Factor Multivariable Polynomials (positive) - Algebra-1

1. Fundamental Concepts

Factoring Multivariable Polynomials refers to the process of decomposing a polynomial with multiple variables into a product of several integral expressions using methods such as extracting the greatest common factor (GCF) and grouping.
During the decomposition, the coefficients are mainly positive. The core is to identify the common factors among the terms of the polynomial or the rules for grouping, thereby simplifying the complex polynomial.

2. Key Concepts

Extracting the GCF: First, check if there is a common monomial factor (including the greatest common divisor of the coefficients and the lowest power of the same variable) among all terms of the polynomial. Extract this common factor and take it as one factor of the polynomial.
Grouping Method: When a polynomial has a relatively large number of terms (usually 4 terms) and there is no uniform common factor among all terms, the polynomial can be divided into two groups. Extract the GCF from each group separately. If the two groups have the same polynomial factor, extract this factor to complete the decomposition.
Judgment of Special Cases: Some multivariable polynomials cannot be factored through conventional grouping (e.g., there are terms that cannot be matched). It is necessary to clearly judge whether they can be factored to avoid ineffective operations.

3. Examples

Easy

Question: Factorize
Solution:
Group the polynomial as .
Extract 2xy from the first group to get , and extract 3 from the second group to get .
Then extract the common factor .
The final result is .

Medium

Question: Factorize
Solution:
First, extract the common factor y from all terms to get .
Then group the expression inside the parentheses as .
Extract from the first group to get , and extract 5 from the second group to get .
Finally, extract .
The result is .

Hard

Question: Factorize
Solution:
Group the polynomial as .
Extract from the first group to get , and extract from the second group to get .
Extract the common factor .
The final result is .

4. Problem-Solving Techniques

Step 1: Observe and Extract the GCF: First, check if there is a common monomial factor among all terms of the polynomial. If there is, extract it first to simplify the structure of the polynomial and lay the foundation for subsequent decomposition.

Step 2: Attempt Grouping: If the polynomial still has 4 terms after extracting the GCF, or if the original polynomial has no uniform common factor, divide it into two groups reasonably (usually grouping the first two terms and the last two terms). Ensure that the GCF can be extracted from each group.

Step 3: Verify and Extract: After grouping, extract the GCF from each group. Check if the two groups have the same polynomial factor. If yes, extract this factor to complete the decomposition. If no, try adjusting the grouping method or judge that the polynomial cannot be factored.

Step 4: Check Completeness: After completing the decomposition, expand the result through integral multiplication and compare it with the original polynomial to verify whether the decomposition is correct and ensure no omissions or errors.