Calculator
Enter an expanded four-term polynomial. The tool combines like terms, tests grouping patterns, and returns grouped factors with verification values.
Formula Used
Grouping rewrites a four-term expression into two pairs. Each pair is factored first, then a shared polynomial is extracted from both groups.
General form: G[A(x)R(x) + B(x)R(x)] = G·R(x)·(A(x) + B(x))
Overall common factor: G = gcd(|a₁|, |a₂|, |a₃|, |a₄|) · xmin(exponents)
Pair factor: For each pair, factor out the coefficient gcd and the smallest exponent. The remaining two-term polynomial must match across both groups.
How to Use This Calculator
- Enter an expanded four-term polynomial, such as
6x^3+9x^2-4x-6. - Choose the variable symbol and select either automatic or manual pairing strategy.
- Keep overall factor extraction enabled to remove any monomial common to all terms first.
- Press Factor Polynomial to place the result beneath the header and above the form.
- Review grouped pairs, final factors, the verification table, and the graph.
- Use the export buttons to save the summary as CSV or PDF.
Example Data Table
| Polynomial | Grouped form | Final factors |
|---|---|---|
| 6x^3 + 9x^2 - 4x - 6 | 3x^2(2x + 3) - 2(2x + 3) | (2x + 3)(3x^2 - 2) |
| 2x^3 + 4x^2 + 2x + 4 | 2[x^2(x + 2) + 1(x + 2)] | 2(x + 2)(x^2 + 1) |
| x^3 - 5x^2 + 2x - 10 | x^2(x - 5) + 2(x - 5) | (x - 5)(x^2 + 2) |
FAQs
1. What does factoring by grouping mean?
It means splitting a four-term polynomial into two smaller groups, factoring each group, then pulling out the same remaining polynomial factor from both parts.
2. Why must the expression have four terms?
Classical grouping works best on four visible terms. This calculator first combines like terms, so the final normalized expression must still contain exactly four nonzero terms.
3. What happens if no shared grouped factor appears?
The tool reports that grouping failed for the tested pattern. Some polynomials need a different ordering, another factoring method, or are not factorable over integers.
4. Why is overall common factor extraction useful?
Removing a common monomial first simplifies the remaining expression. That often reveals matching grouped binomials that are harder to see in the original polynomial.
5. Does the calculator test more than one grouping pattern?
Yes. The automatic mode checks adjacent, cross, and outer-inner pairings after normalization. That helps when the natural order does not show the common grouped factor immediately.
6. Can I enter spaces or implied coefficients?
Yes. Inputs like x^3-5x^2+2x-10 work well. Missing coefficients are treated as 1, and spaces are ignored during parsing.
7. How does the verification table help?
It compares numerical values from the original polynomial and the factored form at sample x-values. Matching outputs confirm the computed factors reproduce the same expression.
8. Can this replace full symbolic algebra software?
No. It focuses on integer-coefficient, one-variable, four-term expressions suited to grouping. More complex symbolic tasks may require complete computer algebra systems.