Factoring Polynomials in Algebra – mathteacherbarbie.com

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, and then we substitute other numbers and this keeps happening no matter which number we choose, then the expressions are 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 expressions only if the variable has that value. When learning this, you may see the phrase equivalent equations, meaning a 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.

The first example is to factor $3x-6$. The terms here are $3x$ and $-6$. (Remember: we can write $3x+(-6)$ to mean exactly the same as the original!) $3x$ has factors $3$ and $x$. $-6$ has factors $3$ and $-2$ (or $-3$ and $2$, but the first will work better for us). The factor $3$ is in common for both of these terms, so we factor out the $3$, “leaving behind” the $x$ and the $-2$: $3x-6=3(x-2)$.

The second example is where we really need to think about negatives: factor $-3x-6$. This time, the terms are $-3x$ and $-6$. The factors of $-3x$ are $-3$ and $x$. So $-3$ and $2$ will be the more useful factor pair of $-6$ this time, since that will make $-3$ be the common factor between them. So we factor out the $-3$, “leaving behind” the $x$ and the $2$: $-3x-6=-3(x+2)$. Notice the $+$ sign before the $2$. If we had left a subtraction sign in there, then when we went to check the factoring, we would have multiplied a negative times a negative, making a positive, not the $-6$ of the original problem.

Let’s factor $2x^4+8x^2+4x$. The terms are $2x^4$, $8x^2$, and $4x$. $2x^4=2\cdot x \cdot x \cdot x \cdot x$, so its factors are 2 and $x$ with four copies of the $x$ factor. $8x^2$ and $4x$ are both divisible by 2 and by $x$. So $2x$ is the greatest common factor (GCF). So when we factor out the $2x$, we end up with $2x^4+8x^2+4x=2x(x^3+4x+2)$. Notice if we were going the other way and multiplied through, $2x\cdot x^3=2x^4$, $2x\cdot 4x=8x^2$, and $2x\cdot 2=4x$, the terms we started with.

Let’s pull the ideas together in one problem by factoring $-2x^4 y+8x^3 y^2-4x^2 y^3$. The terms are $-2x^4 y$, $8x^3 y^2$, and $-4x^2 y^3$. If we look at the coefficients $-2$, $8$, and $-4$, we see that we can factor out either a $-2$ or a $2$. Either is legitimate, though it’s more typical to go with the sign on the first term (here $-2x^4 y$), so we’ll factor out a $-2$. For the variables, each term has an $x$ variable with exponents 4, 3, and 2, respectfully. We’ll use the lowest $x$ exponent to factor out $x^2$. Each term has a $y$ variable with exponents 1, 2, and 3, respectfully. Again, we’ll use the lowest $y$ exponent to factor out $y$ ($y^1$ if we make the “invisible 1” visible). So our GCF is $-2x^2 y$ and our factored form is $-2x^2 y(x^2-4xy+2y^2)$. Notice the sign change inside the parentheses compared to the original polynomial. When we factor out a negative number, all of the terms inside the parentheses must change signs in order to maintain the original meaning when multiplying back through by the negative out front.

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.

Consider the expression $4(x+2)+x(x+2)$. We could distribute the 4 and distribute that second $x$, then combine like terms to get a polynomial. However, if the directions ask us to factor, in this case half the work is done for us.

See those two “$x+2$”s? They match each other. We can treat them like a GCF in the two “terms” $4(x+2)$ and $x(x+2)$. If we do that, we end up with the factored expression $(x+2)(4+x)$.

Multiply and combine like terms for each side to prove to yourself that $4(x+2)+x(x+2)=(x+2)(4+x)$.

The expression $x(x+2)-x-2$ isn’t quite as nice as the “mild” example above. However, it’s still partly done for us. Consider the last two terms, $-x-2$. If I factor out the common factor $-1$, then I see that $-x-2=-1(x+2)$. So I can rewrite $x(x+2)-x-2=x(x+2)-1(x+2)$. Now, it’s just as straightforward as the “mild” example, and I find that $x(x+2)-x-2=x(x+2)-1(x+2)=(x+2)(x-1)$ in its fully-factored form.

