Learning Objectives
- Represent factoring quadratic polynomials geometrically.
- Factor monic quadratic polynomials by grouping.
- Factor non-monic quadratic polynomials by grouping.
Chapter 9 Factoring Quadratic Polynomials
Section 9.1 Lots of Methods, Not All Good
Factoring quadratic polynomials is generally seen as an important marker for a student’s algebra skill level. Because of this, various teachers have tried to design particular methods to help students factor correctly. Unfortunately, many of those methods are answer-oriented, which leads to students getting into the habit of making incorrect algebraic manipulations.
In this section, the focus is going to be on understanding the process of factoring and the ideas behind it. Although getting the right answer is also important, the right answer is the outcome of a proper understanding of the concepts, not the ultimate end in itself.
To emphasize this point, we’re going to use a purely geometric framework and sidestep all of the algebra. A couple sections ago, we introduced algebra tiles. These are geometric images that represent different quantities to help us think about algebraic relationships.
Algebra tiles come with a set of "rules" for how they can be manipulated, which is especially important when you’re working with negative values. However, for the purposes of illustration, we’re just going to focus on the situation where all of the values are positive. This means that we’re going to be thinking about three different tiles (both tiles should be viewed as the same piece, just rotated relative to each other):
A representation of a quadratic polynomial using the algebra tiles is simply a matter of having the right number of pieces:
The challenge is then to arrange all of these pieces into a rectangle where the small squares are at the bottom right, the large square (or squares) are on the top left, and the rectangles fill in the spaces on the top right and lower left. Once a rectangle is found, it can be used to determine the factorization.
Activity 9.1. Factoring a Quadratic by Diagram.
We’re not going to focus on trying to give you a "method" for finding these rectangles. We’re going to trust that your geometric intuition will lead you to the appropriate conclusion.
Try it!
Use a diagram of algebra tiles to factor \(x^2 + 7x + 12\text{.}\) Draw the diagram and write the final equation.
Solution.
Definition 9.1. Monic and Non-Monic Quadratic Polynomials.
A quadratic polynomial of the form \(ax^2 + bx + c\) is called monic quadratic polynomial if \(a = 1\text{.}\) If \(a \neq 1\text{,}\) then we call it a non-monic quadratic polynomial.
Activity 9.2. Factoring a Non-Monic Quadratic by Diagram.
Algebra tiles can be used to factor both monic and non-monic polynomials. It all comes down to finding the right rectangle.
Try it!
Use a diagram of algebra tiles to factor \(2x^2 + 9x + 4\text{.}\) Draw the diagram and write the final equation.
Solution.
While the use of diagrams (or physical manipulatives) may help us to get answers, it doesn’t necessarily help us to think in an organized and logical manner. So we are going to look at a structured algebraic approach to the same challenge.
But before we do that, let’s take a look at why factoring is difficult. Let’s take a look at the steps of expanding out a product.
\begin{equation*}
\begin{aligned}
2x^2 + 7x + 6 \amp = 2x^2 + x + 6x + 6 \\
\amp = 2x^2 + 2x + 5x + 6 \\
\amp = 2x^2 + 3x + 4x + 6
\end{aligned}
\end{equation*}
At this point, you can still factor by grouping to get back the original product.
\begin{equation*}
\begin{aligned}
2x^2 + 7x + 6 \amp = 2x^2 + x + 6x + 6 \\
\amp = 2x^2 + 2x + 5x + 6 \\
\amp = 2x^2 + 3x + 4x + 6
\end{aligned}
\end{equation*}
The challenge arises once the like terms are combined. We no longer have access to the rationale behind uncombining the like terms. In fact, there are lots of ways to uncombine like terms!
\begin{equation*}
\begin{aligned}
2x^2 + 7x + 6 \amp = 2x^2 + x + 6x + 6 \\
\amp = 2x^2 + 2x + 5x + 6 \\
\amp = 2x^2 + 3x + 4x + 6
\end{aligned}
\end{equation*}
Some students learn how to do this with monic polynomials with a guessing method, but it very often falls apart on them when they are working on non-monic polynomials. At the core, most of the problems arise from the students learning to simply "write down" their answer without understanding how or why it worked. We want to avoid that type of method because it does not enhance mathematical thinking.
