Using the Distributive Property in Reverse - Math 6

1. Fundamental Concepts

Factors: Elements that are multiplied together, which can be divided into two categories: numbers and variables/constants. They are the basic multiplicative units that make up an expression.

Essence of Using the Distributive Property in Reverse: Different from the forward distributive property (where a factor is multiplied to each term), the reverse operation involves extracting the common factor from multiple terms. Its core is to "identify the same factor", and it follows the same logic as combining like terms.

Greatest Common Factor (GCF): The "largest factor" that can divide all terms in an expression without a remainder. It includes the greatest common divisor of the coefficients and the lowest power of the common variables.

2. Key Concepts

Applicable Scenarios: For algebraic expressions with 2 or more terms, used to extract the common factor shared by all terms.
Core Logic: After extracting the GCF, the expression is transformed into the form of "GCF × (each original term ÷ GCF)".
Verification Criterion: Redistribute the GCF in the factored result to each term inside the parentheses. If the original expression can be restored, the factoring is correct.
Factoring Principle: Prioritize extracting the GCF to ensure that there are no more common factors (other than 1) among the terms inside the parentheses.

3. Examples

Easy

Problem: Factorize \(3xy + 6x\)
Steps:

Find the GCF: The GCF of the numerical part (3 and 6) is 3; the common unknown factor of the unknown part (xy and x) is x. Thus, the overall GCF is 3x.
Extract the GCF: \(3x(y + 2)\)
Verification: \(3x \times y + 3x \times 2 = 3xy + 6x\), which matches the original expression. The factorization is correct.

Medium

Problem: Factorize \(4x^2y + 8xy^2 + 12x^2y^2\)
Steps:

Find the GCF: The GCF of the numerical part (4, 8, 12) is 4; the common unknown factor of the unknown part (\(x^2y\), \(xy^2\), \(x^2y^2\)) is xy. Thus, the overall GCF is 4xy.
Extract the GCF: \(4xy(x + 2y + 3xy)\)
Verification: \(4xy \times x + 4xy \times 2y + 4xy \times 3xy = 4x^2y + 8xy^2 + 12x^2y^2\), which matches the original expression. The factorization is correct.

Hard

Problem: Factorize \(5a^3b^2c + 10a^2b^3c + 15a^2b^2c^2 + 20a^3b^2\)
Steps:

Find the GCF: The GCF of the numerical part (5, 10, 15, 20) is 5; the common unknown factor of the unknown part (\(a^3b^2c\), \(a^2b^3c\), \(a^2b^2c^2\), \(a^3b^2\)) is \(a^2b^2\) (since c only exists in the first three terms and is not shared by all terms). Thus, the overall GCF is \(5a^2b^2\).
Extract the GCF: \(5a^2b^2(abc + 2bc + 3c^2 + 4a)\)
Verification: \(5a^2b^2 \times abc + 5a^2b^2 \times 2bc + 5a^2b^2 \times 3c^2 + 5a^2b^2 \times 4a = 5a^3b^2c + 10a^2b^3c + 15a^2b^2c^2 + 20a^3b^2\), which matches the original expression. The factorization is correct.

4. Problem-Solving Techniques

Two-Step Method for GCF Extraction:
- Step 1 (Coefficients): Take the absolute values of the coefficients of all terms, decompose them into prime factors, identify the common prime factors and multiply them together to get the GCF of the coefficients. (Note: Keep the sign of the original coefficients; if all terms are negative, the GCF should be negative.)
- Step 2 (Variables): Identify the common variables in all terms, take the lowest power of each common variable, and multiply all the variable parts together to get the GCF of the variables. If there are no common variables, the GCF of the variables is 1.
- Final GCF = GCF of Coefficients × GCF of Variables
Step-by-Step Checking Method:
- Step 1: After extracting the GCF, check if each term inside the parentheses can still be divided by a common factor other than 1. If yes, reconfirm the GCF.
- Step 2: The verification step is mandatory. Restore the original expression using the forward distributive property to eliminate calculation errors.
Technique for Handling Signs: If there are many negative terms in the expression (e.g., 1 out of 2 terms is negative, or 2 out of 3 terms are negative), a GCF with a negative sign can be extracted to simplify the signs of the terms inside the parentheses (Example: )