Factoring polynomials is a hotly-debated topic in algebra. Students and parents (and sometimes teachers) often see little to no point in the activity. To the extent that there is an ongoing conversation among educators as to whether we should even teach the topic or not. You’ll see several arguments below that it is more useful than some claim. However I will admit to straddling the line between the two sides myself as to how much time and emphasis we place upon it in the algebra curriculum.

In reality, factoring is just one tool in the algebra toolbox. Some argue that it is not useful once we leave the context of math textbooks because “real polynomials” are seldom factorable. However, I argue that “real polynomials” that arise in context are more likely to be factorable than theoretical polynomials generated by random-coefficient-generators. Which leads us back to algebraic factoring being sometimes-useful. Thus, read on to get an overview of the whys, hows, and wherefores of factoring in algebra.

This article was originally published on mathteacherbarbie.com. If you are reading it elsewhere, you are reading a stolen copy.

## Why Factor?

Different forms for the same expression are more or less useful in different contexts. We want to be able to write any expression in as many different ways as we can, *while keeping it the same meaning at its core*, so that we have maximum flexibility with it.

Most of algebra in the first years boils down to learning to write *equivalent expressions* (as well as learning to *solve equations*). This idea of *equivalence* is key. Most of the early days of algebra is about finding equivalent forms, often with seemingly no purpose. Soon, however, we learn that different forms make it easier or harder to answer different questions about the function. It’s a little bit like bringing your professional self to work and your competitive self to the soccer game. It’s all still you, you’re just showing, and using, different views of yourself for different purposes.

## Equivalence in Math

Unfortunately, throughout school, we accidentally train students to think that an equal sign ($=$) means something like “the answer comes next” or “the next step is next.” But, in reality, the equal sign is meant to be used * between any two expressions that are equivalent to each other*. It is a means of expressing the idea that the two sides of the equal sign are

*the same in some key way*, even if they look very different.

In algebra, *two expressions are equivalent if substituting any value for the variable gives the same output for both expressions.* If we substitute in a 3 to one expression and a 3 to the other expression and get out the same value after doing the computations,

*then we substitute other numbers and this keeps happening no matter which number we choose, then the expressions are*

**and***likely equivalent*.

I use the word “likely” there intentionally. Substituting in one value for the variable is certainly not enough to prove equivalence. With just one value for the variable, it could be coincidence that the expressions come out the same. However, the more substitutions it actually does work for, the *more evidence* we are gathering that they are equivalent.

We will never be able to fully, completely prove equivalence by plugging in (we’ll need some algebra moves to do that), but we can gather pretty convincing evidence that two expressions are *likely* equivalent if we substitute enough different values for the variable and keep coming up with the same outputs.

[need illustration summarizing previous paragraph here]

Though it’s beyond the scope of the rest of this article, sometimes an algebra problem gives us an equation to start. In this case, we’re being *told* that the expressions on the two sides of the equal sign are *equivalent in this case or problem*, even if they may not always be equivalent. Often the instructions in this case will involve

*solving the*

*equation*, or finding the specific value (or sometimes values) for the variable that indeed makes these

*equivalent*. When learning this, you may see the phrase

**expressions**only if the variable has that value*equivalent*, meaning a

**equations***series of equations that have the same solution or solutions*. Again, this is beyond the scope of this particular article, but I wanted to give the heads up.

In summary, we learn to factor because *sometimes the equivalent factored form is easier to work with or allows us to make our overall problem easier to work with in certain situations.* You can read a little more about why we factor in my short article Why Do We Factor Expressions?

## What Is Factoring?

Factoring is *unmultiplying*. Most algebra classes will teach you to multiply polynomials before you get to factoring (just as, in elementary school, you learned to multiply numbers before learning to factor them). Factoring is the process of *undoing* that multiplication, finding the expressions that may have been multiplied together to create the expression you’re trying to factor. Thus, many of the *factoring strategies* will look like a reversal of the steps of multiplication. In fact, the more you showed your work multiplying polynomials, the more likely you will recognize the steps of factoring as moving through the multiplication process backward.

## Essential Factoring Skills

There are two core skills to be mastered in factoring in algebra (and one is just an extension of the other). All other factoring strategies are just efficiencies that can be used only in special cases. If you master only these two strategies, they will carry you through successfully almost all of the time.