We will be using the "\(ac\) method" of factoring. The goal is to uncover the right way to uncombine like terms so that we can factor by grouping. Here is a representation of the method:
\begin{equation*}
ax^2 + bx + c \longrightarrow \left\{
\begin{array}{l}
\text{Multiply to $ac$} \\
\text{Add to $b$}
\end{array}
\right.
\end{equation*}
As the text states, the goal is to find two numbers that multiply to and add to The values you obtain from this will be the way to uncombine the like terms, which will allow you to factor by grouping.
Activity 9.3. The \(ac\) Method with a Monic Quadratic.
We will start by applying this method to factor \(x^2 - 3x - 10\text{.}\) We will first translate our quadratic into the two target properties.
\begin{equation*}
x^2 - 3x - 10 \longrightarrow
\left\{
\begin{array}{l}
\text{Multiply to -10} \\
\text{Add to -3}
\end{array}
\right.
\end{equation*}
And here is where you simply have to use your experience with numbers to try to find the right combination. What two numbers multiply to -10 and add to -3? You should be able to determine that the values are -2 and 5. These two values are used to uncombine the like terms (\(-3x = -5x + 2x\)), which then allows us to complete the factorization using grouping. Here is the complete presentation.
\begin{equation*}
\begin{aligned}
x^2 - 3x - 10 \amp = x^2 - 5x + 2x - 10 \amp \eqnspacer \amp \text{The $ac$ method} \\
\amp = x(x - 5) + 2(x - 5) \amp \amp \text{Factor by grouping} \\
\amp = (x + 2)(x - 5) \amp \amp \text{Factor out the common factor}
\end{aligned}
\end{equation*}
Try it!
Use the \(ac\) method to factor \(x^2 - 3x - 10\) using a complete presentation, but instead of un-combining the middle term using \(-3x = -5x + 2x\text{,}\) swap the order and un-combine it using \(-3x = 2x - 5x\text{.}\)
Solution.
\begin{equation*}
\begin{aligned}
x^2 - 3x - 10 \amp = x^2 + 2x - 5x - 10 \amp \amp \text{The $ac$ method} \\
\amp = x(x + 2) - 5(x + 2) \amp \amp \text{Factor by grouping} \\
\amp = (x - 5)(x + 2) \amp \amp \text{Factor out the common factor}
\end{aligned}
\end{equation*}
One of the main values of the method is that it works for both monic and non-monic polynomials. This benefit allows us to simplify the process of factoring quadratic polynomials to a single approach that works for all situations. So by adopting the method, we don’t have to create two different sets of methods to accomplish the same goal.
Activity 9.4. The \(ac\) Method with a Non-Monic Quadratic.
We will use this method to factor \(4x^2 + 5x - 6\text{.}\)
\begin{equation*}
4x^2 + 5x - 6 \longrightarrow
\left\{
\begin{array}{l}
\text{Multiply to -24} \\
\text{Add to 5}
\end{array}
\right.
\end{equation*}
With a little bit of thought, you will find the combination of and will satisfy the goal.
\begin{equation*}
\begin{aligned}
4x^2 + 5x - 6 \amp = 4x^2 + 8x - 3x - 6 \amp \amp \text{The $ac$ method} \\
\amp = 4x(x + 2) - 3(x + 2) \amp \amp \text{Factor by grouping} \\
\amp = (4x - 3)(x + 2) \amp \amp \text{Factor out the common factor}
\end{aligned}
\end{equation*}
Try it!
Use the \(ac\) method to factor \(2x^2 + 5x - 12\) using a complete presentation.
