How to Factor Polynomials—Step-by-Step Examples and Tutorial

How to Factor Polynomials—Step-by-Step Examples and Tutorial

How to Factor Polynomials Explained

Step-by-Step Guide: How to Factor Polynomials with 2 Terms, How to Factor Polynomials with 3 Terms, How to Factor Cubic Polynomials

 

Free Step-by-Step Guide: How to factor a polynomial with a specific number of terms

 

In algebra, a polynomial is an expression made up of variables and coefficients separated by the operations of addition and/or subtraction.

Polynomials are a fundamental math topic and understanding how to work with them (including factoring) is essential to being successful in algebra and beyond. Learning how to factor polynomials with 2, 3, or 4 terms involves understanding how to break down a given polynomial into simpler factors.

This free Step-by-Step Guide on How to Factor Polynomials will cover the following topics:

While learning how to factor polynomials can be challenging, it is a learnable skill that can be acquired through practice. The goal of this free guide on how to factor polynomials is to give you plenty of step-by-step practice with factoring polynomials—including polynomials with 4 terms (cubic polynomials)—so that can become more comfortable with factoring all kinds of polynomials.

Before we cover everything you need to know about how to factor a polynomial, let’s quickly recap some key algebra vocabulary terms and phrases that you will need to be familiar with in order to use this guide.

 

Figure 01: How to Factor Polynomials: What is a Polynomial?

 

What is a polynomial?

As previously stated, a polynomial is a math expression comprised of variables, coefficients, and/or constants separated by the operations of addition or subtraction.

The terms of polynomials are individual parts, or monomials, separated by addition or subtraction signs.

For example,

  • 3x² is a monomial

  • 3x² + 6x is a polynomial with 2 terms (3x² and 6x)

  • 3x² + 6x - 15 is a polynomial with 3 terms (3x², 6x, and -15)

  • 9x³ + 3x² + 6x - 15 is a polynomial with 4 terms (9x³, 3x², 6x, and -15)

Figure 01 above illustrates the difference between a monomial and a polynomial.

For an expression to be considered a polynomial, it must have at least two terms, but there is no limit on how many terms a polynomial can have.

When it comes to factoring polynomials, you will most commonly be dealing with polynomials that have 2 terms, 3 terms, or 4 terms:

  • A polynomial with 2 terms is called a binomial

  • A polynomial with 3 terms is called a trinomial

  • A polynomial with 4 terms is called a quadrinomial (also known as a cubic polynomial)

Examples of a polynomial with 2 terms, 3 terms, and 4 terms are shown in Figure 02 below.

 

Figure 02: How to factor polynomials with 4 terms or less.

 

Now that you understand the key terms and the difference between a polynomial with 2 terms, 3 terms, and 4 terms.

For factoring each type of polynomial, we will look at two methods: GCF, direct factoring, and a combination of the two.

Let’s get started!

How to Factor Polynomials with 2 Terms

We will start by learning how to factor polynomials with 2 terms (binomials).

Whenever you are factoring a polynomial with any number of terms, it is always best to start by looking to see if there is a GCF—or greatest common factor—that all of the terms have in common.

For example, consider the following example:

Example #1: Factor 8x + 4

For this example, you should notice that both terms, 8x and 4 are divisible by 4, hence they share a GCF of 4.

Therefore, you can divide out the GCF of 4 from both terms as follows:

  • 8x + 4 → 4 (2x + 1)

So, the factors of 8x + 4 are: 4 and (2x+1).

What we just did was essentially the reverse of the distributive property, as shown in Figure 03 below.

 

Figure 03: How to factor a polynomial with 2 terms using the GCF method.

 

Note that many binomials can be factored using the GCF method, so let’s gain a little more practice with one more example (understanding how to simplify and/or factor a polynomial using the GCF method will come in handy when you start factoring 3 and 4 term polynomials later on).

Example #2: Factor 6x² + 12x

Just like the first example, there is a GCF for both terms. But, in this case, the GCF includes a variable. Why? Because both terms have coefficients that are divisible by 6 and both terms have at least one x variable, so the GCF, in this case, is 6x.

Therefore, you can divide out 6x from both terms as follows:

  • 6x² + 12x → 6x(x + 2)

So, the factors of 6x² + 12x are: 6x and (x+2).

Again, this method of factoring is just the reverse of the distributive property and is illustrated in Figure 04 below.

 

Figure 04: How to factor a polynomial with 2 terms using the GCF method.

 

Next, we will look at a special case of factoring a binomial—when the binomial is a difference of two squares (this is sometimes referred to as DOTS).

Whenever you have a binomial of the form a²-b², the factors will be of the form (a+b)(a-b).

