Overview
This lecture covers how to fully factorize various algebraic expressions, including factoring out common terms, recognizing the difference of squares, and grouping.
Factoring Out Common Factors
- Always check for and factor out the Highest Common Factor (HCF) first.
- For 6x³ - 9x²: HCF is 3x².
- Factor: 6x³ ÷ 3x² = 2x and -9x² ÷ 3x² = -3.
- Full factorization: 3x²(2x - 3).
Difference of Two Squares
- If only 1 is common, consider the difference of squares.
- Recognize 9x² - 49y² as a difference of two squares: (3x)² - (7y)².
- Factor as: (3x + 7y)(3x - 7y).
Factorizing by Grouping
- For four-term expressions, use grouping to factor.
- Group the first two terms: a³ - a²b, HCF is a², factor to get a²(a - b).
- Group the last two terms: 4a - 4b, HCF is 4, factor to get 4(a - b).
- Since both groups have (a - b), final factorization: (a - b)(a² + 4).
Key Terms & Definitions
- Factorize — To write an expression as a product of its factors.
- Highest Common Factor (HCF) — The largest expression that divides all terms.
- Difference of Two Squares — An expression in the form a² - b², which factors as (a + b)(a - b).
- Grouping — A method of factoring by grouping terms with common factors.
Action Items / Next Steps
- Practice more problems on factoring expressions, especially using HCF, difference of squares, and grouping.