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Factoring Polynomials
Factoring Trinomials of the Form by Grouping
Factoring is a fundamental skill in algebra that involves rewriting a polynomial as a product of simpler polynomials. When the leading coefficient a is not 1, the grouping method is a systematic approach to factor trinomials of the form .
Definition: Grouping Method
Grouping method is an alternative technique used to factor trinomials where the leading coefficient .
This method involves splitting the middle term and factoring by grouping.
Steps for Factoring Trinomials by Grouping
Factor out the Greatest Common Factor (GCF): If the trinomial has a GCF other than 1, factor it out first.
Find Two Numbers: For the trinomial , find two numbers whose product is and whose sum is .
Rewrite the Middle Term: Express the middle term as the sum of two terms using the numbers found in step 2.
Factor by Grouping: Group the four terms into two pairs and factor each pair. Then factor out the common binomial factor.
Example 1: Factor by Grouping
Step 1: No GCF other than 1.
Step 2: . Find two numbers whose product is 40 and sum is -14: -4 and -10.
Step 3: Rewrite as :
Step 4: Factor by grouping:
Example 2: Factor by Grouping
Step 1: Factor out GCF 2:
Step 2: . Find two numbers whose product is -30 and sum is -1: -6 and 5.
Step 3: Rewrite as :
Step 4: Factor by grouping:
So,
Example 3: Factor by Grouping
Step 1: No GCF other than 1.
Step 2: . Find two numbers whose product is 8 and sum is 9: 8 and 1.
Step 3: Rewrite as :
Step 4: Factor by grouping:
Example 4: Factor by Grouping
Step 1: No GCF other than 1.
Step 2: . Find two numbers whose product is -24 and sum is 10: 12 and -2.
Step 3: Rewrite as :
Step 4: Factor by grouping:
Summary Table: Steps for Factoring by Grouping
Step | Description |
|---|---|
1 | Factor out the greatest common factor (GCF) if present. |
2 | Find two numbers whose product is and whose sum is . |
3 | Rewrite the middle term using the two numbers found. |
4 | Factor by grouping and extract the common binomial factor. |
Key Terms
Trinomial: A polynomial with three terms, typically written as .
Greatest Common Factor (GCF): The largest factor that divides all terms of a polynomial.
Factoring by Grouping: A method of factoring polynomials by grouping terms with common factors and factoring each group separately.
Additional info:
Factoring trinomials is a foundational skill for solving quadratic equations, simplifying rational expressions, and analyzing polynomial functions in College Algebra.
