Factoring polynomials involves breaking down a polynomial into simpler expressions (factors) that, when multiplied together, equal the original polynomial. Here's a breakdown of the key rules:
Core Principles of Factoring Polynomials
These steps outline a systematic approach to factoring, which is crucial for simplifying algebraic expressions and solving equations.
1. Check for Common Factors (Greatest Common Factor - GCF)
-
Identify the GCF: Look for the largest factor that divides all terms in the polynomial.
-
Factor out the GCF: Divide each term by the GCF and write the expression as a product of the GCF and the remaining polynomial. This is the first step in nearly all factoring problems.
- Example: In the polynomial 6x² + 9x, the GCF is 3x. Factoring this out results in 3x(2x + 3).
2. Determine the Number of Terms
The number of terms in the polynomial often determines the appropriate factoring method:
-
Two Terms:
- Difference of Squares: a² - b² = (a + b)(a - b)
- Example: x² - 4 = (x + 2)(x - 2)
- Sum of Cubes: a³ + b³ = (a + b)(a² - ab + b²)
- Difference of Cubes: a³ - b³ = (a - b)(a² + ab + b²)
- Difference of Squares: a² - b² = (a + b)(a - b)
-
Three Terms:
- Simple Trinomial (ax² + bx + c, where a=1): Find two numbers that multiply to 'c' and add up to 'b'.
- Example: x² + 5x + 6 = (x + 2)(x + 3)
- General Trinomial (ax² + bx + c, where a ≠ 1): Can be factored by grouping or trial and error, often requiring factoring by grouping.
- Example: 2x² + 7x + 3 = (2x + 1)(x + 3)
- Simple Trinomial (ax² + bx + c, where a=1): Find two numbers that multiply to 'c' and add up to 'b'.
-
Four or More Terms: Factoring by grouping is often used.
- Example: x³ + 2x² + 3x + 6 = (x³ + 2x²) + (3x + 6) = x²(x+2) + 3(x+2) = (x²+3)(x+2)
3. Look for Factors that Can Be Factored Further
After each factoring step, examine the resulting factors to see if they can be factored again using the same techniques. This is an iterative process until the polynomial cannot be factored any further.
4. Check by Multiplying
To ensure the factoring is correct, multiply the resulting factors back together. If the product equals the original polynomial, then the factoring is correct. This is a vital step to verify your work.
Summary of the Factoring Process
| Step | Description | Example |
|---|---|---|
| 1. GCF | Factor out the greatest common factor (GCF) if any. | 4x² + 8x -> 4x(x + 2) |
| 2. Terms | Determine the number of terms to select appropriate factoring methods | 2 terms: Difference of squares; 3 terms: trinomial, 4+ terms: grouping |
| 3. Factor Further | Look for opportunities to factor further | (x² - 4) -> (x+2)(x-2), factor again if needed |
| 4. Check | Multiply the factors to verify the result | (x + 2)(x - 3) -> x² - x - 6, confirms previous factoring if product matches initial polynomial |
By applying these rules, you can systematically factor a wide variety of polynomial expressions.