I can extend this idea to four-term polynomials that might be factorable. In particular, grouping the terms into pairs and factoring out common factors within each pair to hopefully find a more general common factor like the above examples.

Let’s look at $6x^3-9x^2-4x+6$. I can group this into pairs as $6x^3-9x^2$ and $-4x+6$, so $6x^3-9x^2-4x+6=(6x^3-9x^2)+(-4x+6)$. Then I use factoring-out-the-GCF on each pair to find that $6x^3-9x^2=3x^2(2x-3)$ and $-4x+6=-2(2x-3)$. So, altogether, $6x^3-9x^2-4x+6=(6x^3-9x^2)+(-4x+6)=3x^2(2x-3)-2(2x-3)=(x-3)(3x^2-2)$ in its finally, fully-factored form.

Almost all of math, whether classroom learning or cutting-edge research, is skills to turn new problems into problems we already know how to solve. This example is no different. If I want to factor $10x^2+29x+21$, one way to approach it is to try to “expand” it into a four-term polynomial that I can group. To this end, we’re going to “expand” the $29x$ into two terms: $1x+28x$ or $2x+27x$ or $3x+26x$ or…

Hopefully one of the pairs on that list will “work”. We could try them all and find out, or we can apply a strategy to help us figure out which pair is the best to try.

Remember that factoring is unmultiplying? We can take our clues from the process of double-distributing (which some teachers still call FOILing). The OI of FOIL gives us our clue. Quite often (not always, but quite often) those two terms end up being like terms that can combine to make the middle term. In order to use factoring by grouping, we’re going to look for the two terms that combined and split them back apart.

We do that by multiplying the leading coefficient by the constant (or final coefficient). In this case: $10\cdot 21=210$. We then search for factors of that product that add up to the middle coefficient. In this case $14\cdot 15=210$ and $14+15=29$, so we choose to rewrite $29x=14x+15x$ and that gives us our best shot at this strategy working out.

In the end, the final process looks like this: $10x^2+29x+21=10x^2+14x+15x+21=(10x^2+14x)+(15x+21)=2x(5x+7)+3(5x+7)=(5x+7)(2x+3)$. We can check this by multiplying out to make sure we end back up where we started.

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-undistributing (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 unmultiplying? 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 unmultiplying, 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.

The “Zero Product Principle” is typically the first application of factoring that students encounter. The Principle itself states that you can never get zero as the product of two numbers multiplied together unless at least one of the numbers is itself zero. Factoring comes into play when we try to solve an equation of the form “polynomial = 0”. If the polynomial is factorable, then we can factor it, set each factor equal to zero, and because of the Zero Product Principle, we would know that the solutions to each “factor=0” are also solutions to the equation as a whole.

In practice and in isolation, most textbook applications of this in and of itself are highly contrived. We know that a lot fewer polynomials are factorable in the world than in math textbooks. Since we have other tools (eg, completing the square, quadratic formula, graphing technology, etc), it’s also not as essential a skill as the attention paid to it might seem. However, this idea is key to several of the other uses of factoring below, some of which are somewhat less contrived and more likely to be ultimately useful.

If we’re trying to graph a function or expression by hand, there are a few key pieces of information needed: $x$- and $y$-intercepts, maximums and minimums (points where the graph “turns around”), how the graph is shaped near each of these points, how the graph is shaped near each $x$ where it’s undefined, and how the graph is shaped at its “ends” are key among them.

Factoring can sometimes be useful in finding the $x$-intercepts. Since the $x$-intercepts happen whenever the graph crosses or touches the $x$ axis, it is neither “up” nor “down” from baseline. So the $y$ coordinate (output) at each of these points is zero. In other words, we can set the “expression=0” and solve using whatever strategies are best. In textbooks, a very good strategy is often factoring and applying the Zero Product Principle. In reality, this may not always be the best strategy (especially since we really only teach “factoring over the integers” rather than across all real numbers) — but occasionally it could work.

