6.1 Greatest Common Factor and Factor by Grouping

Greatest Common Factor and Factor by Grouping

Factoring is the reverse of multiplying. Instead of expanding expressions, you're breaking a polynomial back down into the product of simpler expressions. This skill is essential for solving equations, simplifying rational expressions, and working through most of what comes next in algebra.

Two foundational techniques get you started: finding the greatest common factor (GCF) and factoring by grouping. The GCF method handles polynomials where every term shares a common factor. Grouping tackles longer expressions (usually four terms) where no single factor divides everything. You'll often use both together.

Factoring Basics

A few definitions to keep straight before diving in:

• Factoring means rewriting a polynomial as a product of two or more simpler expressions
• Like terms share the same variables raised to the same powers (e.g., $3x^2$ and $7x^2$)
• A binomial has two terms; a trinomial has three terms
• Polynomials with four or more terms don't have a special name, but they show up often in grouping problems

Greatest Common Factor in Polynomials

The greatest common factor (GCF) of a polynomial is the largest expression that divides every term evenly. Finding and pulling out the GCF is always the first step in any factoring problem.

How to find the GCF:

1. Look at the coefficients (the numbers in front). Find their GCF using prime factorization or by listing factors. For example, the GCF of 12 and 18 is 6.
2. Look at the variables. For each variable that appears in every term, take the lowest power. If your terms have $x^3$ and $x^2$, the variable part of the GCF is $x^2$.
3. Multiply the results from steps 1 and 2 together. That's your GCF.

How to factor it out:

1. Divide each term of the polynomial by the GCF.
2. Write the result as the GCF times the remaining terms in parentheses.
3. Check your work by distributing the GCF back through. You should get the original polynomial.

Example: Factor $12x^3 + 18x^2$

• Coefficients: GCF of 12 and 18 is 6
• Variables: both terms have $x$, and the lowest power is $x^2$
• GCF = $6x^2$
• Divide each term: $12x^3 \div 6x^2 = 2x$ and $18x^2 \div 6x^2 = 3$
• Factored form: $6x^2(2x + 3)$

Factor by Grouping Technique

When a polynomial has four terms and no single GCF across all of them, factoring by grouping is your go-to method. The idea is to split the polynomial into two pairs, factor each pair separately, and then pull out the common binomial.

Steps for factoring by grouping:

1. Group the four terms into two pairs (usually the first two and the last two).
2. Factor the GCF out of each pair separately.
3. Check whether the expressions left in parentheses match. If they do, that matching binomial is a common factor.
4. Factor out the common binomial. The final answer is the product of two binomials.

If the parentheses don't match after step 2, try rearranging the terms or adjusting signs before grouping again.

Example: Factor $6x^2 + 3x - 4x - 2$

• Group: $(6x^2 + 3x) + (-4x - 2)$
• Factor each group: $3x(2x + 1) - 2(2x + 1)$
• The binomial $(2x + 1)$ appears in both groups
• Factor it out: $(3x - 2)(2x + 1)$

Notice the second group needed a $-2$ factored out (not $+2$) so that the binomial $(2x + 1)$ would match. Watch your signs carefully here; sign errors are the most common mistake in grouping problems.

GCF vs. Factor by Grouping: When to Use Which

• Use GCF when every term in the polynomial shares a common factor. This works on binomials, trinomials, or any size polynomial.
  • Example: $15x^2 - 25x = 5x(3x - 5)$
• Use grouping when the polynomial has four terms and there's no GCF across all of them.
  • Example: $2x^3 - 3x^2 + 4x - 6$
    • Group: $(2x^3 - 3x^2) + (4x - 6)$
    • Factor each group: $x^2(2x - 3) + 2(2x - 3)$
    • Factor out $(2x - 3)$: $(x^2 + 2)(2x - 3)$
• Use both together when a GCF exists across all terms and the leftover polynomial still has four terms. Always pull out the GCF first, then group what remains.

Recognizing which method fits the problem saves you time. Get in the habit of checking for a GCF before trying anything else.