Factoring by grouping is a technique used to simplify algebraic expressions by breaking them down into smaller, more manageable parts. When faced with a complex expression, factoring by grouping can make it easier to solve or manipulate the expression. In this article, we will explore the step-by-step process of factoring by grouping.

Step 1: Identify the Common Factors

The first step in factoring by grouping is to identify any common factors within the expression. Look for terms that share a common factor or have a pattern that can be simplified. An important tip is to keep an eye out for pairs of terms that have common factors, as this will make it easier to apply the next steps.

Step 2: Group the Terms

Once you’ve identified common factors, you can now group the terms with these shared elements together. Depending on the complexity of the expression, there may be more than one way to group the terms. It’s essential to choose groupings that simplify the overall expression.

Step 3: Apply the Distributive Property

Now that your terms are grouped appropriately, it’s time to apply the distributive property within each group to simplify further and rewrite your expression. The distributive property states that:

a(b + c) = ab + ac

By applying this property, you can factor out a common element in each group and rewrite your expression accordingly.

Step 4: Factor Out a Common Factor from Each Group

With each group rewritten using the distributive property, you can now compare them and look for another common factor among all groups. If such a factor exists, factor it out from each group and move it in front of remaining parentheses.

Step 5: Check Your Work

Before considering your factoring by grouping complete, double-check your work. Ensure that your simplified expression would expand back into the original expression if distributed. If there seems to be a mistake, return to the previous steps and try factoring again.

Example:

Let’s see how to factor by grouping works in practice with a real example. Consider the following algebraic expression:

4x^2 + 2x – 12x – 6

1. Identify common factors: In this case, we can observe that “2x” is common in the first two terms and “-6” is common in the last two terms.

2. Group the terms: (4x^2 + 2x) + (-12x – 6)

3. Apply the distributive property: 2x(2x + 1) – 6(2x + 1)

4. Factor out a common factor: (2x – 6)(2x + 1)

The expression has now been simplified using factoring by grouping. You can verify that expanding your simplified expression will result in the original expression.

In conclusion, factoring by grouping is a powerful technique for simplifying complex algebraic expressions. By following these steps and practicing, you’ll become adept at using this method in your mathematical endeavors.