Factor by Grouping Calculator
Factor 4-term polynomials and trinomials by grouping with complete step-by-step solutions. Handles the AC method for trinomials automatically.
Example Polynomials
Enter Coefficients
ax³ + bx² + cx + d
Step-by-Step
Factored Form
When to Use Grouping
- - 4-term polynomials (most common use case)
- - Trinomials when a ≠ 1 (via AC method)
- - After pulling out GCF from all terms
- - When the polynomial has a common binomial factor
Factor by Grouping - Common Examples
| Polynomial | Grouping | Factored Form |
|---|---|---|
| x³ + 2x² + 3x + 6 | (x³+2x²) + (3x+6) | (x² + 3)(x + 2) |
| x³ - x² - x + 1 | (x³-x²) + (-x+1) | (x² - 1)(x - 1) |
| 2x³ + 6x² + x + 3 | (2x³+6x²) + (x+3) | (2x² + 1)(x + 3) |
| x³ + 3x² - 4x - 12 | (x³+3x²) + (-4x-12) | (x² - 4)(x + 3) |
| 3x³ - 9x² + 2x - 6 | (3x³-9x²) + (2x-6) | (3x² + 2)(x - 3) |
Factoring by Grouping in Depth
Factoring by grouping is a technique that extends the simple GCF (greatest common factor) method to polynomials with four or more terms. The core idea is to find subgroups within a polynomial where each subgroup shares a GCF, and after factoring those GCFs, the remaining binomial factors match - allowing a final factoring step.
The method is most commonly applied to 4-term polynomials of the form ax³ + bx² + cx + d. The standard approach groups the first two and last two terms: (ax³ + bx²) + (cx + d). After factoring GCFs from each group, you ideally get something like x²(ax + b) + k(ax + b), which factors to (x² + k)(ax + b). The key requirement is that the binomials in parentheses must match after GCF extraction.
When to Try Different Groupings
The standard 1-2 and 3-4 grouping does not always work. Some polynomials need to be grouped as 1-3 and 2-4, or 1-4 and 2-3. If the initial grouping fails to produce matching binomials, systematically try these alternatives before concluding the polynomial is irreducible. In some cases, terms need to be rearranged - for example, x³ + 3x + x² + 3 can be rewritten as x³ + x² + 3x + 3 before grouping as (x³ + x²) + (3x + 3) = x²(x + 1) + 3(x + 1) = (x² + 3)(x + 1).
The AC Method Connection
For trinomials ax² + bx + c with a greater than 1, factoring by grouping is actually the mechanism that makes the AC method work. After finding the two numbers m and n (where m × n = a × c and m + n = b), you rewrite the trinomial as ax² + mx + nx + c - a 4-term polynomial. Grouping (ax² + mx) + (nx + c) then yields the factored form. This connection means understanding grouping deeply gives you mastery of trinomial factoring as well.
Always Factor Out the GCF First
Before attempting grouping, always check for and remove any GCF common to all terms. Skipping this step can make grouping appear to fail when it would succeed on the simplified polynomial. For example, 4x³ + 8x² + 6x + 12 has a GCF of 2: factoring gives 2(2x³ + 4x² + 3x + 6). The 4-term polynomial inside then groups as 2[(2x³ + 4x²) + (3x + 6)] = 2[2x²(x + 2) + 3(x + 2)] = 2(2x² + 3)(x + 2).