What if the groups do not share a common binomial factor?

Rearrange the terms of the original polynomial. If no rearrangement yields a common binomial factor, the polynomial cannot be factored by grouping over the rational numbers.

Can grouping be used for polynomials with more than four terms?

Yes. A polynomial with six terms can be grouped into two groups of three, or three groups of two, provided a common factor emerges from each group.

Do I always group the first two and last two terms?

Usually, this works if the polynomial is in standard form. However, the commutative property allows rearranging terms if the standard consecutive grouping fails.

How to factor a polynomial by grouping

Factoring by grouping is a technique used to factor polynomials by dividing terms into pairs and extracting a common binomial factor. It applies primarily to polynomials with four terms where a global greatest common factor (GCF) does not exist, but local GCFs can be factored out of paired terms.

The setup

Verify the polynomial has exactly four terms. Ensure there is no overall GCF across all four terms; if there is, factor it out first. Arrange the polynomial in standard descending order of degree.

The steps

Group the first two terms together and the last two terms together. 2. Factor out the GCF from the first group. 3. Factor out the GCF from the second group. If the third term is negative, factor out a negative GCF. 4. Identify the common binomial factor in the resulting expression. 5. Factor out the common binomial to write the final expression as a product of two polynomials.

Checking the result

Multiply the resulting factors using the distributive property (FOIL). The expanded result must match the original polynomial exactly. Check signs carefully.

Common errors

Failing to factor out a negative sign from the second group when the third term is negative, resulting in mismatched binomials. Stopping after extracting the local GCFs and leaving the expression as a sum rather than a product of factors.

Worked example

Factor the polynomial $x^3 + 2x^2 - 3x - 6$ by grouping.

Group the terms: $(x^3 + 2x^2) + (-3x - 6)$ . Factor the GCF $x^2$ from the first group: $x^2(x + 2)$ . Factor the GCF $-3$ from the second group: $-3(x + 2)$ . Combine into a single expression: $x^2(x + 2) - 3(x + 2)$ . Factor out the common binomial $(x + 2)$ : $(x + 2)(x^2 - 3)$ .

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References: OpenStax Intermediate Algebra, Chapter 6 · Khan Academy Unit: Polynomial factorization · Algebra and Trigonometry by Sullivan