Solution.
\begin{equation*}
\begin{aligned}
2x^2 + 5x - 12 \amp = 2x^2 - 3x + 8x - 12 \amp \amp \text{The $ac$ method} \\
\amp = x(2x - 3) + 4(2x - 3) \amp \amp \text{Factor by grouping} \\
\amp = (x + 4)(2x - 3) \amp \amp \text{Factor out the common factor}
\end{aligned}
\end{equation*}
Section 9.2 Worksheets
PDF Version of these Worksheets
1
external/worksheets/09-Worksheets.pdfWorksheet Worksheet 1
1.
Translate the diagram of algebra tiles into an equation.
2.
Use a diagram of algebra tiles to factor \(x^2 + 8x + 15\text{.}\) Draw the diagram and write the final equation.
3.
Fill in the appropriate value into the boxes, then use the method to factor the given quadratic polynomial using a complete presentation.
\begin{equation*}
x^2 + 8x + 15 \longrightarrow \left\{
\begin{array}{l}
\text{Multiply to $\boxed{\strut \phantom{.....}}$} \\
\text{Add to $\boxed{\strut \phantom{.....}}$}
\end{array}
\right.
\end{equation*}
Worksheet Worksheet 2
1.
Use a diagram of algebra tiles to factor \(2x^2 + 9x + 4\text{.}\) Draw the diagram and write the final equation.
2.
Fill in the appropriate value into the boxes, then use the method to factor the given quadratic polynomial using a complete presentation.
\begin{equation*}
x^2 + 2x - 8 \longrightarrow \left\{
\begin{array}{l}
\text{Multiply to $\boxed{\strut \phantom{.....}}$} \\
\text{Add to $\boxed{\strut \phantom{.....}}$}
\end{array}
\right.
\end{equation*}
3.
Fill in the appropriate value into the boxes, then use the method to factor the given quadratic polynomial using a complete presentation.
\begin{equation*}
2x^2 - 3x - 27 \longrightarrow \left\{
\begin{array}{l}
\text{Multiply to $\boxed{\strut \phantom{.....}}$} \\
\text{Add to $\boxed{\strut \phantom{.....}}$}
\end{array}
\right.
\end{equation*}
Worksheet Worksheet 3
1.
Fill in the appropriate value into the boxes, then use the method to factor the given quadratic polynomial using a complete presentation.
\begin{equation*}
x^2 - 8x + 7 \longrightarrow \left\{
\begin{array}{l}
\text{Multiply to $\boxed{\strut \phantom{.....}}$} \\
\text{Add to $\boxed{\strut \phantom{.....}}$}
\end{array}
\right.
\end{equation*}
2.
Fill in the appropriate value into the boxes, then use the method to factor the given quadratic polynomial using a complete presentation.
\begin{equation*}
2x^2 + 5x + 2 \longrightarrow \left\{
\begin{array}{l}
\text{Multiply to $\boxed{\strut \phantom{.....}}$} \\
\text{Add to $\boxed{\strut \phantom{.....}}$}
\end{array}
\right.
\end{equation*}
3.
Use the method to factor \(x^2 - 7x + 10\) using a complete presentation.
Worksheet Worksheet 4
1.
Fill in the appropriate value into the boxes, then use the method to factor the given quadratic polynomial using a complete presentation.
\begin{equation*}
3x^2 - 10x - 8 \longrightarrow \left\{
\begin{array}{l}
\text{Multiply to $\boxed{\strut \phantom{.....}}$} \\
\text{Add to $\boxed{\strut \phantom{.....}}$}
\end{array}
\right.
\end{equation*}
2.
Use the \(ac\) method to factor \(x^2 + 6x + 9\) using a complete presentation.
3.
Use the \(ac\) method to factor \(x^2 - 3x - 40\) using a complete presentation.
4.
