Factoring by Grouping

Factoring by grouping is a method used to factor polynomials that have four terms. The method involves grouping pairs of terms and factoring out the greatest common factor from each pair.

Group the terms: Arrange the terms of the polynomial into two pairs.
Factor out the GCF: Factor out the greatest common factor from each pair of terms.
Factor out a common binomial factor: Factor out the common binomial factor from the grouped terms.
Check for further factorization: Look for opportunities to factor the resulting quadratic expression further, if possible.

Let's factor the polynomial 2x + 6 + 3x + 9 by grouping:

Step 1: Group the terms

2x + 6 + 3x + 9

Step 2: Factor out the GCF from each pair

2x + 6 = 2(x + 3) 3x + 9 = 3(x + 3)

Step 3: Factor out the common binomial factor

2(x + 3) + 3(x + 3) = (x + 3)(2 + 3) = (x + 3)(5)

Step 4: Further factorization (if possible)

The resulting expression (x + 3)(5) is already fully factored.

Here are some key points to remember when factoring by grouping:

Make sure the polynomial has four terms before using the factoring by grouping method.
Look for common factors within each pair of terms and factor them out first.
Pay attention to the signs when factoring out the greatest common factor from each pair.
Always check to see if the resulting expression can be further factored after grouping.

Practice factoring by grouping with various examples to master this technique.