Factoring by grouping is the go-to method for polynomials with four terms — the kind that have no obvious single greatest common factor. The idea is simple: pair the terms, factor each pair separately, and a shared binomial should appear. When it does, one final factorization finishes the job. Here are the steps with three worked examples, including the negative-sign cases that cause most errors.
The 4-Step Grouping Method
- Form two pairs. Group the first two terms and the last two terms.
- Factor the GCF out of each pair. Pull the greatest common factor from each group separately.
- Check for a matching binomial. Both groups should now contain an identical binomial factor.
- Factor out the common binomial. Combine the outside factors into one binomial and multiply by the shared one.
Example 1: Basic 4-Term Polynomial
Factor completely:
x³ + 3x² + 2x + 6
Step 1: Form two separate pairs
(x³ + 3x²) + (2x + 6)
Step 2: Extract the GCF from each group
- From (x³ + 3x²), the common factor is x², leaving (x + 3).
- From (2x + 6), the common factor is 2, leaving (x + 3).
Rewritten: x²(x + 3) + 2(x + 3)
Step 3: Identify the shared binomial
Both groups contain the identical binomial (x + 3).
Step 4: Extract the common binomial
Pull (x + 3) out and combine the outside terms (x² + 2):
(x² + 2)(x + 3)
Verify with FOIL: x²·x = x³; x²·3 = 3x²; 2·x = 2x; 2·3 = 6 → x³ + 3x² + 2x + 6 ✓
Example 2: Factoring with Negative Signs
Negative signs are the most common source of grouping errors. Factor:
2x³ − 6x² − 5x + 15
Step 1: Group the terms carefully
Treat the minus sign as part of the third term:
(2x³ − 6x²) + (−5x + 15)
Step 2: Factor out the GCF
- From (2x³ − 6x²), factor out 2x², leaving (x − 3).
- From (−5x + 15), factor out −5 (not positive 5) so the signs inside reverse: −5x ÷ −5 = x, 15 ÷ −5 = −3, leaving (x − 3).
Rewritten: 2x²(x − 3) − 5(x − 3)
Step 3: Extract the matching binomial
Factor out (x − 3):
(2x² − 5)(x − 3)
Verify: 2x³ − 6x² − 5x + 15 ✓
When the second pair starts with a minus, factor out a negative GCF. Pulling out positive 5 from (−5x + 15) gives (x + 3) — and the binomials will never match.
Example 3: Rearranging Terms Before Grouping
Sometimes the terms do not cooperate in their original order. Factor:
xy + 6 + 2y + 3x
Grouping the first two terms (xy + 6) goes nowhere — they share nothing useful. So rearrange the polynomial, placing terms with matching variables side by side:
xy + 3x + 2y + 6
Now proceed with standard grouping:
- Group into pairs: (xy + 3x) + (2y + 6)
- Factor out x from the first pair: x(y + 3)
- Factor out 2 from the second pair: 2(y + 3)
- Combine into factored form: (x + 2)(y + 3)
Verify: x·y + x·3 + 2·y + 2·3 = xy + 3x + 2y + 6 ✓
3 Critical Mistakes to Avoid
- Distributing negative signs incorrectly: when factoring a negative GCF out of the second pair, remember every sign inside flips.
- Giving up before rearranging: if the first and second terms share no GCF, try swapping the order of the second and third terms before concluding the polynomial cannot be grouped.
- Forcing it when the binomials do not match: if Step 2 produces, say, (x − 4) and (x + 4), you cannot factor out the binomial. Recheck your signs or try a different pairing order.
After every factoring problem, multiply the factors back out. If you do not get the original polynomial, something went wrong — usually a sign.
Quick Practice (with answers)
- x³ + 2x² + 5x + 10
- 3x³ − 9x² + 2x − 6
Answers: 1) x²(x + 2) + 5(x + 2) = (x² + 5)(x + 2); 2) 3x²(x − 3) + 2(x − 3) = (3x² + 2)(x − 3).
Mastering grouping gives you a reliable tool for four-term polynomials and a solid foundation for factoring higher-degree expressions, simplifying rational expressions, and analyzing curves in calculus.