Factoring

  • Look for GCF first
  • Count the number of terms
  • Identify special patterns
  • Check coefficient signs

Special Patterns

  • Difference of squares: \(a^2 - b^2\)
  • Perfect square: First & last = squares
  • Sum of cubes: \(a^3 + b^3\)
  • Difference of cubes: \(a^3 - b^3\)

After Factoring

  • Verify by multiplying out
  • Check each factor for more factoring
  • Ensure completely factored
  • Apply to solve if equation

Common Mistakes

  • Forgetting to factor GCF
  • Sign errors with negatives
  • Not factoring completely
  • Skipping verification step

Factoring: The Foundation of Advanced Algebra

Factoring transforms complicated expressions into products of simpler factors, revealing structure and enabling solutions. While it may seem like algebraic manipulation for its own sake, factoring is fundamental to advanced mathematics and its applications. Engineers factor polynomials to analyze system stability. Economists factor equations to find equilibrium points. Physicists factor expressions describing motion to find when objects reach maximum height or return to ground level. The Zero Product Property—setting factored equations equal to zero to find solutions—underpins everything from designing parabolic satellite dishes to calculating optimal launch angles. Master factoring not just for the SAT, but as essential preparation for calculus, where factoring rational expressions and finding limits requires instant pattern recognition. The techniques you learn here—spotting difference of squares, recognizing perfect square trinomials, systematically approaching general trinomials, and checking whether factors can be factored further—represent systematic problem-solving strategies applicable far beyond algebra. When you can look at \(x^2 - 16\) and instantly see \((x+4)(x-4)\), or recognize that \(x^2 + 10x + 25\) is \((x+5)^2\), you're demonstrating the pattern recognition and algebraic fluency that defines mathematical maturity.