How do you factor by grouping?

Factoring by grouping is a technique used to factor polynomials, typically with four terms, by strategically pairing terms and extracting common factors.

Here's a breakdown of the process:

Steps for Factoring by Grouping

1. Grouping: Begin by grouping the first two terms together and the last two terms together. This essentially creates two separate binomials.

   • Example: For the expression ax + ay + bx + by, group as (ax + ay) + (bx + by).

2. Factoring out the GCF: Next, factor out the greatest common factor (GCF) from each of the binomials you created in the previous step.

   • Example: From (ax + ay) + (bx + by), factor out 'a' from the first group and 'b' from the second group: a(x + y) + b(x + y).

3. Factoring out the Common Binomial: Observe if there's a common binomial factor in the expression. If so, factor it out. This is the key step in factoring by grouping.

   • Example: In the expression a(x + y) + b(x + y), the common binomial is (x + y). Factor it out to get (x + y)(a + b).

Example Problem Walkthrough

Let's factor the polynomial: x³ + 3x² + 4x + 12

1. Grouping: (x³ + 3x²) + (4x + 12)
2. Factoring out the GCF:
   • From (x³ + 3x²) factor out x²: x²(x + 3)
   • From (4x + 12) factor out 4: 4(x + 3)
   • The expression now becomes: x²(x + 3) + 4(x + 3)
3. Factoring out the Common Binomial: The common binomial is (x + 3). Factor it out: (x + 3)(x² + 4)

Therefore, the factored form of x³ + 3x² + 4x + 12 is (x + 3)(x² + 4).

When Does Factoring by Grouping Work?

Factoring by grouping typically works best when:

• The polynomial has four terms (although it can sometimes be adapted for polynomials with more terms).
• There is a common binomial factor that emerges after factoring out the GCF from the initial groupings.