Overview
This lecture introduces factoring special product polynomials, focusing on the difference of two squares and the sum and difference of cubes, including methods, patterns, and sample problems.
Perfect Squares and Cubes
- Perfect squares are numbers or expressions raised to the power of 2 (e.g., 3² = 9, y⁶ = (y³)²).
- Perfect cubes are numbers or expressions raised to the power of 3 (e.g., 8 = 2³).
Multiplying and Factoring Polynomials
- The product of (a + b)(a - b) simplifies to a² - b², known as the difference of two squares.
- Factoring is the reverse process of multiplying: given a² - b², factors are (a + b)(a - b).
Factoring the Difference of Two Squares
- A difference of two squares appears as x² - y² and factors to (x + y)(x - y).
- Example: 81 - p² = (9 + p)(9 - p).
- Both terms must be perfect squares and separated by subtraction.
- If terms aren't perfect squares, check for a greatest common factor first, then factor further if possible.
- Example: 3w² - 48 = 3(w² - 16) = 3(w + 4)(w - 4).
Factoring the Difference and Sum of Two Cubes
- The difference of cubes: a³ - b³ = (a - b)(a² + ab + b²).
- The sum of cubes: a³ + b³ = (a + b)(a² - ab + b²).
- Signs in the formula follow the "SOP" mnemonic: Same, Opposite, Positive.
- The binomial uses the same sign as the original expression.
- The first sign in the trinomial is the opposite.
- The last sign in the trinomial is always positive.
- Example (difference): 125 - x³ = (5 - x)(25 + 5x + x²).
- Example (sum): 40k³ + 5 = 5(2k + 1)(4k² - 2k + 1).
Sample Problems and Answers
- d² - 25 factors to (d + 5)(d - 5).
- 25e² - 16 factors to (5e + 4)(5e - 4).
- 20c² - 45d² factors to 5(2c + 3d)(2c - 3d).
Key Terms & Definitions
- Perfect Square — an expression that can be written as (x)².
- Perfect Cube — an expression that can be written as (x)³.
- Difference of Two Squares — a polynomial of the form a² - b².
- Difference of Two Cubes — a polynomial of the form a³ - b³.
- Sum of Two Cubes — a polynomial of the form a³ + b³.
- Greatest Common Factor (GCF) — the largest common monomial factor in all terms.
Action Items / Next Steps
- Complete the self-learning module activities on factoring special products (difference of squares, sum/difference of cubes).
- Practice additional examples to reinforce the steps and patterns in factoring.
- Prepare for the next lesson on factoring trinomials.