Factor Polynomials
Complete Notes & Formulae for Eleventh Grade (Algebra 2)
1. Understanding Factors
What is Factoring?
Factoring is the reverse process of multiplication - breaking down an expression into products of simpler expressions.
Example:
Multiply: \( (x + 2)(x + 3) = x^2 + 5x + 6 \)
Factor: \( x^2 + 5x + 6 = (x + 2)(x + 3) \)
What are Factors?
Factors are expressions that when multiplied together produce the original expression.
Single-Variable Factors:
Factors of \( x^2 - 9 \): \( (x + 3) \) and \( (x - 3) \)
Multi-Variable Factors:
Factors of \( x^2y - xy^2 \): \( xy \) and \( (x - y) \)
2. Factor Out a Monomial (Greatest Common Factor)
Method:
Always check for GCF FIRST before using any other method!
Steps:
1. Find the greatest common factor of all terms
2. Divide each term by the GCF
3. Write as: GCF × (remaining polynomial)
Examples:
Example 1:
Factor: \( 6x^3 + 9x^2 - 12x \)
GCF = \( 3x \)
Divide each term: \( \frac{6x^3}{3x} = 2x^2, \; \frac{9x^2}{3x} = 3x, \; \frac{-12x}{3x} = -4 \)
Answer: \( 3x(2x^2 + 3x - 4) \)
Example 2:
Factor: \( 15a^2b^3 - 10ab^2 + 5ab \)
GCF = \( 5ab \)
Answer: \( 5ab(3ab^2 - 2b + 1) \)
3. Factor Quadratics
General Form:
\[ ax^2 + bx + c = (mx + p)(nx + q) \]
Case 1: When a = 1
Method: Find two numbers that:
• Multiply to give \( c \)
• Add to give \( b \)
Example:
Factor: \( x^2 + 7x + 12 \)
Need two numbers that multiply to 12 and add to 7
Numbers: 3 and 4 (3 × 4 = 12, 3 + 4 = 7)
Answer: \( (x + 3)(x + 4) \)
Case 2: When a ≠ 1 (AC Method)
Steps:
1. Multiply \( a × c \) (product)
2. Find two numbers that multiply to \( ac \) and add to \( b \)
3. Split the middle term using these numbers
4. Factor by grouping
Example:
Factor: \( 3x^2 + 10x + 8 \)
Step 1: \( ac = 3 × 8 = 24 \)
Step 2: Numbers that multiply to 24 and add to 10: 4 and 6
Step 3: Split: \( 3x^2 + 4x + 6x + 8 \)
Step 4: Group: \( (3x^2 + 4x) + (6x + 8) \)
Factor each: \( x(3x + 4) + 2(3x + 4) \)
Answer: \( (3x + 4)(x + 2) \)
4. Factor Using Quadratic Patterns (Special Products)
Pattern 1: Difference of Squares
\[ a^2 - b^2 = (a + b)(a - b) \]
Recognition:
• Two perfect squares
• Subtraction between them
Examples:
• \( x^2 - 25 = (x + 5)(x - 5) \)
• \( 4x^2 - 9 = (2x + 3)(2x - 3) \)
• \( 16a^2 - 49b^2 = (4a + 7b)(4a - 7b) \)
Pattern 2: Perfect Square Trinomial
\[ a^2 + 2ab + b^2 = (a + b)^2 \]
\[ a^2 - 2ab + b^2 = (a - b)^2 \]
Recognition:
• First and last terms are perfect squares
• Middle term = \( 2 × \sqrt{\text{first}} × \sqrt{\text{last}} \)
Examples:
• \( x^2 + 6x + 9 = (x + 3)^2 \)
• \( 4x^2 - 12x + 9 = (2x - 3)^2 \)
• \( 25a^2 + 20a + 4 = (5a + 2)^2 \)
5. Factor by Grouping
When to Use:
Use grouping when you have 4 terms or when factoring quadratics where \( a ≠ 1 \)
Steps:
Step 1: Group terms in pairs
Usually first two and last two terms
Step 2: Factor out GCF from each group
Step 3: Factor out the common binomial
Examples:
Example 1:
Factor: \( x^3 + 3x^2 + 2x + 6 \)
