Factor Multivariable Polynomials (negative) - Algebra-1

1. Fundamental Concepts

Factoring Multivariable Polynomials (Negative) refers to the decomposition of polynomials that contain multiple variables and have negative coefficients. Through methods such as extracting common factors (including negative common factors) and grouping, these polynomials are converted into a product of several integral expressions.
The core lies in handling the sign issues caused by negative coefficients to ensure the decomposition process is logically consistent and the results are accurate.

2. Key Concepts

Extracting negative common factors: When the signs of the coefficients of each term in the polynomial are different or there are negative coefficients, the negative greatest common divisor can be extracted as a common factor. After extraction, attention should be paid to adjusting the signs of the remaining terms to simplify subsequent decomposition.
Sign handling in grouping: When grouping, if one group needs to form the same factor as the other group after extracting the common factor, it may be necessary to adjust the signs within or between groups (e.g., grouping negative terms together and extracting the negative sign) to ensure the two groups have the same polynomial factor.
Avoiding sign errors: Throughout the decomposition process, close attention must be paid to sign changes, especially when extracting negative common factors or adjusting the signs of grouped terms, to prevent incorrect decomposition results due to sign mistakes.

3. Examples

Easy

Question: Factorize
Solution:
Group the polynomial as .
Extract from the first group to get , and extract from the second group to get .
Then extract the common factor .
The result is (or further extract w to get ).

Medium

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 result is (or extract the negative sign to get ).

Hard

Question: Factorize
Solution:
First, extract the common factor y to get .
Group the expression inside the parentheses 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: Prioritize handling signs and common factors: First, observe the coefficients of the polynomial. If there are negative coefficients and potential common factors among the terms, prioritize determining whether to extract a negative common factor. After extraction, adjust the signs of the remaining terms to simplify the polynomial.

Step 2: Group reasonably and adjust signs: Group terms based on their characteristics (e.g., grouping terms with the same variable together). If one group cannot form the same factor as the other group after grouping, try adjusting the signs of the terms within the group (e.g., moving negative terms to one group and extracting the negative sign) before extracting the common factor.

Step 3: Verify signs and factor consistency: After each step of decomposition, check whether the signs of the remaining terms after extracting the common factor are correct and whether the two groups have the same factor. If not, re-adjust the grouping or the sign of the common factor.

Step 4: Check the decomposition result: Expand the factored integral expressions by multiplication, and compare the signs of the coefficients and each term with the original polynomial to confirm that the result has no sign errors or missing terms, ensuring the decomposition is correct.