Example #3: Factor x² - 49

For example, if you wanted to factor the binomial: x² - 49, you would notice that both x² and 49 are squares:

  • x² = (x)(x)

  • 49=(7)(7)

So, another way to write (x²- 49) is (x²- 7²)

Therefore, you can use the DOTS method for factoring binomials. In this case, a = x and b = 7, so:

  • (a²-b²) = (a+b)(a-b) → (x²- 7²) = (x-7)(x+7)

You can now conclude that the factors of x²- 49 are (x-7) and (x+7) using the DOTS method.

This process is illustrated in Figure 05 below:

 

Figure 05: How to factor a polynomial that is the difference of two squares.

 

If you want to learn more about the DOTS method for factoring polynomials that are the difference of two squares, check out this free video tutorial on YouTube for more practice.

Otherwise, let’s continue onto the next section where you will learn how to factor polynomials with 3 terms.

How to Factor Polynomials with 3 Terms

Moving on, we will now look at polynomials with 3 terms, typically referred to as trinomials

Learning how to factor polynomials with 3 terms involves a more involved factoring process that we will explore in this section.

The trinomials that we will cover will be of the form ax² + bx + c (where c is a constant). The strategies that we will use will depend on whether a (the leading coefficient) equals one or not. Therefore, the first two examples in this section will be factoring trinomials when a=1 and the second two examples will be when a≠1.

How to Factor Polynomials with 3 Terms when a=1

Example #1: Factor x² + 6x + 8

For the first example, we have to factor the trinomial: x² + 6x + 8

 

Figure 06: How to factor polynomials with 3 terms (when a=1)

 

Again, the leading coefficient, a, is equal to 1 in this example. This is important to note because the following method for factoring a trinomial only works when a=1.

Now we are ready to factor this trinomial in 3 easy steps:

Step One: Identify the values of b and c.

In this example, the values of b and c in the trinomial are: b=6 and c=8

Step Two: Figure out two numbers that both ADD to b and MULTIPLY to c.

The second step often involves some of trial-and-error as you pick numbers and see if they meet both conditions (the two numbers have to add together to make b and multiply together to make c).

  • 5 + 1 =6 (the value of b) ✓

  • 5 x 1 ≠ 8 (the value of c) ✘

For example, lets say that you chose the numbers 5 and 1. While 5+1=6 is true (satisfying the first condition), 5x1=5 (not 8), therefore, they do not satisfy the second condition. So, 5 and 1 do not work.

But, if you picked the numbers 2 and 4, you can see that:

  • 2 + 4 =6 (the value of b) ✓

  • 2 x 4 = 8 (the value of c) ✓

Since 2 and 4 satisfy both conditions, you can stop searching and move onto the third step.

Step Three: Use your numbers from step two to write out the factors

In this case, you can conclude that the factors of x² + 6x + 8 are (x+2) and (x+4).

 

Figure 07: The factors of x² + 6x + 8 are (x+2) and (x+4).

 

You can verify that these are the correct factors by performing double distribution as follows:

  • (x+2)(x+4) = x² + 2x + 4x + 7 = x² + 6x + 8

Notice that you ended up with the trinomial that you started with! Now, lets work through one more example of how to factor polynomials with 3 terms when a=1.

Example #2: Factor x² - 3x - 40

For this next example, we have to factor the trinomial: x² - 3x - 40

 

Figure 08: How to factor 3rd degree polynomials

 

Notice that, in this case, the trinomial includes subtraction signs, which will affect how you perform step two below.

Step One: Identify the values of b and c.

For this trinomial, b= -3 and c= -40

Step Two: Figure out two numbers that both ADD to b and MULTIPLY to c.

Again, you have to find two numbers that add to make -3 and that multiply together to make -40.

This part can be tricky when both of the values for b and c are negative (like in this example). You have to recall that a negative number times another negative number will lead to a positive result, so you can’t have two negatives (since you need to find two numbers that multiply together to make -40).

Eventually, after some trial-and-error, you should find that -8 and +5 satisfy both conditions:

  • -8 + 5 =-3 (the value of b) ✓

  • -8 x 5 = -40 (the value of c) ✓

Step Three: Use your numbers from step two to write out the factors

Finally, you can conclude that the factors of x² - 3x -40 are (x-8) and (x+5).

(You make sure that this answer is correct, you can perform double distribution on (x-8)(x+5) to make sure that the result is equal to the original trinomial).

 

Figure 09: The factors of x² - 3x -40 are (x-8) and (x+5).

 

If you want more practice factoring trinomials when a=1, check out our free step-by-step guide on how to factor trinomials to gain some more practice.

Otherwise, you can continue on to learn how to factor polynomials with 3 terms when a≠1.

