Factoring by grouping is a method for rewriting a polynomial with four or more terms as a product of simpler factors. It is especially useful when there is no single greatest common factor for every term, but pairs of terms have common factors. The main goal is to create a shared binomial factor that can be pulled out.
This skill matters because factoring helps solve equations, simplify expressions, and recognize algebraic structure.
The method starts by splitting the polynomial into groups, usually pairs, and factoring the greatest common factor from each group. If the remaining binomials match, that binomial becomes a common factor of the whole expression. Then the outside factors combine into the second factor.
Sometimes terms must be rearranged or a negative greatest common factor must be factored out to make the binomials match.
Key Facts
- Factoring by grouping often starts with four terms: ax + ay + bx + by.
- Group terms in pairs: (ax + ay) + (bx + by).
- Factor each group: a(x + y) + b(x + y).
- Factor the shared binomial: a(x + y) + b(x + y) = (x + y)(a + b).
- If the binomials differ only by signs, factor out a negative GCF to make them match.
- Always check by multiplying the factors back together using distribution.
Vocabulary
- Polynomial
- A polynomial is an algebraic expression made of terms added or subtracted, where variables have whole-number exponents.
- Term
- A term is one part of an expression separated by plus or minus signs.
- Greatest common factor
- The greatest common factor is the largest factor shared by two or more terms.
- Binomial
- A binomial is a polynomial with exactly two terms.
- Common binomial factor
- A common binomial factor is the same two-term expression that appears in each grouped part of a polynomial.
Common Mistakes to Avoid
- Forgetting to factor each group completely, which can hide the shared binomial and stop the method from working.
- Using groups that do not create matching binomials, which means the expression may need to be rearranged before factoring.
- Not factoring out a negative when needed, which leaves binomials such as (x + 3) and (-x - 3) looking different even though they can be matched.
- Stopping at a partially factored expression, which is wrong because a result like 2x(x + 5) + 3(x + 5) should be finished as (x + 5)(2x + 3).
Practice Questions
- 1 Factor by grouping: 3x + 6 + 5xy + 10y.
- 2 Factor by grouping: 2x^3 - 8x^2 + 5x - 20.
- 3 Explain why factoring out a negative greatest common factor can be necessary when using grouping.