Factoring out the greatest common divisor (GCD) involves identifying the largest number that divides evenly into all terms of an expression and then rewriting the expression as a product of the GCD and the remaining terms.
Here's a step-by-step breakdown:
1. Find the Greatest Common Divisor (GCD):
- List the factors of each term: Write down all the numbers that divide evenly into each term.
- Identify the common factors: Find the factors that appear in the lists for all terms.
- Choose the largest common factor: The largest number among the common factors is the GCD.
Example: Find the GCD of 12x and 18y.
- Factors of 12x: 1, 2, 3, 4, 6, 12, x, 2x, 3x, 4x, 6x, 12x
- Factors of 18y: 1, 2, 3, 6, 9, 18, y, 2y, 3y, 6y, 9y, 18y
- Common factors: 1, 2, 3, 6
- Greatest common factor: 6
2. Divide Each Term by the GCD:
Divide each term in the original expression by the GCD you found in step 1.
Example: Using the expression 12x + 18y and the GCD of 6:
- 12x / 6 = 2x
- 18y / 6 = 3y
3. Rewrite the Expression:
Rewrite the original expression as the product of the GCD and the results of the division in step 2.
Example:
12x + 18y = 6(2x + 3y)
More Complex Examples:
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Example 1: Factor out the GCD from 24a2b + 36ab2
- GCD of 24 and 36: 12
- GCD of a2b and ab2: ab
- Therefore, the GCD of 24a2b and 36ab2 is 12ab
- 24a2b / 12ab = 2a
- 36ab2 / 12ab = 3b
- Final factored expression: 12ab(2a + 3b)
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Example 2: Factor out the GCD from 5x3 - 10x2 + 15x
- GCD of 5, -10, and 15: 5
- GCD of x3, x2, and x: x
- Therefore, the GCD of 5x3, -10x2, and 15x is 5x.
- 5x3 / 5x = x2
- -10x2 / 5x = -2x
- 15x / 5x = 3
- Final factored expression: 5x(x2 - 2x + 3)
Key Points:
- Always look for the greatest common divisor.
- Be sure to include variables in the GCD if they are present in all terms.
- After factoring, you can always check your work by distributing the GCD back into the parentheses to see if you get the original expression.
Factoring out the greatest common divisor simplifies expressions and is a fundamental skill in algebra.
