What Is Factoring in Math? A Beginner’s Guide

Think of multiplication like baking a cake. You start with ingredients such as flour, eggs, and sugar, and combine them to make something new. Two (or more) numbers go in, and out comes a single, bigger result. Simple enough.

But what if you're given the cake and asked to figure out what went into it?

How much sugar? How many eggs? What were the original parts?

In math, that’s called factoring, and this guide will walk you through exactly what that means. 

Read on for clear definitions, step-by-step instructions, practice exercises, and answers to common questions, all designed to help you learn and master factoring with confidence.

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Let’s Review First: What Are Factors?

To understand what factoring means, it helps to start with its root word: factor. 

At Mathnasium, we like to say that a factor is one of the numbers that makes up a product through multiplication.

So, how do we find the factors of a number?

Let’s try it with 12.

What’s the smallest whole number that divides evenly into 12?

That’s right, 1. And what do you multiply it by to get 12?

1 × 12 = 12, so both 1 and 12 are factors.

What’s the next smallest number?

That’s 2. Does 2 go evenly into 12?

Yes! 2 × 6 = 12, so we can add 2 and 6 to our list.

How about 3?

3 × 4 = 12, so 3 and 4 are factors too.

What comes next?

After 3, we reach 4, but we’ve already seen that pair. Once our factor pairs start to repeat in reverse, we know we’ve found them all.

So, what are the complete factors of 12?

1, 2, 3, 4, 6, and 12

Pretty simple, right?

What Is Factoring in Math? 

Factoring is the process of breaking down a number or expression into its building blocks, its factors.

We can also think of it as the reverse of multiplication. 

When you multiply, you take smaller numbers and combine them to make something bigger.

When you factor, you start with the bigger result and figure out which smaller numbers were multiplied to get it.

Let’s try factoring a new number: 16.

What are the factor pairs of 16?

• 1 × 16

• 2 × 8

• 4 × 4

From those pairs, we can list all the factors of 16:

1, 2, 4, 8, and 16

That’s factoring in action.

See how Mathnasium’s proprietary teaching approach, the Mathnasium Method™, helps students learn and master any math topic, including factoring.

Factoring Expressions

So far, we’ve seen how factoring works with whole numbers. But as we mentioned earlier, factoring also applies to expressions, especially in algebra.

This becomes especially useful when expressions include both numbers and variables, and even more important later in your math journey, when you’ll be working with polynomials.

Let’s try factoring a simple expression:

2x + 4

Can we break this down into smaller parts, just like we did with numbers?

Look closely: both terms (2x and 4) have something in common.

Can you spot it?

They both have a factor of 2!

That means we can factor out the 2:

2x + 4 = 2(x + 2)

If you’ve seen the distributive property before, you might remember using it to multiply a number into parentheses.

For example:

2(x + 2) = 2x + 4

What we just did is the reverse! We started with 2x + 4 and factored it into 2(x + 2).

So we could say that factoring is also undoing the distributive property.

Solved Examples of Factoring Numbers & Expressions

Factoring is a skill that gets easier with practice. Let’s go over a few examples to build your confidence and sharpen your understanding.

Example 1: Factoring a Number

Question: Find all the factors of 28.

Let’s start with the smallest whole number and work our way up:

• 1 × 28 = 28 \(\rightarrow\) So, 1 and 28 are factors.

• 2 × 14 = 28 \(\rightarrow\) Add 2 and 14 to the list.

• 3? Doesn’t divide evenly into 28.

• 4 × 7 = 28 \(\rightarrow\) 4 and 7 are factors too.

• 5 and 6? Neither divides evenly into 28.

• 7? Already used it in a pair (4 × 7).

Once factors start reappearing in reverse, we know we’re done.

So, all factors of 28 are:

1, 2, 4, 7, 14, and 28

Example 2: Factoring a Simple Expression

Question: Factor the expression: 4x + 12

Let’s start by looking for a common factor in both terms:

• The first term is 4x.

• The second term is 12.

What’s the greatest number that divides evenly into both 4 and 12?

That’s right, 4.

Now we use the distributive property in reverse to factor out the 4:

4x + 12 = 4(x + 3)

We factored the expression by pulling out the greatest common factor (GCF).

Now it's written as a product: 4 (x + 3)

To check your work, you can expand it again using the distributive property:

4(x + 3) = 4x + 12

Example 3: Factoring with a Negative Term

Question: Factor the expression: 6x − 12

Let’s break it down:

• The first term is 6x

• The second term is −12

Step 1: Find the greatest common factor (GCF)

Both terms share a numerical factor of 6.

Do they both include a variable?