Use the \(ac\) method to factor \(2x^2 - 5x - 3\) using a complete presentation.
Worksheet Worksheet 5
1.
Use the \(ac\) method to factor \(x^2 + 9x + 20\) using a complete presentation.
2.
Use the \(ac\) method to factor \(4x^2 - 4x - 3\) using a complete presentation.
3.
Factor \(x^2 + 10x + 16\) and \(x^2 - 10x + 16\text{,}\) then compare the results. What do you notice about the factorizations?
4.
Factor \(x^2 - 5x - 14\) and \(x^2 + 5x - 14\text{,}\) then compare the results. What do you notice about the factorizations?
Worksheet Worksheet 6
1.
Suppose you are trying to factor a quadratic that has the following condition:
\begin{equation*}
\left\{
\begin{array}{l}
\text{Multiply to a positive number} \\
\text{Add to a positive number}
\end{array}
\right.
\end{equation*}
What information can you conclude about the signs of the two numbers you’re looking for?
2.
Suppose you are trying to factor a quadratic that has the following condition:
\begin{equation*}
\left\{
\begin{array}{l}
\text{Multiply to a positive number} \\
\text{Add to a negative number}
\end{array}
\right.
\end{equation*}
What information can you conclude about the signs of the two numbers you’re looking for?
3.
Suppose you are trying to factor a quadratic that has the following condition:
\begin{equation*}
\left\{
\begin{array}{l}
\text{Multiply to a negative number} \\
\text{Add to a positive number}
\end{array}
\right.
\end{equation*}
What information can you conclude about the signs of the two numbers you’re looking for?
4.
Suppose you are trying to factor a quadratic that has the following condition:
\begin{equation*}
\left\{
\begin{array}{l}
\text{Multiply to a negative number} \\
\text{Add to a negative number}
\end{array}
\right.
\end{equation*}
What information can you conclude about the signs of the two numbers you’re looking for?
Section 9.3 Deliberate Practice: Factoring Quadratic Polynomials
Algebra is a skill, which means it requires practice to become proficient. But it will take more than rote repetition to get there. Deliberate practice is the thoughtful repetition of a task. For each of these sections, you will be given a list of specific skills or ideas to focus on as you practice thinking through the problems.
Focus on these skills:
- Write the original expression.
- Visualize the grids, but try to do the calculations without drawing them.
- Mentally state the "Multiply to" and "Add to" properties. You can write them out as scratch work, if necessary.
- Present your work legibly.
Worksheet Worksheet
Instructions: Use the \(ac\) method to factor using a complete presentation.
1.
Use the \(ac\) method to factor \(x^2 + 2x - 15\text{.}\)
2.
Use the \(ac\) method to factor \(x^2 - 4x + 4\text{.}\)
3.
Use the \(ac\) method to factor \(x^2 + 7x + 12\text{.}\)
4.
Use the \(ac\) method to factor \(x^2 - 9\text{.}\)
5.
Use the \(ac\) method to factor \(x^2 + 8x + 16\text{.}\)
6.
Use the \(ac\) method to factor \(x^2 - x - 42\text{.}\)
7.
Use the \(ac\) method to factor \(2x^2 + 5x - 3\text{.}\)
8.
Use the \(ac\) method to factor \(3x^2 - 13x - 10\text{.}\)
9.
Use the \(ac\) method to factor \(2x^2 - 3x - 20\text{.}\)
10.
Use the \(ac\) method to factor \(4x^2 + 4x - 15\text{.}\)
Section 9.4 Closing Ideas
As mentioned earlier, factoring quadratic polynomials is viewed as an important marker of your algebraic skill level. But the skill isn’t just in getting the answer. In fact, the bulk of mathematics is not about getting the answer. If you look back over everything we’ve done so far, you will see that there is a heavy emphasis on understanding and communicating the thought processes involved in performing these algebraic manipulations.
