Lesson 1 • 45 min read
Factoring Polynomials
In This Lesson
1Greatest Common Factor (GCF)
Always Try This First!
Find the largest factor that divides evenly into all terms.
Steps:
- Find the GCF of all coefficients (numbers)
- Find the lowest power of each common variable
- Factor out the GCF and write remaining terms in parentheses
Example 1:
6x² + 9x
GCF = 3x
= 3x(2x + 3)
Example 2:
12x³ - 8x² + 4x
GCF = 4x
= 4x(3x² - 2x + 1)
2Difference of Two Squares
Pattern: a² - b² = (a + b)(a - b)
How to Recognize:
- Two terms only
- Both terms are perfect squares
- Connected by subtraction (-)
Example 1:
x² - 25
a = x, b = 5
= (x + 5)(x - 5)
Example 2:
4x² - 9
a = 2x, b = 3
= (2x + 3)(2x - 3)
Example 3:
16x⁴ - 1
a = 4x², b = 1
= (4x² + 1)(4x² - 1)
Example 4:
x² - y²
a = x, b = y
= (x + y)(x - y)
⚠️ Important:
a² + b² (sum of squares) CANNOT be factored with real numbers!
3Perfect Square Trinomials
Patterns
a² + 2ab + b² = (a + b)²
a² - 2ab + b² = (a - b)²
How to Check:
- First term is a perfect square (a²)
- Last term is a perfect square (b²)
- Middle term is 2 × a × b
Example 1:
x² + 6x + 9
a = x, b = 3, check: 2(x)(3) = 6x ✓
= (x + 3)²
Example 2:
x² - 10x + 25
a = x, b = 5, check: 2(x)(5) = 10x ✓
= (x - 5)²
Example 3:
4x² + 12x + 9
a = 2x, b = 3, check: 2(2x)(3) = 12x ✓
= (2x + 3)²
Example 4:
9x² - 24x + 16
a = 3x, b = 4, check: 2(3x)(4) = 24x ✓
= (3x - 4)²
4General Trinomial (x² + bx + c)
Find Two Numbers Method
Find two numbers that:
- • Multiply to give c (the constant term)
- • Add to give b (the middle coefficient)
Example 1: x² + 7x + 12
Find numbers that multiply to 12 and add to 7
3 × 4 = 12 and 3 + 4 = 7 ✓
= (x + 3)(x + 4)
Example 2: x² - 5x + 6
Multiply to +6, add to -5
(-2) × (-3) = 6 and (-2) + (-3) = -5 ✓
= (x - 2)(x - 3)
Example 3: x² + 2x - 15
Multiply to -15, add to +2
5 × (-3) = -15 and 5 + (-3) = 2 ✓
= (x + 5)(x - 3)
Example 4: x² - x - 12
Multiply to -12, add to -1
(-4) × 3 = -12 and (-4) + 3 = -1 ✓
= (x - 4)(x + 3)
5Sum/Difference of Cubes
Formulas
Sum of Cubes
a³ + b³ = (a + b)(a² - ab + b²)
Difference of Cubes
a³ - b³ = (a - b)(a² + ab + b²)
Memory Trick: SOAP
- Same sign as the original
- Opposite sign
- Always Positive
Example 1: x³ + 8
a = x, b = 2
= (x + 2)(x² - 2x + 4)
Example 2: x³ - 27
a = x, b = 3
= (x - 3)(x² + 3x + 9)
Perfect Cubes to Remember:
1³=1, 2³=8, 3³=27, 4³=64, 5³=125, 10³=1000