Factor by Grouping Calculator
Enter a polynomial to factor it by grouping into a product of factors.
Use ^ for powers. Grouping works best on four-term polynomials.
Factoring by grouping works on polynomials, often with four terms, by splitting them into groups, factoring the greatest common factor from each group, and then factoring out the shared binomial. For example, x³ + 2x² + 3x + 6 groups into x²(x + 2) + 3(x + 2) = (x + 2)(x² + 3).
How to Use This Calculator
Enter a polynomial, most often one with four terms, using ^ for powers. The calculator returns the factored form, which for a groupable polynomial is a product of a binomial and another factor.
Factoring by grouping is the go-to technique when a polynomial has four terms and no single common factor across all of them, but the terms pair up in a way that reveals a shared binomial.
How the Calculation Works
Grouping splits the polynomial into two pairs. From each pair you factor out its greatest common factor. If the technique applies, both pairs are left multiplying the same binomial, which you then factor out once.
The result is a product of that common binomial and the factors pulled from each group. If the two groups do not share a binomial, the polynomial does not factor by grouping in that arrangement, and the terms may need reordering.
ax + ay + bx + by = a(x + y) + b(x + y) = (x + y)(a + b)
Worked Example
Factor x³ + 2x² + 3x + 6. Group as (x³ + 2x²) + (3x + 6). From the first group factor x², giving x²(x + 2); from the second factor 3, giving 3(x + 2).
Both share (x + 2), so factor it out: (x + 2)(x² + 3). Expanding confirms it returns the original polynomial.
Assumptions and Limitations
- Works best on four-term polynomials that group into a common binomial
- Some polynomials need their terms reordered before grouping works
- Not all polynomials are factorable by grouping
- Check the result by expanding it back
When to Use a Factor by Grouping Calculator
Algebra students use this to factor a four-term polynomial by grouping, with the factors shown. It is the go-to method when a polynomial has no common factor across all terms but pairs of terms do. Enter the expression to see it grouped and factored step by step.
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Open calculator →Frequently Asked Questions
What is factoring by grouping?
It is a technique for factoring polynomials, usually with four terms, by splitting them into groups, factoring each group's common factor, then factoring out the shared binomial that results.
When should I use grouping?
Use it when a polynomial has four terms with no factor common to all of them, but the terms pair up so each pair shares a factor and the pairs reveal a common binomial.
What if the groups don't share a binomial?
Try reordering the terms before grouping. If no arrangement produces a common binomial, the polynomial cannot be factored by grouping and may need another method or be irreducible.
How do I check my factoring?
Multiply the factors back together. If expanding them returns the original polynomial, the factoring is correct. This tool's result can be verified with the polynomial calculator's expand mode.
Can grouping factor three-term polynomials?
Grouping is designed for four terms, but a three-term quadratic can be factored by splitting the middle term into two, which turns it into a four-term grouping problem.
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