What is Factorization?
Factorization (or factoring) is the process of writing a polynomial as a product of simpler polynomials (factors). It's the reverse of multiplication.
Key Algebraic Identities for Factorization:
| Identity | Formula | Example |
|---|---|---|
| **Perfect Square (Difference)** | (x - y)² = x² - 2xy + y² | (x - 5)² = x² - 10x + 25 |
| **Difference of Squares** | x² - y² = (x + y)(x - y) | x² - 16 = (x + 4)(x - 4) |
| **Product of Binomials** | (x + a)(x + b) = x² + (a + b)x + ab | (x + 3)(x + 4) = x² + 7x + 12 |
Steps to Factor a Quadratic Trinomial (x² + bx + c):
- Find two numbers whose product = c and sum = b
- Write as (x + m)(x + n) where m × n = c and m + n = b
Special Cases:
- If a ≠ 1 in ax² + bx + c, factoring requires more steps (splitting middle term)
- Difference of squares: a² - b² = (a - b)(a + b)
- Sum of squares a² + b² cannot be factored with real numbers