Common Factor Method of Factorization
Definition:
The common factor method involves factoring out the greatest common factor (GCF) from the terms of an algebraic expression. This is similar to finding the highest common factor (HCF) in numbers.
Steps for Common Factor Method:
- Identify common factors in each term of the expression.
- Factor out the common term from the expression.
- Rewrite the expression in factored form.
Examples:
Example 1: Factorizing a Simple Expression
Factorize: 6x + 9
Step 1: Find the common factor of 6x and 9.
- 6x = 2 × 3 × x
- 9 = 3 × 3
- The common factor is 3.
Step 2: Factor out 3:
6x + 9 = 3(2x + 3)
Thus, the factored form is 3(2x + 3).
Example 2: Factorizing an Expression with Variables
Factorize: 4xy + 8x
Step 1: Identify the common factors.
- 4xy = 2 × 2 × x × y
- 8x = 2 × 2 × 2 × x
- The common factor is 4x.
Step 2: Factor out 4x:
4xy + 8x = 4x(y + 2)
Thus, the factored form is 4x(y + 2).
Example 3: Factorizing a Three-Term Expression
Factorize: 12x²y + 18xy² + 24xy
Step 1: Identify the common factors in all terms.
- 12x²y = 2 × 2 × 3 × x × x × y
- 18xy² = 2 × 3 × 3 × x × y × y
- 24xy = 2 × 2 × 2 × 3 × x × y
- The common factor is 6xy.
Step 2: Factor out 6xy:
12x²y + 18xy² + 24xy = 6xy(2x + 3y + 4)
Thus, the factored form is 6xy(2x + 3y + 4).
Conclusion:
The Common Factor Method is a simple and effective way to factorize expressions by taking out the greatest common factor (GCF). It simplifies expressions and makes algebraic operations easier.