No, only the first term has x, so we don’t factor out a variable.

So the GCF here is just 6.

Step 2: Factor out the GCF

We factor out 6 from both terms:

6x − 12 = 6(x − 2)

Notice that the negative sign in −12 stays inside the parentheses as −2.

That’s important, factoring doesn’t change the sign of a term, it just pulls out what’s common.

Why does this work?

You can check it by expanding:

6(x − 2) = 6x − 12

So when factoring expressions with a negative term, just be sure to carry the sign through correctly inside the parentheses.

Example 4: Factoring an Expression with an Exponent

Question: Factor the expression: 3x² + 6x

Let’s look at both terms:

• The first term is 3x², which means 3 × x × x

• The second term is 6x, which means 6 × x

Step 1: Find the greatest common factor (GCF)

Numerically, both terms share a factor of 3.

Now let’s look at the variable part:

• x² means we have two x’s multiplied: x × x

• x means we have just one x

So what do they have in common?

They both include at least one x.

Since x is the highest power that appears in both terms, that’s what we can factor out.

So, the GCF for this expression is 3x.

Step 2: Factor out the GCF

Now we pull out 3x from each term:

3x² + 6x = 3x(x + 2)

We’ve now rewritten the expression as a product: 3x(x + 2).

Want to verify? Try expanding it again:

3x(x + 2) = 3x² + 6x

Do It Yourself: Factor These Numbers and Expressions

Now it’s your turn to practice! Try factoring the numbers and expressions below.

1. Find all the factors of 36

2. Factor the expression: 5x + 10

3. Factor the expression with a negative term: 8x − 16

4. Factor the expression with an exponent: 6x² + 9x

FAQs About Factoring in Math

Factoring comes up throughout a student’s math journey. It starts with basic number work in elementary school and continues into algebra and beyond. Because factoring shows up in so many places, students often have questions as they learn how it works.

Here are a few questions we frequently hear at Mathnasium of Blue Ash, along with answers that can help you understand factoring more clearly.

1. When do students learn about factoring?

Most students first encounter factoring in 4th or 5th grade when they begin working with factors and multiples. They learn how to break down numbers and find which ones divide evenly into others.

In middle school, students start to factor algebraic expressions, looking for what’s shared between terms and using the distributive property in reverse.

In algebra, factoring becomes a common method for simplifying expressions and solving equations. Students often factor expressions with exponents and, later, polynomials in Algebra 1 and Algebra 2.

2. How do I know if I factored an expression completely?

An expression is fully factored when all the terms inside have nothing else in common. If you factor something out and can’t find anything else that’s shared, you’re done. You can check your work by expanding it back out.

3. What if there’s nothing in common between the terms?

If there’s no shared factor other than 1, then the expression can’t be factored using basic methods. That means it’s already in its simplest form.

4. What’s the difference between a factor and a term?

A factor is one of the parts being multiplied in a product.

A term is one of the parts being added or subtracted in an expression.

For example, in 3x + 9, the terms are 3x and 9.

When we factor it as 3(x + 3), we’ve rewritten the expression as a product, 3 is a factor, and (x + 3) is another.

5. Is factoring the same as dividing?

Not exactly. When you divide, you find how many times one number fits into another. When you factor, you’re breaking something down into parts that can be multiplied to get back to the original.

At Mathnasium of Blue Ash, we encourage students to ask questions and think out loud, building deep understanding of math topics like factoring and beyond.     

Master Factoring at Mathnasium of Blue Ash

At Mathnasium of Blue Ash, we help K–12 students of all skill levels build a deep understanding of math, including foundational skills and topics like factoring.

Our specially trained tutors provide face-to-face instruction in a caring and fun group environment, both in-center and online. Whether your student is factoring numbers in upper elementary or working through algebraic expressions and equations in middle or high school, we’re here to support them every step of the way.

Each student begins their Mathnasium journey with a diagnostic assessment that helps us identify their current skills, strengths, and knowledge gaps. Using these insights, we create a personalized learning plan that puts them on the best path toward math mastery.

Whether your student is looking to catch up, keep up, or get ahead in their math class, schedule an assessment and enroll at Mathnasium of Blue Ash today!

Schedule your free assessment at Mathnasium of Blue Ash today!

Pssst! Check Your Answers Here

If you’ve given our practice exercises a go, check your answers here:

1. Factors of 36:

1, 2, 3, 4, 6, 9, 12, 18, 36

2. Factored form of 5x + 10:

5(x + 2)

3. Factored form of 8x − 16:

8(x − 2)

4. Factored form of 6x² + 9x:

3x(2x + 3)