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Algebra 2Algebra 219 views·Updated May 27, 2026·2 pages

Understanding the Sum and Difference of Cubes

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Chelsea@zuncho

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

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

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 $64a^6 - b^6$, 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 $2a^4 + 16a,firstfactorout, first factor out 2atoget to get 2aa3+8a^3+8.Thentheexpressioninsideparenthesesisasumofcubesthatfactorsas. Then the expression inside parentheses is a sum of cubes that factors as 2aa+2a+2a2−2a+4a^2-2a+4$.

2
of 2
480/81 Day 5 Notes ## 7.4 NOTES - SUM AND DIFFERENCE OF CUBES **OBJECTIVES:** 1) Factor the sum and difference of cubes. | DIFFERENCE OF

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 $81-x^2+2xy-y^2,groupthelastthreeterms:, group the last three terms: 81-x2−2xy+y2x^2-2xy+y^2,whichsimplifiesto, which simplifies to 81-x−yx-y^2.Thisisadifferenceofsquaresthatfactorsas. This is a difference of squares that factors as 9+x−y9+x-y9−x+y9-x+y$.

🔍 Strategy Alert: When factoring expressions with four terms, try grouping them into pairs. For example, in $7a^2x-6a^2-7x+6,grouptermswithcommonfactorstoget, group terms with common factors to get a^27x−67x-6-17x−67x-6,whichequals, which equals a2−1a^2-17x−67x-6$.

For trickier expressions like $49y^2-x^2-10x-25,recognizethatthelastthreetermsform, recognize that the last three terms form x+5x+5^2withanegativesign.Rearranginggives with a negative sign. Rearranging gives 49y^2-x+5x+5^2,whichfactorsas, which factors as 7y+x+57y+x+57y−x−57y-x-5$.

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Algebra 2Algebra 219 views·Updated May 27, 2026·2 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.

1
of 2
480/81 Day 5 Notes ## 7.4 NOTES - SUM AND DIFFERENCE OF CUBES **OBJECTIVES:** 1) Factor the sum and difference of cubes. | DIFFERENCE OF

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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 $64a^6 - b^6$, 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 $2a^4 + 16a,firstfactorout, first factor out 2atoget to get 2aa3+8a^3+8.Thentheexpressioninsideparenthesesisasumofcubesthatfactorsas. Then the expression inside parentheses is a sum of cubes that factors as 2aa+2a+2a2−2a+4a^2-2a+4$.

2
of 2
480/81 Day 5 Notes ## 7.4 NOTES - SUM AND DIFFERENCE OF CUBES **OBJECTIVES:** 1) Factor the sum and difference of cubes. | DIFFERENCE OF

Sign up to see the content. It's free!

  • Access to all documents
  • Improve your grades
  • Join milions of students

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 $81-x^2+2xy-y^2,groupthelastthreeterms:, group the last three terms: 81-x2−2xy+y2x^2-2xy+y^2,whichsimplifiesto, which simplifies to 81-x−yx-y^2.Thisisadifferenceofsquaresthatfactorsas. This is a difference of squares that factors as 9+x−y9+x-y9−x+y9-x+y$.

🔍 Strategy Alert: When factoring expressions with four terms, try grouping them into pairs. For example, in $7a^2x-6a^2-7x+6,grouptermswithcommonfactorstoget, group terms with common factors to get a^27x−67x-6-17x−67x-6,whichequals, which equals a2−1a^2-17x−67x-6$.

For trickier expressions like $49y^2-x^2-10x-25,recognizethatthelastthreetermsform, recognize that the last three terms form x+5x+5^2withanegativesign.Rearranginggives with a negative sign. Rearranging gives 49y^2-x+5x+5^2,whichfactorsas, which factors as 7y+x+57y+x+57y−x−57y-x-5$.

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.

Most popular content in Algebra 2

9

Most popular content

9

Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.6/5App Store
4.7/5Google 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 SiOS 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 KlichAndroid 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.

AnnaiOS user