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Factoring Polynomials Using Grouping
Definition of Factoring by Grouping
Factoring by grouping involves rewriting an expression into separate groups.
Each group may have a Greatest Common Factor (GCF) that can be factored out.
Steps to Factor by Grouping
Grouping Terms: Use parentheses to combine terms into different groups.
Finding GCF: Factor each of the grouped terms separately, identifying the GCF of each group.
Factor Out Common GCF: Factor the overall polynomial by taking out the common GCF of all terms.
Example of Factoring by Grouping
Expression: ax + ay + bx + by
Step 1: Grouping Terms
Grouped as: (ax + ay) + (bx + by)
Step 2: Finding GCF
For (ax + ay), GCF = a
For (bx + by), GCF = b
Rewritten: a(x + y) + b(x + y)
Step 3: Factoring Common GCF
Final Factor: (x + y)(a + b)
Limitations of Factoring by Grouping
Factoring by grouping is essential when polynomials do not share a common GCF overall.
Additional Example
Expression: pq - 3q + 4p - 12
Step 1: Grouping Terms
(pq - 3q) + (4p - 12)
Step 2: Finding GCFs
GCF of first group = q, GCF of second group = 4
Rewritten: q(p - 3) + 4(p - 3)
Step 3: Common Factor Extraction
Final Factor: (p - 3)(q + 4)
Important Notes
Always be cautious of signs, especially when factoring out negatives.
Example: mx - my - nx + ny
Proper grouping would involve: (mx - my) - (nx - ny).
Factor with GCF First
If all the terms share a GCF, factor out the GCF before grouping.
Example: 12m³n² - 8m²n² + 9m³n³ - 6m²n³
GCF is m²n², leading to:
m²n²(12m - 8 + 9mn - 6n) = m²n²(3m - 2)(4m + 3n)
Review Questions
Factor the following expressions:
x² + 5x + xy + 5y
3fm - gm + 6fn - 2gn
10a² + 14a - 15ab - 21b
5ac - 15ad - bc + 3bd
2m + 7am - 6n - 21an
30am² - 40amn + 16bmn - 12bm²
Answers to Factor Examples
(x + 5)(x + y)
(3f - g)(m + 2n)
(5a + 7)(2a - 3b)
(c - 3d)(5a - b)
(2 + 7a)(m - 3n)
2m(3m - 2)(5a - 2b)