On the last worksheet page, instead of having you do more factorizations, you were asked to seek out patterns in the factorizations you’ve already done. You were only asked to identify the information you could conclude in each situation, but you were hopefully also able to discover the underlying logic. That logic goes all the way back to basic arithmetic, with ideas such as the following:
- A positive number multiplied by a positive number is a positive number.
- A negative number multiplied by a negative number is a positive number.
- A positive number multiplied by a negative number is a negative number.
- A negative number multiplied by a positive number is a negative number.
There are also ideas that perhaps we don’t have "memorized" phrases for, but should make sense when you think about it.
- A positive number plus a positive number is a positive number.
- A negative number plus a negative number is a negative number.
- When adding a positive and a negative number together, the sign of the result will match the sign of the number that is larger (in absolute value).
Factoring quadratic polynomials is not an end in and of itself. It is a tool that allows you to do advanced algebraic manipulations, such as solving more complicated equations. So we usually do not use it in complete isolation. There is an important property of numbers that can be a useful supplemental tool.
Theorem 9.2. Zero Product Property.
The zero product property states that if \(a \cdot b = 0\text{,}\) then we must have either \(a = 0\) or \(b = 0\) (or possibly both).
The combination of factoring with the zero product property allows you to take an equation with higher degree and convert it into multiple equations of lower degree. Here is an example of solving the equation \(x^2 + 3x - 4 = 0\text{:}\)
\begin{equation*}
\begin{array}{ccl}
\begin{aligned}
x^2 + 3x - 4 \amp = 0 \\
(x-1)(x+4) \amp = 0
\end{aligned} \amp \phantom{.}\hspace{1cm}\phantom{.} \amp
\begin{aligned}
\\
\amp \text{Factor} \\
\end{aligned} \\
\begin{array}{rlcrl}
x - 1 \amp = 0 \amp \phantom{.}\hspace{5mm} \text{or} \hspace{5mm}\phantom{.} \amp x + 4 \amp = 0 \\
x \amp = 1 \amp \amp x \amp = -4
\end{array} \amp \amp
\begin{aligned}
\amp \text{The zero product property} \\
\amp \text{Solve each equation} \\
\end{aligned}
\end{array}
\end{equation*}
Notice how the factorization was reduced to just a single step and that none of the steps for solving the equation were justified. When you get further along in your math courses, you will be expected to know how to do many of those steps for yourself, which is why we’re strongly emphasizing the steps here. We are creating a foundation for you to build on in the future.
In this section, we can see the full scaffolded nature of algebraic reasoning. The skill of factoring is right in the middle of the tower. Below us, we see that everything we’re doing is premised on having basic fluency with algebraic concepts such as the distributive property, and it also requires us to have enough experience with numbers to search through different combinations to find the one we want. Above us, we see that there’s an entirely new set of equations that we can solve once we add in some more ideas. Every time you learn a new idea in mathematics, you should take a step back to see how it fits into the bigger picture.
Section 9.5 Going Deeper: Special Factorizations
There are certain factoring patterns that are useful to learn to recognize. It’s not that you would be unable to factor these without recognizing them, but they come up so frequently that they have special names so that we can identify them when they happen:
The basic idea of using these patterns to factor is that you need to match your expression with one of the above formulas and identify the appropriate values of \(a\) and \(b\text{.}\) For example, suppose you want to factor \(x^2 + 6x + 9\) and you want to check whether this fits one of the patterns above. By counting the number of terms and looking at the signs, we can see that there’s only one formula that has a chance.