How to Factor Polynomials with 3 Terms when a≠1

Example #1: Factor 2x² - x - 6

For the first example, we have to factor the trinomial: 2x² - x - 6

 

Figure 10: How to factor polynomials with 3 terms when a≠1

 

For starters, notice that you can not pull out a GCF.

So, to solve trinomials of the form ax² + bx + c when a≠1, you can use the AC method as follows:

Step One: Identify the values of a and c and multiply them together

In this case, a=2 and c=-6, so

  • a x c = 2 x -6 = -12

Step Two: Factor and replace the middle term

The second step requires you to use the result from step one to factor and replace the middle term.

The middle term is currently -1x and note that:

  • -12 = -4 x 3; and

  • -4 + 3 = -1

We chose -4 and 3 as factors because the sum of -4 and 3 equals negative 1, so we can rewrite the original trinomial as 2x² - 4x +3x - 6

 

Figure 11: Factor and replace the middle term

 

Step Three: Split the new polynomial down the middle and take the GCF of each side

Note that we are now working with a polynomial that actually has four terms: 2x² - 4x + 3x - 6

In this third step, you have to split the polynomial down the middle to essentially create two separate binomials that you can simplify by dividing GCF’s out of as follows:

  • First Half: 2x² - 4x = 2x(x-2)

  • Second Half: 3x - 6 = 3(x-2)

This third step is illustrated in Figure 12 below:

 

Figure 12: Split the new polynomial down the middle and take the GCF of each side

 

Step Four: Identify the Factors

Finally, you are ready to identify the factors.

The result from the previous step was 2x(x - 2) + 3(x -2). Hidden within this expression are your two factors, which you can see by looking at Figure 13 below.

 

Figure 13: The final step is to identify the factors

 

Finally, you can conclude that the factors of 2x² - x - 6 are (2x+3) and (x-2).

Clearly, factoring a trinomial when a≠1 can be a tricky and there are several steps along the way, but, the more that you practice this process, the better you will become at factoring polynomials with 3 terms like the one in this past example. To give you a little more practice, lets work through one more example before we move on to learning how to factor cubic polynomials.

Example #2: Factor 4x² - 15x + 9

 

Figure 14: Factor the trinomial where a=4, b=-15, and c=9

 

Step One: Identify the values of a and c and multiply them together

In this example, a=4 and c=9, so

  • a x c = 4 x 9 = 36

Step Two: Factor and replace the middle term

For the next step, note that the middle term is -15x, so you will need to find two numbers that multiply to 36 and add to -15:

  • 36 = -12 x -3; and

  • -12 + -3 = -15

Now, we can rewrite the original trinomial as 4x² -12x -3x +9

Step Three: Split the new polynomial down the middle and take the GCF of each side

For step three, you have to split the polynomial into two separate binomials and divide a GCF out of each one as follows:

  • First Half: 4x² -12x = 4x(x-3)

  • Second Half: -3x+9 -3(x-3)

Step Four: Identify the Factors

The last step is to identify the factors as shown in Figure 15 below.

 

Figure 15: The factors are (4x-3)(x-3)

 

Now, you can conclude that the factors of 4x² - 15x + 9 are (4x-3) and (x-3).

You can again use double distribution on (4x-3)(x-3) to verify that your solution is correct.

If you need more step-by-step help with how to factor polynomials with 3 terms when a does not equal 1, check out our free YouTube video tutorial.

Otherwise, continue on to the final section where you will learn how to factor polynomials with 4 terms.

How to Factor Polynomials with 4 Terms

The last section of this guide will cover how to factor polynomials with 4 terms and how to factor cubic polynomials.

In this section, we are going to apply a grouping method for how to factor a cubic polynomial that is very similar to the way that you factored trinomials when the leading coefficient, a, did not equal one in the last section. So, you may want to review that section before moving onto the 4 term polynomial factoring examples, however, it is not completely necessary, as we will be taking a step-by-step approach to solving two examples of factoring cubic polynomials.

Now, lets go ahead and work through our first example on how to factor cubic polynomials.

 

Figure 16: Polynomials with 4 terms are referred to as cubic polynomials.

 

Example #1: Factor 2x³ - 3x² + 18x - 27

For the first example, we have to factor the cubic polynomial: 2x³ - 3x² + 18x - 27

Step One: Split the cubic polynomial into groups of two binomials.

To factor this 4 term polynomial, we are going to apply what is called the grouping method, which requires you to split the polynomial into two groups (two separate binomials) with the goal of factoring a GCF out of each one.

 

Figure 17: How to Factor Cubic Polynomials by Grouping: The first step is to split the polynomial into two groups of binomials.

 

Remember that the goal is to create two separate binomials that have a GCF. If there is no apparent GCF, you have the option of swapping the positions of the middle terms (- 3x² and 18x), but that is not necessary for factoring this 4 term polynomial.