Step 1: Group: \( (x^3 + 3x^2) + (2x + 6) \)
Step 2: Factor each: \( x^2(x + 3) + 2(x + 3) \)
Step 3: Common binomial: \( (x + 3) \)
Answer: \( (x + 3)(x^2 + 2) \)
Example 2:
Factor: \( 6xy - 9x + 10y - 15 \)
Group: \( (6xy - 9x) + (10y - 15) \)
Factor each: \( 3x(2y - 3) + 5(2y - 3) \)
Answer: \( (2y - 3)(3x + 5) \)
6. Factor Sums and Differences of Cubes
Sum of Cubes Formula:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
Memory Aid: "SOAP"
Same sign (as original)
Opposite sign
Always positive
Perfect squares
Example:
Factor: \( x^3 + 8 \)
\( a = x, \; b = 2 \) (since \( 2^3 = 8 \))
Apply formula: \( (x + 2)(x^2 - 2x + 4) \)
Answer: \( (x + 2)(x^2 - 2x + 4) \)
Difference of Cubes Formula:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
Example:
Factor: \( 27x^3 - 64 \)
\( a = 3x, \; b = 4 \) (since \( (3x)^3 = 27x^3 \) and \( 4^3 = 64 \))
Apply formula: \( (3x - 4)((3x)^2 + (3x)(4) + 4^2) \)
Simplify: \( (3x - 4)(9x^2 + 12x + 16) \)
Answer: \( (3x - 4)(9x^2 + 12x + 16) \)
Common Perfect Cubes:
• \( 1 = 1^3 \)
• \( 8 = 2^3 \)
• \( 27 = 3^3 \)
• \( 64 = 4^3 \)
• \( 125 = 5^3 \)
• \( 216 = 6^3 \)
7. Complete Factoring Strategy
General Factoring Procedure:
Follow these steps IN ORDER:
Step 1: Factor out GCF (if any)
Always check for greatest common factor first!
Step 2: Count the terms
• 2 terms → Check for difference of squares or sum/difference of cubes
• 3 terms → Try quadratic factoring or perfect square trinomial
• 4+ terms → Try grouping
Step 3: Factor completely
Check if any factor can be factored further
Step 4: Check your answer
Multiply the factors to verify you get the original polynomial
Complete Example:
Factor completely: \( 2x^4 - 32 \)
Step 1: Factor out GCF
GCF = 2
\( 2(x^4 - 16) \)
Step 2: Recognize pattern
\( x^4 - 16 \) is difference of squares: \( (x^2)^2 - 4^2 \)
\( 2(x^2 + 4)(x^2 - 4) \)
Step 3: Factor further if possible
\( x^2 - 4 \) is also difference of squares!
\( 2(x^2 + 4)(x + 2)(x - 2) \)
Final Answer: \( 2(x^2 + 4)(x + 2)(x - 2) \)
Note: \( x^2 + 4 \) cannot be factored over real numbers
8. Quick Reference - Factoring Formulas
| Pattern | Formula |
|---|---|
| Difference of Squares | \( a^2 - b^2 = (a+b)(a-b) \) |
| Perfect Square Trinomial | \( a^2 ± 2ab + b^2 = (a ± b)^2 \) |
| Sum of Cubes | \( a^3 + b^3 = (a+b)(a^2-ab+b^2) \) |
| Difference of Cubes | \( a^3 - b^3 = (a-b)(a^2+ab+b^2) \) |
| General Quadratic | \( ax^2 + bx + c \) → Find factors or use AC method |
Remember:
✓ Always factor out GCF first
✓ Factor completely (check if factors can be factored again)
✓ Verify by multiplying factors back together
✓ Sum of squares \( a^2 + b^2 \) CANNOT be factored over real numbers
📚 Study Tips
✓ Master the GCF method - it's used in almost every problem
✓ Memorize special product formulas to recognize patterns quickly
✓ Use SOAP to remember sum/difference of cubes
✓ Always check your work by multiplying factors back
✓ Practice identifying which method to use by counting terms