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Understanding the Sum and Difference of Cubes

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Chelsea

11/29/2025

Algebra 2

Sum and Difference of Cubes

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Algebra 2

Factoring

Factoring Using Structure

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Nov 29, 2025

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2 pages

Understanding the Sum and Difference of Cubes

user profile picture

Chelsea

@zuncho

Factoring polynomial expressions is like solving math puzzles that reveal... Show more

480/81 Day 5 Notes 7.4 NOTES - SUM AND DIFFERENCE OF CUBES OBJECTIVES: 1) Factor the sum and difference of cubes. DIFFERENCE OF SQUARES a²-b

Sum and Difference of Cubes

Ever wondered how to break down complex cubic expressions? The key is memorizing two powerful formulas. For the difference of cubes, use: a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)

For the sum of cubes, the pattern is slightly different: a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2)

Let's see this in action! When factoring x3−27x^3 - 27, identify it as x3−33x^3 - 3^3. Using the difference of cubes formula, this becomes (x−3)(x2+3x+9)(x-3)(x^2+3x+9). The first factor is simple, while the second never factors further.

💡 Quick Tip: When factoring expressions like 64a6−b664a^6 - b^6, you can either factor as a difference of cubes first or as a difference of squares. Both approaches work, but choosing wisely can save you time!

Remember that sometimes you'll need to factor out common terms first. For example, with 2a4+16a2a^4 + 16a, first factor out 2a2a to get 2a(a3+8)2a(a^3+8). Then the expression inside parentheses is a sum of cubes that factors as 2a(a+2)(a2−2a+4)2a(a+2)(a^2-2a+4).

480/81 Day 5 Notes 7.4 NOTES - SUM AND DIFFERENCE OF CUBES OBJECTIVES: 1) Factor the sum and difference of cubes. DIFFERENCE OF SQUARES a²-b

Factoring by Grouping Terms

Factoring by grouping is like solving a puzzle by rearranging pieces. This technique is super useful when expressions don't immediately look like standard patterns.

Start by looking for ways to rewrite the expression as a difference of squares (a+b)(a−b)(a+b)(a-b) or as a perfect square minus another term. For example, with x2+6x+9−y2x^2+6x+9-y^2, recognize that the first three terms form a perfect square trinomial: (x+3)2−y2(x+3)^2-y^2. This becomes (x+3+y)(x+3−y)(x+3+y)(x+3-y).

Sometimes you'll need to rearrange terms to spot the pattern. For expression 81−x2+2xy−y281-x^2+2xy-y^2, group the last three terms: 81−(x2−2xy+y2)81-(x^2-2xy+y^2), which simplifies to 81−(x−y)281-(x-y)^2. This is a difference of squares that factors as (9+x−y)(9−x+y)(9+x-y)(9-x+y).

🔍 Strategy Alert: When factoring expressions with four terms, try grouping them into pairs. For example, in 7a2x−6a2−7x+67a^2x-6a^2-7x+6, group terms with common factors to get a2(7x−6)−1(7x−6)a^2(7x-6)-1(7x-6), which equals (a2−1)(7x−6)(a^2-1)(7x-6).

For trickier expressions like 49y2−x2−10x−2549y^2-x^2-10x-25, recognize that the last three terms form (x+5)2(x+5)^2 with a negative sign. Rearranging gives 49y2−(x+5)249y^2-(x+5)^2, which factors as (7y+x+5)(7y−x−5)(7y+x+5)(7y-x-5).



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The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️

Thomas R

iOS user

Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!

Paul T

iOS user

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️

Thomas R

iOS user

Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!

Paul T

iOS user

Subjects

Algebra 2

Factoring

Factoring Using Structure

 

Algebra 2

•

16

•

Nov 29, 2025

•

2 pages

Understanding the Sum and Difference of Cubes

user profile picture

Chelsea

@zuncho

Factoring polynomial expressions is like solving math puzzles that reveal hidden patterns. This guide will help you master factoring the sum and difference of cubes, along with related factoring techniques that make algebra problems much easier to solve.

480/81 Day 5 Notes 7.4 NOTES - SUM AND DIFFERENCE OF CUBES OBJECTIVES: 1) Factor the sum and difference of cubes. DIFFERENCE OF SQUARES a²-b

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Sum and Difference of Cubes

Ever wondered how to break down complex cubic expressions? The key is memorizing two powerful formulas. For the difference of cubes, use: a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)

For the sum of cubes, the pattern is slightly different: a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2)

Let's see this in action! When factoring x3−27x^3 - 27, identify it as x3−33x^3 - 3^3. Using the difference of cubes formula, this becomes (x−3)(x2+3x+9)(x-3)(x^2+3x+9). The first factor is simple, while the second never factors further.

💡 Quick Tip: When factoring expressions like 64a6−b664a^6 - b^6, you can either factor as a difference of cubes first or as a difference of squares. Both approaches work, but choosing wisely can save you time!

Remember that sometimes you'll need to factor out common terms first. For example, with 2a4+16a2a^4 + 16a, first factor out 2a2a to get 2a(a3+8)2a(a^3+8). Then the expression inside parentheses is a sum of cubes that factors as 2a(a+2)(a2−2a+4)2a(a+2)(a^2-2a+4).

480/81 Day 5 Notes 7.4 NOTES - SUM AND DIFFERENCE OF CUBES OBJECTIVES: 1) Factor the sum and difference of cubes. DIFFERENCE OF SQUARES a²-b

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Factoring by Grouping Terms

Factoring by grouping is like solving a puzzle by rearranging pieces. This technique is super useful when expressions don't immediately look like standard patterns.

Start by looking for ways to rewrite the expression as a difference of squares (a+b)(a−b)(a+b)(a-b) or as a perfect square minus another term. For example, with x2+6x+9−y2x^2+6x+9-y^2, recognize that the first three terms form a perfect square trinomial: (x+3)2−y2(x+3)^2-y^2. This becomes (x+3+y)(x+3−y)(x+3+y)(x+3-y).

Sometimes you'll need to rearrange terms to spot the pattern. For expression 81−x2+2xy−y281-x^2+2xy-y^2, group the last three terms: 81−(x2−2xy+y2)81-(x^2-2xy+y^2), which simplifies to 81−(x−y)281-(x-y)^2. This is a difference of squares that factors as (9+x−y)(9−x+y)(9+x-y)(9-x+y).

🔍 Strategy Alert: When factoring expressions with four terms, try grouping them into pairs. For example, in 7a2x−6a2−7x+67a^2x-6a^2-7x+6, group terms with common factors to get a2(7x−6)−1(7x−6)a^2(7x-6)-1(7x-6), which equals (a2−1)(7x−6)(a^2-1)(7x-6).

For trickier expressions like 49y2−x2−10x−2549y^2-x^2-10x-25, recognize that the last three terms form (x+5)2(x+5)^2 with a negative sign. Rearranging gives 49y2−(x+5)249y^2-(x+5)^2, which factors as (7y+x+5)(7y−x−5)(7y+x+5)(7y-x-5).

We thought you’d never ask...

What is the Knowunity AI companion?

Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.

Where can I download the Knowunity app?

You can download the app in the Google Play Store and in the Apple App Store.

Is Knowunity really free of charge?

That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.

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Algebra 2Algebra 2

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Students love us — and so will you.

4.9/5

App Store

4.8/5

Google Play

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️

Thomas R

iOS user

Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!

Paul T

iOS user

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️

Thomas R

iOS user

Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!

Paul T

iOS user