In this example, by the end of step one, you now have two groups to factor:

  • (2x³ - 3x²)

  • (18x - 27)

Step Two: Factor each binomial by pulling out a GCF

Now, go ahead and divide a GCF out of each binomial as follows:

  • (2x³ - 3x²) → x²(2x - 3)

  • (18x - 27) → 9(2x - 3)

This step is illustrated in Figure 18 below.

 

Figure 18: How to factor a cubic polynomial by grouping.

 

Step Three: Identify the factors

Notice that both results have a (2x-3) term. This is important and expected. If both results do not share a same term, then you either made a mistake or the polynomial with 4 terms is not factorable.

But, since we were able to factor each group by pulling out a GCF that resulted in both groups sharing a common factor of (2x-3), we know that we can factor out the other terms (x² and +9), so now have our factors: (x²+9) and (2x-3)

Final Answer: The factors of 2x³ - 3x² + 18x - 27 are (x²+9) and (2x-3)

The entire process of how to factor polynomials by 4 terms by grouping is illustrated in Figure 19 below.

 

Figure 19: The factors of 2x³ - 3x² + 18x - 27 are (x²+9) and (2x-3)

 

Example #2: Factor 3y³ + 18y² + y + 6

Let’s gain some more practice with how to factor a cubic polynomial by grouping by solving one more example problem.

In this case, we have to factor the cubic polynomial 3y³ + 18y² + y + 6 using the same grouping method as the previous example.

Step One: Split the cubic polynomial into groups of two binomials.

Start by splitting the cubic polynomial into two groups (two separate binomials).

 

Figure 20: Split the cubic polynomial into two groups of binomials and check to see if they can both be factored by pulling out a GCF.

 

As shown in Figure 20 above, by completing step one, you are left with these two groups

  • (3y³ +18y²)

  • (y+6)

Hold on! Before moving onto the next step, you should notice that the second group (y+6) cannot be factored by pulling out a GCF (because there is no greatest common factor between 1y and 6).

However, notice that we can swap the middle terms of the cubic polynomial (18y² and +y) as shown in Figure 21 below.

Now, we can factor a new 4 term polynomial 3y³ + y + 18y² + 6 that is equivalent to the original 4 term polynomial since the commutative property of addition allows you to rearrange the terms.

 

Figure 21: The commutative property of addition allows you to rearrange the middle terms when you are unable to group and GCF the original cubic polynomial.

 

Notice that you can split this new polynomial into two binomials that can be factored by pulling out a GCF:

  • (3y³ + y)

  • (18y² + 6)

 

Figure 22: After rearranging the original cubic polynomial, you can split it into two binomial groups that can be factoring by pulling out a GCF.

 

Step Two: Factor each binomial by pulling out a GCF

As illustrated in Figure 22 above, after rearranging the original cubic polynomial, you can split it into two binomial groups that can be factoring by pulling out a GCF as follows:

  • (3y³ + y) → y(3y² + 1)

  • (18y² + 6) → 6(3y² + 1)

Step Three: Identify the factors

Now, you can see that both factors have a (3y² + 1) term, which means that you have factored correctly.

Final Answer: The factors of 3y³ + 18y² + y + 6 are (y+6) and (3y² + 1)

The entire process of how to factor polynomials a cubic polynomial like the one in this example is illustrated in Figure 23 below.

 

Figure 23: How to factor cubic polynomials by grouping (step-by-step).

 

How to Factor Polynomials: Conclusion

Learning how to factor a polynomial is an important algebra skill that every math student must learn at some point.

While factoring polynomials can be tricky, there are several useful and effective strategies that you can use to factor polynomials. The strategy that you choose will depend on how many terms a polynomial has (as you will often be dealing with factoring polynomials with 2, 3, or 4 terms).

The best way to get better at factoring polynomials (especially cubic polynomials that have 4 terms) is by working through practice problems step-by-step. If you feel like you need more practice, we highly recommend working through the examples in this guide several times to gain more experience.

Keep Learning:

March Madness Perfect Bracket Odds—What are the Chances?

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March Madness Perfect Bracket Odds—What are the Chances?

What are the March Madness Perfect Bracket Odds?

The Chances of a Perfect March Madness Bracket are Crazier Than You Might Think

 

Is this possible? What are the March Madness perfect bracket odds?

 

The annual NCAA College Basketball Tournament—known as March Madness—is one of the most exciting times of the year for college basketball fans. Every year, millions of fans across the United States fill out their tournament brackets, hoping to predict the winners of all 63 games. But what are the chances of filling out a perfect March Madness bracket? Is it even possible?

Let’s explore the math behind why a predicting a perfect March Madness bracket is essentially impossible by diving into the answers to the following questions:

 

Every year, millions of fans fill out a tournament bracket. What what are the chances of a perfect March Madness bracket?

