Overview
This lecture covers special products in polynomials, with a focus on how to recognize and factor difference of squares, as well as sum and difference of cubes, using their unique patterns.
Perfect Squares and Cubes
- Perfect squares are numbers or expressions that can be written as something raised to the second power (e.g., 9 = 3², y⁶ = (y³)²).
- Perfect cubes are numbers or expressions written as something raised to the third power (e.g., 8 = 2³).
Multiplying and Factoring Special Products
- The product (a + b)(a - b) results in a² - b², called the difference of two squares.
- Factoring a² - b² gives (a + b)(a - b).
- If terms are not perfect squares, factor out the greatest common monomial factor (GCF) first before applying the difference of squares pattern.
Examples: Factoring Difference of Two Squares
- 81 - p² factors as (9 + p)(9 - p).
- 16a⁶ - 25b² factors as (4a³ + 5b)(4a³ - 5b).
- 3w² - 48 factors as 3(w + 4)(w - 4) after factoring out 3 and applying the difference of squares.
Factoring Sum and Difference of Cubes
- The difference of cubes a³ - b³ factors as (a - b)(a² + ab + b²).
- The sum of cubes a³ + b³ factors as (a + b)(a² - ab + b²).
- Use the mnemonic "SOP" (Same, Opposite, Positive) to remember the signs in trinomial factors.
Examples: Factoring Cubes and Using GCF
- 125 - x³ factors as (5 - x)(25 + 5x + x²).
- 40k³ + 5 factors as 5(2k + 1)(4k² - 2k + 1) after factoring out 5, then applying the sum of cubes.
Practice Questions Recap
- d² - 25 factors as (d + 5)(d - 5).
- 25e² - 16 factors as (5e + 4)(5e - 4).
- 20c² - 45d² factors as 5(2c + 3d)(2c - 3d).
Key Terms & Definitions
- Perfect Square — a number or expression that is the product of an integer or variable with itself.
- Perfect Cube — a number or expression that is the product of an integer or variable multiplied by itself three times.
- Difference of Two Squares — an expression of the form a² - b² that factors as (a + b)(a - b).
- Sum/Difference of Cubes — expressions a³ + b³ or a³ - b³, factored using specific binomial and trinomial patterns.
- Greatest Common Monomial Factor (GCF) — the largest monomial that divides each term of a polynomial.
Action Items / Next Steps
- Complete the self-learning module activity on factoring difference of two squares and cubes.
- Review examples and practice more on factoring special products.
- Prepare for the next lesson on factoring trinomials.