### Factoring out the Greatest Common Factor (GCF)

Factoring out the GCF is essentially * undistributing*. Elementary curricula now use the

*distributive property*a lot in learning to perform arithmetic. It’s often key to mental math and is at the core of the area model. When I talk about

*distributing*here, it’s the same phenomenon, just using variables and distributing

*across algebraic terms*instead of across place values.

So, let’s do a quick review of what I mean by “algebraic terms” and “distributing.” Hopefully I will have future articles describing each of these in detail, but in the meantime, if this quick review isn’t enough, a quick web search should get you the information you need.

#### Review of “terms” and “distributive property”

A * term* in algebra is an expression that may contain a number, a variable, and the operations of multiplication, division, and/or exponentiation. In general,

*algebraic terms*do not involve addition or subtraction (though the definition loosens up a bit occasionally to allow for describing more complex situations). Instead, addition or subtraction is typically the operation that separates one term from the next.

** Distributing** in this sense is technically “distributing multiplication across addition,” but since this is the most common type of distributing, we usually don’t say the whole phrase.

I like to think of the operations as having personalities: addition and subtraction are shy. They kind of hang back until it’s “their turn.” So either someone (like parentheses) can come and protect/surround the addition and subtraction, making the more forceful multiplication wait its turn — eg, $3(2+7)$ — or the multiplication can gently approach each term and multiply each separately before leaving the field — eg, $3\dot 2 + 3\dot 7$.

#### Back to factoring out a common factor

There are three key prerequisite ideas that can determine how difficult or easy this is for a given student:

- Factoring numbers
- Understanding exponents as repeated multiplication
- Remembering and paying attention to the “rules” of multiplying positive and negative numbers

To *factor out the GCF*, first look at the terms separately and think about the *factors* of each term. You may be able to hold these in your head, or you may wish to write them down as you think about them.* Either way is okay!*

Once you identify the terms and the factors of each of those terms, look for factors that are found in *every* term. Those are the factors that you will * undistribute*, sometimes casually referred to as “pulling out” of the expression. The factors that are

*not*in common will remain “behind”, now inside parentheses.

### Factoring by Grouping

*Factoring by Grouping* is an extension of factoring out a common factor, where the common factor may be an algebraic expression. The strategy is not limited to four-term polynomials, but that is generally the context where it’s taught. In essence, we “group” a polynomial with four terms into two groups of two terms each and then factor out the GCF twice. Sometimes we use this with a three-term trinomial (or even a two-term binomial) by expanding the middle term into the sum of two strategically-chosen terms.

## What Makes Factoring Difficult?

### Lack of fluency with basic multiplication facts

One of the best things you can do for your future math self is to get comfortable with the multiplication tables. Find *your* strategies for knowing multiplication facts, and practice over and over. Whether you draw the facts from memory or use strategies to multiply in the moment, make it as automatic and routine as you can. There are a lot of resources to help with this. I’ve noted a few in my post How to Learn Multiplication Facts: A Roundup.

### Operations with negative numbers

One of our examples in the GCF section of this post involved factoring out a negative number as the greatest common factor. As part of this process, it seemed as though a couple of signs changed. If you’re not comfortable with multiplying and dividing negative numbers, this change feels arbitrary and random. However, in this case, it arose from the idea that a negative $\div$ a negative $=$ a positive. The more automatically you know these “rules” of negative number operations, the less arbitrary the sign changes will feel and the better you’ll predict them yourself.

### Weak understanding of exponent properties

If you don’t know that $x^5=x^3\cdot x^2$, then you’ll struggle to see why $3x^5+6x^3=3x^3(x^2+2)$. Exponent properties arise out of interpreting exponents as repeated multiplication: $x^5=x\cdot x\cdot x\cdot x\cdot x=(x\cdot x\cdot x)\cdot (x\cdot x)=x^3\cdot x^2$.

### Recognizing the differences between addition and multiplication

I’ve seen a lot of students so anxious to “finish” the problem and move on that they forget there’s a real difference between addition and multiplication, or sometimes simply fail to pay attention to which operation is involved. Most often this results in leaving out an addition (or subtraction) sign where it should be or inserting one where it wasn’t before.

Starting in algebra, we often leave off multiplication signs while still meaning multiplication. In this, multiplication takes its place as a sort of “default” operation: the operation that’s assumed if none other is indicated. “Dropping,” forgetting, or leaving out a $+$ or $-$ sign whimsically (and almost always incorrectly) removes that operation which then becomes multiplication by default. But multiplication is not the same. It doesn’t have the same effect on numbers, it takes precedence in the order of operations, and more. In general, students should pay attention to the operations meant, slow down if they have a tendency toward this action, and carefully articulate exactly what they mean. (In some cases, this may mean re-introducing multiplication signs if you need whenever multiplication is the default! There’s nothing wrong with that.)