 

What are the March Madness Perfect Bracket Odds?

Filling out a perfect March Madness bracket means correctly predicting the winner of all 63 NCAA basketball tournament games. Having a perfect bracket means that must be 100% correct in all of your picks, including all of the inevitable upsets (when a lower ranked team defeats a higher ranked team) that occur every year.

The mathematical probability of a perfect March Madness bracket are:

1 : 9,200,000,000,000,000,00.

You read that correctly. The chances of correctly predicting a perfect March Madness bracket are astronomically low. In fact, some mathematical models estimate the odds of having a perfect March Madness bracket at one in 9.2 quintillion.

To put it another way, you are more likely to win the lottery multiple times in a row or get struck by lightening twice on the same day than to fill out a perfect bracket.

 

Given the low odds of a perfect March Madness bracket, is it even possible? Photo by Hannah Gibbs on Unsplash

 

Is a March Madness perfect bracket even possible?

The short answer is no, it is not possible to fill out a perfect March Madness bracket given the insanely high odds of correctly predicting the winner of all 63 games.

The reason for saying that a March Madness perfect bracket is impossible is simple—there are just too many variables at play in every game that make predicting winners incredibly difficult.

Even the most talented and experienced basketball analysts and sports statistics experts can not predict with complete accuracy how a given tournament game will play out. Factors such as injuries to key players, momentum swings, and elements of chance can all play a role in the outcome of a basketball game. And when you consider that there are 63 games in the tournament, the chances of getting correctly predicting the winner of every single game is virtually zero.

Of course, the fact that a perfect March Madness bracket is essentially impossible should not discourage you from filling out a bracket and attempting to correctly predict the outcome of as many games as possible. Why? Because filling out a bracket and chasing this impossible dream is what makes March Madness such a fun time of year for die-hard and casual basketball fans alike.

Even if you don't have a perfect bracket, you can still compete with your friends and colleagues to see who can get the most games right. And who knows? You might just run into some luck and make some accurate predictions that others did not.

 

Has there ever been a perfect March Madness bracket? Photo by Jacob Rice on Unsplash

 

Has There Ever Been a Perfect March Madness Bracket?

Millions of basketball fans have been filling out March Madness brackets since the tournament’s inception in 1939, but has anyone ever had a perfect March Madness bracket?

The answer is no.

Remember that the odds of predicting the winners of all 63 tournament games are astronomically low, with some estimates placing the odds of a perfect bracket at around 1 in 9.2 quintillion.

Even with the rise of advanced analytics and machine learning algorithms, no one person has ever been able to achieve a perfect March Madness bracket, and they likely never will.

How Can You Use Math to Improve Your Bracket Predictions?

Mathematically speaking, filling out a perfect bracket is virtually impossible. But, what are your chances of getting a certain number of games right?

Let’s take a look at some statistics that will shed some light on this question.

According to the NCAA, the chances of correctly predicting the outcome of the first round (the 32 games played during the first two days of the tournament) are about 1 in 4.3 billion. While those odds are not very much in your favor, they are still way better than the odds of predicting a perfect March Madness bracket. In fact, there have been several occasions where someone correctly predicted the winners of all 32 first-round games.

 

The chances of predicting the winner of any NCAA tournament game depend on many factors including matchups, injuries, and past performance. Photo by Markus Spiske on Unsplash

 

But, as the tournament continues on into the later rounds, the odds of correctly predicting every winner become exponentially smaller. By the time you get to the third round (known as the Sweet 16), the chances of predicting all the games correctly are about 1 in 75.6 million.

As for the point in the tournament when only four teams remain (known as the Final Four), the odds of predicting all games correctly drop to about 1 in 2.4 million.

These odds are only estimates, and the actual chances of getting each game right can vary depending on many factors. The point is that, while the chances of a perfect March Madness bracket are essentially zero, it is still possible to correctly predict winners and have fun competing with others.

In fact, if you were to simply guess the winner of every March Madness game at random, without any knowledge or analysis of college basketball, you would be mathematically expected to correctly predict around half of the total games (about 31.5 out of the 63 games).

So, what should you keep in mind when filling out your March Madness bracket? The best advice we can give is to focus more on making educated based on information such as a teams’ record, matchups, current injuries, and recent performance. You can also take advice from trusted sports analysts and experts that can offer valuable insights that will help you make decisions when filling out your bracket.

 

Given that the March Madness perfect bracket odds are pretty much impossible, you can let yourself off the hook and simply have fun filling out your bracket. Photo by Ben Hershey on Unsplash

 

Another thing to keep in mind about filling out your bracket is the occurrence of upsets—when a lower ranked team wins against a higher ranked team. Upsets are incredibly common in the NCAA tournament and they are one of the most fun and exciting aspects of March Madness.