Once we know that we should be focusing on the square of a binomial sum, we can try to identify the \(a\) and the \(b\) by looking at the terms on the end and then check to see if the term in the middle matches. We can see that we can take \(a = x\) and \(b = 3\text{,}\) which means that the middle term should be \(2ab = 6x\text{.}\) And this all matches perfectly.
\begin{equation*}
x^2 + 6x + 9 = x^2 + 2 \cdot x \cdot 3 + 3^2 = (x + 3)^2
\end{equation*}
The nice part about these formulas is that it gives us insights into other factorizations that we may not immediately see using the method. Here’s an example:
\begin{equation*}
x^2 + 2 \sqrt{2} x + 2 = x^2 + 2 \cdot x \cdot \sqrt{2} + ( \sqrt{2})^2 = (x + \sqrt{2})^2
\end{equation*}
It’s not that the method can’t be applied here. Here’s how it would look.
\begin{equation*}
x^2 + 2 \sqrt{2} x + 2 \longrightarrow \left\{
\begin{array}{l}
\text{Multiply to $2\sqrt{2}$} \\
\text{Add to $2$}
\end{array} \right.
\end{equation*}
It’s technically true that \(\sqrt{2} + \sqrt{2} = 2 \sqrt{2}\) and \(\sqrt{2} \cdot \sqrt{2} = 2\text{,}\) and we would be able to do the factorization by grouping:
\begin{equation*}
\begin{aligned}
x^2 + 2 \sqrt{2} x + 2 \amp = x^2 + \sqrt{2} x + \sqrt{2} x + 2 \amp \eqnspacer \amp \text{The $ac$ method} \\
\amp = x(x + \sqrt{2}) + \sqrt{2} (x + \sqrt{2}) \amp \amp \text{Factor by grouping} \\
\amp = (x + \sqrt{2})(x + \sqrt{2}) \amp \amp \text{Factor out the common factor}
\end{aligned}
\end{equation*}
But we can see from this that the \(ac\) method is really designed for thinking through integer factorizations. Recognizing the factorization with square roots is just not something we should expect students at this level to do.
The reason we use these special factorizations is because they are a special pattern that we can learn to recognize with practice. And that’s pretty much all this is. By learning to recognize these patterns, you have access to a larger collection of factorizations that you might otherwise miss. Here are some examples:
Another aspect of these formulas is that your awareness of them can help you to avoid a few very specific algebraic errors. If you are familiar with the formulas for the square of a binomial sum and difference, you would immediately recognize what these expressions should be. These are errors that are so common that one of them has been given the nickname of "the freshman’s dream."
You might have noticed that that there’s a formula for the square of a binomial sum and difference, but only a formula for the difference of squares and not a sum of squares. It turns out that the sum of squares cannot be factored using the tools that we have developed so far. We can use logic to prove that we can’t do this. Let’s try to factor \(x^2 + 9\) as an example. Using the method, we have to find values that do the following:
\begin{equation*}
x^2 + 9 \longrightarrow \left\{
\begin{array}{l}
\text{Multiply to $9$} \\
\text{Add to $0$}
\end{array} \right.
\end{equation*}
In order to get two numbers to multiply to a positive number, they must either both be positive or both be negative. But if they are both positive or both negative, then when you add them together, you can’t get zero because both numbers have the same sign. And so there aren’t any numbers that will do this for us.
The special factorization formulas don’t stop there. Factorization is such an important concept in mathematics that there are all sorts of factorizations that can be helpful at various times and various situations. Here are a few examples, all of which you can check by multiplying everything out:
- The sum of cubes: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)
- The difference of cubes: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)
- The difference of fourth powers: \(a^4 - b^4 = (a - b)(a^3 + a^2b + ab^2 + b^3)\)
- A "magic" formula: \(x^2 + bx + c = \left( x - \left( - \frac{b}{2} + \sqrt{\frac{b^2}{4} - c} \right) \right) \left( x + \left( - \frac{b}{2} - \sqrt{\frac{b^2}{4} - c} \right) \right)\)
The last formula is not one that you’re likely to find anywhere, but it may look similar to something you might remember from your past mathematical experiences. And it’s actually related to an approach to factoring that was known by mathematicians over 3500 years ago that was introduced to the modern world by Dr. Po-Shen Loh in 2019.
