Factoring by grouping is a method used to simplify algebraic expressions, particularly polynomials, by grouping terms in such a way that common factors can be factored out. This method is particularly useful when working with four-term polynomials, but it can be adapted to other situations as well.

Steps for Factoring by Grouping:

1. Group Terms: Divide the polynomial into two (or more) groups of terms that might share common factors.
2. Factor Out Common Factors: For each group, factor out the greatest common factor (GCF).
3. Identify a Common Binomial Factor: After factoring out the GCF from each group, check if the resulting terms have a common binomial factor.
4. Combine into Factored Form: If there’s a common binomial factor, factor it out, and write the expression as a product of two factors.

Example:

Factor $3x^3 + 6x^2 + 2x + 4$ by grouping:

1. Group terms: $(3x^3 + 6x^2) + (2x + 4)$

2. Factor out the GCF from each group: $3x^2(x + 2) + 2(x + 2)$

3. Identify the common binomial factor: Both groups contain $(x + 2)$.

4. Combine into factored form: $(3x^2 + 2)(x + 2)$

This is the fully factored expression.

Would you like additional examples or explanations?

Related Questions:

1. What types of polynomials are best suited for factoring by grouping?
2. How does factoring by grouping relate to the distributive property?
3. Can factoring by grouping work for polynomials with more than four terms?
4. What happens if the groups don't have a common factor after grouping?
5. How can factoring by grouping be applied in solving quadratic equations?

Tip:

When grouping terms, experiment with different groupings if the first attempt doesn't yield a common factor. Reordering terms might help!