With this in mind, don't be afraid to pick a few lower-seeded teams to defeat a higher-seeded opponent, especially during the first two rounds.

On a final note, given that the March Madness perfect bracket odds are impossible, you can let yourself off the hook and simply have fun filling out your bracket, knowing that it will likely get busted during the first round.

The unpredictability of the tournament is what contributes to the 'Madness'—so have fun and good luck!

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Exploring Positive and Negative Numbers: Elevation Activity

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Exploring Positive and Negative Numbers: Elevation Activity

How to Use Positive and Negative Numbers to Represent Quantities in the Real-World

Exploring Elevation Above/Below Sea Level is Great Way to Explore Positive and Negative Numbers

 
 

Are you looking for a fun way to teach your students about the relationship between positive and negative numbers in a real-world context?

On its own, a mathematical topic such as how positive and negative numbers can be used together to describe opposite directions or values can be confusing and uninteresting to kids who often struggle to see how a topic applies to their world.

Relating mathematics to the real world helps students to understand the practical everyday applications of math, which makes learning math fun, engaging, and, most importantly, meaningful. Plus, understanding the connections between mathematics and real life can spark student interest in math and allow them to see how useful the subject truly is.

In this post, you will learn about a fun activity that explores the relationship between positive and negative numbers in the context of their elevation above or below sea level in comparison to other geographical locations.

(Do you want free K-8 math resources and activities in your inbox every week? Click here to sign up for our free math education email newsletter)

 

Positive and Negative Numbers Activity: What is My Current Elevation? (Image: Mashup Math FP)

 

What is My Current Elevation?

Math Activity: What is My Current Elevation Above/Below Sea Level Compared to Other Locations?

Grade Level(s): 5th Grade, 6th Grade, 7th Grade

Learning Standard(s): Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

This activity helps students understand the concept of elevation, especially in the context of U.S. cities like New Orleans and Miami, which are at or below sea level, and Denver, known for being thousands of feet above sea level.

 

Click the link below to download the free pdf guide that accompanies this lesson.

 

The following activity lets students take a deeper look into finding the elevation of geographic locations with sea level as a reference point. In this case, the measurement tool will be a vertical number line where sea level is at zero. All locations above sea level will have positive elevation values and all locations below sea level will have negative elevation values.

To find the difference in elevation between two locations, students will have to add positive and negative numbers.

For example, to find the difference in elevation between Denver, Colorado (5,690 feet elevation) and New Orleans, Louisiana (-3 feet elevation), students would have to perform the following calculation:

  • 5,690 - (-3) = 5,690 + 3 = 5,693 foot difference in elevation

Why is Denver, Colorado called the Mile High City?

Photo by Nils Huenerfuerst on Unsplash

Why is New Orleans at such high risk of flooding?

Photo by Stella He on Unsplash

The lesson is interactive and calls on kids to use Google Maps to research their own elevation and figure out how many feet higher or lower they are than my location in Rocky Mountain National Park in Colorado, which is where the video that corresponds with this lesson was filmed!

The lesson also has real world extension questions related to science, geology, and climate change.

Click the video link below to watch the free What is Your Elevation video guide for students and click here to download the free pdf student guide that accompanies the video lesson.

 
 
 
 

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Combining Like Terms Explained—Examples, Worksheet Included

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Combining Like Terms Explained—Examples, Worksheet Included

Combining Like Terms Explained

How to Combine Like Terms Quickly and Easily (Worksheet Included)

 

Free Step-by-Step Guide: Combining Like Terms

 

Combining like terms is an important math skill that involves simplifying an expression by combining terms that have the same variables and/or exponents. You combine like terms when you want to make an expression simpler and easier to read and solve.

This free Step-by-Step Guide to Combining Like Terms will walk you through how to quickly and easily combine like terms and it includes examples and a free practice worksheet. This guide will cover the following topics:

Before we dive into how to combine like terms, lets do a quick review of some important math vocabulary related to combining like terms (this quick recap will help you to better understand the examples in this guide).

What are Terms in Math?

In math, a term is one single number or algebraic expression that is part of a larger expression or equation.

For example, the mathematical expression 6x + 8y - 4z + 5 has four terms:

  • 6x

  • 8y

  • -4z

  • 5

 

What are Terms in Math?

 

Each of the first three terms consists of a coefficient (a number in front of a variable) and variable letter (x, y, or z in this example). The fourth term is called a constant because it is a single number without a variable.

It is also important to note that a term can have multiple variables as well. For example, 6xy or 10xyz could be terms (more on this later on).

What are terms in math? The important thing to understand right now is that each individual part of an expression (separated by mathematical operation signs like + or -) is a term and, when terms are like, they can be combined to create a simpler result that is easier to work with.

