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.
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.
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.
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: 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
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: \(3x(2x - 1)(x + 3)\)
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: \((x + 7)(x - 7)\)
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 \((x + 6)(x + 6)\)
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: \((x + 3)(x - 8)\)
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 \(x = 2\)
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: \((3x + 2)(x + 3)\)
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: \((x^2 + 9)(x + 3)(x - 3)\)
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.
