Factoring by Grouping — Definition, Formula & Examples
Factoring by grouping is a method where you split a polynomial (usually with four terms) into two pairs, factor out the greatest common factor from each pair, and then factor out the shared binomial.
Given a polynomial , factoring by grouping rewrites it as , then extracts the common binomial factor to produce . The method extends to polynomials where strategic grouping reveals a common factor across groups.
How It Works
Start with a four-term polynomial. Split the terms into two groups of two—choose groups so that each pair shares a common factor. Factor out the GCF from each group. If both groups now contain the same binomial factor, factor that binomial out. The result is a product of two factors. If the binomial factors do not match, try rearranging the terms and grouping differently.
Worked Example
Problem: Factor x³ + 3x² + 2x + 6.
Group the terms: Split into two pairs.
Factor each group: Pull out the GCF from each pair: x² from the first group and 2 from the second.
Factor out the common binomial: Both groups share the factor (x + 3), so factor it out.
Answer:
Why It Matters
Factoring by grouping is essential for polynomials that do not fit simple trinomial patterns, especially cubics and four-term expressions. It also appears inside the AC method for factoring harder quadratics like , where you split the middle term and then group.
Common Mistakes
Mistake: Getting two different binomial factors after grouping, such as , and trying to factor anyway.
Correction: The binomial factors must be identical. If they differ, rearrange the original terms and try a different grouping before concluding the polynomial cannot be factored this way.