Combining Like Terms Definition

Now that you understand what are terms in math, you are ready to learn how to combine like terms.

Combining Like Terms Definition: Combining like terms is the act of simplifying an algebraic expression by either adding or subtracting terms that have the same variables and/or exponents.

Before we look at a simple numerical example, lets take a look at the fruit expression in Figure 01 below.

 

Figure 01: The fruit expression has three terms: pomegranates, avocados, and lemons.

 

The fruit expression has three terms: pomegranates, avocados, and lemons.

You can read this expression verbally as: 14 pomegranates plus 8 lemons plus 5 avocados minus 6 pomegranates plus 4 avocados minus 2 lemons.

By looking at this expression, it should be clear that you can make it simpler by combining like fruits. We can easily do this by color coding the terms as follows:

  • Highlight pomegranates in pink

  • Highlight lemons in yellow

  • Highlight avocados in green

Now, you can make the following combinations of like fruits:

  • Pomegranates: 14 + - 6 = 8 pomegranates*

  • Lemons: 8 + -2 = 6 lemons

  • Avocados: 5 + 4 = 9 avocados

*Note that you simplify an expression like 8+-2 as 8-2.

So, after combining like terms, the new expression would be: 8 pomegranates plus 6 lemons plus 9 avocados

Figure 02 below illustrates how we just combined like terms (fruit) to simplify a complicated expression by using color-coding.

 

Figure 02: Combine Like Terms Using Color Coding

 

How to Combine Like Terms

The fruit expression is a good first introduction to combining like terms. Obviously, in math, you won’t be dealing with fruit, but variables like x, y, and z instead.

For example, consider the expression 3x + 2y + 5x -3z + 7y + z

To combine like terms, we can color coordinate the x-terms, y-terms, and z-terms and then combine them together group by group as follows:

  • x-terms: 3x + 5x = 8x

  • y-terms: 2y + 7y = 9y

  • z-terms: -3z + z = -2z

After combining like terms, the new expression would be: 8x +9y - 2z.

This example is illustrated in Figure 03 below.

 

Figure 03: How to combine like terms.

 

Now, lets work through 3 more step-by-step examples of combining like terms.

Combining Like Terms Examples

Example #1: Combine Like Terms 5x + 9 + 2x

For this first example, you have to combine like terms 5x + 9 + 2x

This expression has 2 terms. To combine like terms, we can color coordinate the x-terms and constant terms, and then combine them together group by group as follows:

  • x-terms: 5x + 2x = 7x

  • constant terms: 9

After combining like terms, the new expression would be: 7x + 9

This example is illustrated in Figure 04 below.

 

Figure 04: Combine Like terms 5x + 9 + 2x

 

Example #2: Combine Like Terms: 7q + 5r -4 +3s -3q +5r - 3

This expression has 4 terms. To combine like terms, we can color coordinate the q-terms, r-terms, s-terms, and constant terms, and then combine them together group by group as follows:

  • q-terms: 7q + (-3q) = 4q

  • r-terms: 5r + 5r=10r

  • s-terms: 3s

  • constant terms: -4 + (-3) = -7

Notice that there is only one s term, so you can not combine it with another like term.

After combining like terms, the new expression would be: 4q+10r+3s-7

This example is illustrated in Figure 05 below.

 

Figure 05: How to Combine Like Terms

 

Example #3: Combine Like Terms: -5xy^2 +2x^2 + 4xy^2 - 2x^2 - y

This expression has 3 terms. To combine like terms, we can color coordinate the xy^2-terms, x^2 terms, and y-terms, and then combine them together group by group as follows:

  • xy^2-terms: -5xy^2 + 4xy^2 = -1xy^2

  • x^2-terms: 2x^2 + (-2x^2) = 0 (they cancel each other out)

  • y-terms: -y

After combining like terms, the new expression would be: -xy^2 - y

This example is illustrated in Figure 06 below.

 

Figure 06: Combining Like Terms

 

Free Combining Like Terms Worksheet

Combining Like Terms Worksheet Preview

Are you looking for more practice with combining like terms?

Click the link below to download our free Combining Like Terms worksheet as a pdf file (full answer key included). We highly recommend using colored pencils or highlighters to color code the terms in each example.

→ Download your free Combining Like Terms Worksheet PDF File (with Answer Key)

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Is a Rhombus a Parallelogram? Yes or No?

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Is a Rhombus a Parallelogram? Yes or No?

Is a Rhombus a Parallelogram? Yes or No?

Explore the unique features of a rhombus and a parallelogram to determine whether or not a rhombus is both

 

Is a rhombus a parallelogram? Is a parallelogram a rhombus? Let’s explore!