Factoring is a primary tool for determining the behavior near each $x$-intercept. The number of factors that give rise to the same intercept tells us about the shape of the graph near that intercept. For any given $x$-intercept of a graph, if it arises from an odd number of factors in the expression, then it “passes through” the $x$ axis — starts either above or below and ends on the opposite side. If it arises from an even number of factors in the expression, then it “bounces off” the $x$-axis: starting either above or below and then returning back toward the direction it came from after touching the axis. The greater the number of factors, the “flatter” the graph appears near the $x$-intercept.

To detail the maximums and minimums by hand, we typically need a tool from calculus, namely, derivatives. However, even then, a key step in finding these points is to set the derivative equal to zero and then solve. If the derivative is a polynomial or rational function, then the same tools of the Zero Product Principle may be useful. Similarly, determining the graph’s shape near and between these points uses the second derivative and solving that expression equal to zero.

Determining where an expression is undefined, and thus where its graph might have either a hole or a vertical asymptote, often involves factoring some portion of the expression. Functions are undefined for any input (usually $x$) value that would cause an undefined result when computing the output. This might be any input that causes a negative number under a square root sign, or an input value that creates a zero in the denominator, for example. Since most of these errors involve a zero in some way (either equalling zero or because zero is the tipping point between positive and negative numbers), solving an expression=0 is a critical step in the process of finding almost all of these undefined inputs. Factoring and the Zero Product Principle, as before, can be one of the tools we use in finding these.

However, even when factoring is not the ideal tool to find the actual undefined inputs, it is a primary tool for finding the shape or behavior of the graph near those inputs, especially for functions that involve rational expressions. In particular, if we simplify the rational expression (below) and the factor associated with that undefined input “cancels out,” then the graph has a “hole” in it and otherwise behaves straightforwardly around that hole. If the factor associated with the undefined input does not “cancel out,” however, we end up with vertical asymptote behavior, and the graph approaches that input value from the left and right with runaway behavior, either increasing or decreasing infinitely very close to that input ($x$) value.

A rational expression in algebra is a polynomial divided by a polynomial, generally written in fraction form. We can work with them in much the same way we work with rational numbers (eg, fractions) in arithmetic. In fact, as is the case for all of algebra as generalized arithmetic, they are just generalized fractions.

Perhaps you remember reducing or simplifying numeric fractions by factoring and dividing (often called “cancelling”) out the common factors in the numerator and denominator. We can simplify or reduce rational expressions the same way. Factor the numerator and denominator as far as you can, then look for matching factors that divide out of the expression.

As noted in the section before, these “cancelled” factors become indicators of holes in the graph. The factors that remain after simplifying as much as possible tell us about the general shape and broad-strokes behavior of the graph.

Because of the nature of multiplication, division, and simplifying, factoring at the start can also allow us to divide out factors even before we perform multiplication of rational expressions, simplifying the multiplication or division processes as well.

Dividing polynomials puts us in the realm of rational expressions, which were discussed above. However, there may occasionally be non-polynomial situations wherein we can similarly find identical factors that divide each other out, in effect “cancelling” each other in the division process.

As mentioned in the graphing application above, most domain restrictions have something or other to do with zero: either the expression is undefined when some portion of it is equal to zero, or when some portion of it is less than zero. Whenever this happens, we may well end up needing to solve “subexpression=0” as part of the domain-finding process. In these cases, it’s often helpful to remember the Zero Product Principle and factoring as a means of solving (especially for textbook problems, where factorable polynomials tend to be over-represented).

This one’s a little more esoteric and not often taught directly. However, I could envision it as possibly being one of the more useful thought-processes around factoring (though it’s not clear how often it’s truly useful). Its utility would arise out of how specific real-world polynomials could be generated, particularly those that arise out of proportional relationships. If you can factor a real-world polynomial, the two factors may well have a real-world interpretation themselves. Remember that multiplying means finding the total out of a given number of groups of equal size. You may (or may not) be able to use that interpretation of multiplying to gather more knowledge of the real-world phenomenon based on what the factors come out to be.

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!