### Belief that all, or even most, polynomials can be factored

To be fair, this is an understandable conclusion given how much time we spend teaching factoring and how many of our textbook problems can be factored. Statistically, the number of *possible* polynomials that can be factored is essentially zero percent. In reality, the factorable polynomials “in the world” are somewhere in between.

Not all polynomials are equally likely to occur. In particular, multiplication is a common relationship between ideas and variables. Relationships like $d=rt$ (distance, rate, and time) and $R=pn$ (revenue, price, and number sold) are particularly common. (We call these “proportional relationships.” There are oodles of these.) Any time a polynomial arises out of these relationships, it will necessarily be factorable because it was created by a multiplicative relationship. So the “real world” will disproportionately generate factorable polynomials because of the vast array of proportional relationships.

However, there are also oodles of non-proportional relationships (such as $P=R-C$ (profit, revenue, and cost), which may well generate polynomials that are not factorable. Thus, we still significantly overrepresent the number of factorable polynomials in our math curricula.

### Understanding “equivalence”

This is so important and so misunderstood that I wrote a whole section of this post about it.

### Attitude and Frustration

It’s so hard to keep going when we’re frustrated! However, if it’s always easy, then you’re not actually learning very much. To really learn, we need to work at the edge of what we know, not solidly within our own ease.

Perseverance is often the key to getting a new skill or understanding a new idea. If you’re looking for ideas on how to keep moving forward even when you’re stuck, check out the perseverance section of my post my post on this important Math Practice.

### The feeling of “working backwards”

For some people, this will help them visualize the steps (reverse from multiplying) as well as the context (unmultiplying, or unentangling the factors from each other). For others, this sense of working backwards can be extremely disorienting. For many of us, which of these we fall into will depend on how well we really, truly know the multiplication algorithms for polynomials. If we’re still relying on old acronyms (such as FOIL) to multiply polynomials, this idea of working backwards is very unlikely to help.

Avoiding the acronyms, I like to teach polynomial multiplication as double- (or triple- or quadruple-, etc.) distributing. Then factoring becomes double-*un*distributing (or triple- or quadruple-, etc). You can readily see this happen in the factoring by grouping examples.

## What More Should I Learn About Factoring?

### Special cases and patterns for quick-factoring

Certain factorable polynomials follow patterns we can learn in order to factor them more quickly. These include quadratic trinomials with leading coefficient $1$, difference of squares binomials, perfect square trinomials, and certain cubic (third-power) polynomials. (These are the patterns most often taught. There are more, but they are less efficient to spot and use.)

### Using algebra tiles

Remember how I called factoring *un*multiplying? This also means it’s really closely related to multiplication and division. These two operations have to do with making a number of *equal-sized* groups out of a whole. This basic idea of “how many equal-sized groups and what is the size?” questioning is key to using algebra tiles in factoring.

In short, to factor using algebra tiles, your goal is to arrange the tiles that represent your starting polynomial into a perfect rectangle, then “read” the lengths of each side of the rectangle. The two sides are your two factors. You can think of one side as the “size” of the equal groups and the other side as the “number” of equal groups.

Since not all polynomials factor, not all polynomials will arrange into rectangles using algebra tiles. A few, however, may work with a few tweaks. In mid-level problems, you’ll need to introduce zero-pairs of tiles or “split” tiles in half to make the rectangle happen (this latter is a good argument for virtual tiles). And, of course, this strategy also runs into all the standard limitations of algebra tiles.

### Using area models

As *un*multiplying, factoring is almost a variation of division. In fact, we can think of it as division where we are * required* to state our answer all together as the related multiplication problem at the end. We can use this relationship to factor using area model division routines.

### Contexts where we use factoring

If we remember the “why” of factoring, we can start to explore the reasons we learn and teach factoring in algebra. Namely, the purpose of factoring in algebra is to change the *form* of the expression without changing the *content* of the expression in order to observe new characteristics and get to know it better.

## Conclusion

This was a lot of words about a topic that might or might not be overemphasized in modern algebra curriculum. Hopefully it gives you some insights into the overall ideas of factoring, what it is, how to approach it, and why we do still teach it. Feel free to dig deeper into specific ideas by following the links. And remember…

**You’ve Got This!**