 

It’s time to explore and answer the controversial math question: Is a rhombus a parallelogram? Yes or No? This short guide will help you to understand the following concepts:

What is a Rhombus?

Why was the geometry student late for school? Because he took the rhombus!

Okay, jokes aside, let’s quickly explore the features and characteristics of a rhombus and a parallelogram.

In geometry, a rhombus is a four-sided figure (quadrilateral) where all four sides have the same length and opposite sides are parallel to each other. Additionally, the opposite interior angles of any rhombus are equal to each other (congruent).

 

Is a rhombus a parallelogram? Start by knowing exactly what a rhombus is.

 

What is a Parallelogram?

In geometry, a parallelogram is a four-sided figure (quadrilateral) where opposite sides have the same length and are parallel to each other, but all four sides do not have to be equal. Similar to a rhombus, the opposite interior angles of any parallelogram are congruent.

The key feature of a parallelogram is that the opposite sides are both equal in length and parallel to each other—hence the name parallelogram.

 

Is a rhombus a parallelogram? Continue by learning the definition of a parallelogram.

 

What is the Difference Between a Rhombus and a Parallelogram?

Now that you know the key characteristics of a rhombus and a parallelogram, let’s explore what sets them apart.

For starters, a rhombus and a parallelogram are similar in that they are both quadrilaterals, meaning that they are four sides figures. They also both have opposite sides that are equal in length to each other and they both have sets of opposite interior angles that are congruent to each other.

However, the key differences between a rhombus and a parallelogram are that a parallelogram has two sets of parallel sides that are equal in length, while a rhombus has four sides that are all equal in length.

So, a parallelogram is like a rectangle that doesn’t have right angles, while a rhombus is like a square that doesn’t have right angles.

 

What is the Difference Between a Rhombus and a Parallelogram?

 

The key takeaway concerning the difference between a rhombus and a parallelogram is that they both have opposite sides that are parallel and congruent (and opposite interior angles that are congruent), but a rhombus has to have four equal sides, but a parallelogram does not.

Is a Rhombus a Parallelogram? Yes or No?

The answer is YES—by definition, a rhombus is a parallelogram.

Again, by definition, a parallelogram is quadrilateral with two pairs of parallel sides with opposite interior angles that are congruent. A rhombus is just a special type of parallelogram where all four of the sides are equal in length. As for interior angles, the opposite interior angles of a rhombus are congruent just as they are in a parallelogram.

Is a rhombus always a parallelogram?

To say that every rhombus is a parallelogram would be true.

Since the properties of a rhombus satisfy the definition of a parallelogram and we can conclude that:

  • Is a rhombus always a parallelogram? Yes!

  • A rhombus is a special type of parallelogram—namely, one where all four sides are equal in length.

  • All rhombuses are parallelograms.

 

Is a rhombus always a parallelogram? Yes, a rhombus is a special type of parallelogram—namely, one where all four sides are equal in length.

 

It is important to note that the conclusion that we just made is not an opinion, but a mathematical fact.

All rhombuses are parallelograms. In fact, a rhombus is a subset of a parallelogram in that it is special because it is a parallelogram with four congruent sides.

This explanation is very similar to our exploration of the question is a square a rectangle?, where we similarly concluded that a square is a special type of rectangle and that all squares are rectangles.

Next, let’s consider if the opposite is true: is a parallelogram a rhombus?

 

Is a Parallelogram a Rhombus? No. All Oreos are cookies, but not all cookies are Oreos (because Oreos are a special type of cookie). Similarly, every rhombus is a parallelogram, but every parallelogram is not necessarily a rhombus.

 

Is a Parallelogram a Rhombus? Yes or No?

We have just applied the mathematical definition of a rhombus and parallelogram to prove that a rhombus is indeed a parallelogram. We concluded that a rhombus is a special type of parallelogram that has all of the characteristics of a parallelogram and four congruent sides.

But, is a parallelogram a rhombus?

No. By definition, a parallelogram does not necessarily have to be a rhombus.

Remember that a parallelogram is a quadrilateral with two pairs of sides that have to be parallel and equal in length, but it does not need to have all four sides be congruent.

Again, a rhombus is like a square and a parallelogram is like a rectangle. Similarly, all squares are rectangles, but not all rectangles are squares.

To say that every parallelogram is a rhombus would be false.

Therefore, while every rhombus is a parallelogram, not all parallelograms are rhombuses.

Conclusion: Is a Rhombus a Parallelogram?

In this short lesson, you learned that both a rhombus and a parallelogram are quadrilaterals with opposite sides that are equal in length and opposite interior angles that are congruent. You also learned that a rhombus is a special type of parallelogram—namely one that has four congruent sides.

As for the question, is a rhombus a parallelogram? The answer is yes! All rhombuses are parallelograms, but not all parallelograms are rhombuses.

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