Factorization, in essence, is the process of breaking down a mathematical expression into simpler parts, usually by finding expressions that, when multiplied together, produce the original expression. Instead of rules, one should think of it as utilizing various algebraic identities and strategies. Here are some key algebraic identities and their corresponding factorization principles:
Common Factorization Identities
Here are the most commonly used factorization identities:
Table of Factorization Identities
| Identity | Factorization Principle |
|---|---|
| (a + b)² = a² + 2ab + b² | Perfect square trinomial to binomial squared |
| (a − b)² = a² − 2ab + b² | Perfect square trinomial to binomial squared |
| a² – b² = (a + b)(a – b) | Difference of two squares |
| (a + b)³ = a³ + 3a²b + 3ab² + b³ = a³ + b³ + 3ab (a + b) | Sum of two cubes |
| (a - b)³ = a³ - 3a²b + 3ab² - b³ = a³ – b³ – 3ab (a – b) | Difference of two cubes |
| (a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴ | Power of 4 expansion |
| (a − b)⁴ = a⁴ − 4a³b + 6a²b² − 4ab³ + b⁴ | Power of 4 expansion |
| (a + b + c)² = a² + b² +c² + 2ab + 2ac + 2bc | Square of a trinomial |
Examples
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Perfect Square Trinomials:
- Example: x² + 4x + 4 can be factored as (x + 2)², since it matches the pattern a² + 2ab + b² where a = x and b = 2.
-
Difference of Two Squares:
- Example: y² - 9 can be factored as (y + 3)(y - 3), since it matches the pattern a² – b² where a = y and b = 3.
-
Sum and Difference of Cubes:
- Example: x³ + 8 can be factored as (x+2)(x²-2x+4)
- Example: x³ - 8 can be factored as (x-2)(x²+2x+4)
Factorization by Grouping
Sometimes, terms in a larger expression can be grouped to reveal common factors, facilitating factorization.
- Example: Consider the expression ax + ay + bx + by.
- First, group the terms: (ax + ay) + (bx + by)
- Then, factor out the common terms: a(x + y) + b(x + y)
- Now, (x+y) is common, so we can rewrite it as (x + y)(a + b)
Practical Insights
-
Always look for a greatest common factor (GCF) before applying other techniques. For example, 6x² + 12x has a GCF of 6x so it can be factored to 6x(x+2).
-
Use reverse FOIL (First, Outer, Inner, Last) for quadratic expressions. This involves testing factors of the constant term that will sum to the middle term.
-
Factorization is reversible: To check your factorization, expand the factored terms back. If it returns the original expression, your factorization is correct.
In summary, factorization is less about strict rules, but a toolbox of techniques utilizing algebraic identities, common factor extraction, and strategic grouping. Mastering these methods allows for the simplification of complex expressions, making them easier to handle in further mathematical operations.
