Overview
This lecture explains key methods for factoring polynomials: finding the greatest common factor (GCF), factoring trinomials, factoring by grouping, and factoring the difference of squares.
Factoring Out the GCF
- For 6x - 12, factor out 6: 6(x - 2).
- For 3x³ - 9x², factor out 3x²: 3x²(x - 3).
- For 4x² - 12x, factor out 4x: 4x(x - 3).
Factoring Trinomials (Leading Coefficient 1)
- x² + 7x + 12: Find numbers that multiply to 12 and add to 7 (3 and 4): (x + 3)(x + 4).
- x² + 2x - 15: Find numbers that multiply to -15 and add to 2 (5 and -3): (x + 5)(x - 3).
Factoring Trinomials (Leading Coefficient ≠ 1)
- 2x² - 6x - 56: Factor out 2 → 2(x² - 3x - 28), then (x - 7)(x + 4).
- 3x² - 18x + 24: Factor out 3 → 3(x² - 6x + 8), then (x - 4)(x - 2).
Factoring Difference of Squares
- x² - 16: (x + 4)(x - 4)
- x² - 64: (x + 8)(x - 8)
- 4x² - 25: (2x + 5)(2x - 5)
- 9x² - 49: (3x + 7)(3x - 7)
Factoring by Grouping
- 2x² - 5x - 3: Multiply 2 × -3 = -6; numbers are -6 and 1. Split: 2x² - 6x + 1x - 3. Group: (2x² - 6x) + (1x - 3) → 2x(x - 3) + 1(x - 3) → (x - 3)(2x + 1).
- 6x² + x - 15: Multiply 6 × -15 = -90; numbers are -9 and 10. Split: 6x² - 9x + 10x - 15 → 3x(2x - 3) + 5(2x - 3) → (2x - 3)(3x + 5).
Factoring Polynomials with Four Terms (Grouping)
- 3x³ - 2x² - 12x + 8: Group as (3x³ - 2x²) + (-12x + 8). Factor: x²(3x - 2) - 4(3x - 2) → (3x - 2)(x² - 4) → (3x - 2)(x + 2)(x - 2).
Key Terms & Definitions
- Greatest Common Factor (GCF): Largest factor shared by all terms.
- Trinomial: A polynomial with three terms.
- Difference of Squares: Expression a² - b², factors as (a + b)(a - b).
- Factoring by Grouping: Pairing terms and factoring out common factors.
Next Steps
- Practice more factoring problems using these methods.
- Review difference of squares and grouping techniques.