Factoring by Grouping: A Step-by-Step Guide

Factoring by grouping is a useful technique for factoring quadratic equations, especially when the leading coefficient is greater than 1. It involves grouping terms together and then factoring out common factors. This method can also be applied to factoring polynomials with four terms.

Understanding the Concept

The key to factoring by grouping is to recognize that a four-term polynomial can be rewritten as a product of two binomials. This is done by grouping the terms into pairs and then factoring out the greatest common factor (GCF) from each pair. Here's a breakdown:

1. Group the terms: Group the first two terms together and the last two terms together.
2. Factor out the GCF: Factor out the GCF from each group.
3. Identify the common binomial: Notice that both groups now have a common binomial factor.
4. Factor out the common binomial: Factor out the common binomial, leaving the remaining factors as the other binomial.

Step-by-Step Example

Let's factor the quadratic expression 2x² + 5x + 3 using the grouping method:

1. Group the terms: (2x² + 5x) + (3)
2. Factor out the GCF: x(2x + 5) + 1(2x + 5)
3. Identify the common binomial: (2x + 5) is the common binomial.
4. Factor out the common binomial: (2x + 5)(x + 1)

Therefore, the factored form of 2x² + 5x + 3 is (2x + 5)(x + 1).

Factoring Polynomials with Four Terms

The same principle applies to factoring polynomials with four terms. For example, let's factor the polynomial 3x³ + 9x² + 2x + 6:

1. Group the terms: (3x³ + 9x²) + (2x + 6)
2. Factor out the GCF: 3x²(x + 3) + 2(x + 3)
3. Identify the common binomial: (x + 3) is the common binomial.
4. Factor out the common binomial: (x + 3)(3x² + 2)

Thus, the factored form of 3x³ + 9x² + 2x + 6 is (x + 3)(3x² + 2).

Practice Problems

Here are some practice problems to solidify your understanding of factoring by grouping:

1. 4x² + 10x - 6
2. 6x³ + 15x² - 2x - 5
3. 2x² - 7x + 3

Remember, factoring by grouping is a powerful technique that can help you simplify expressions and solve equations. With practice, you'll become proficient in using this method.