Algebra 221 views·Updated Jul 8, 2026·2 pages

Understanding the Sum and Difference of Cubes

Chelsea@zuncho

Add to Google sources

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

of 2

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: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

For the sum of cubes, the pattern is slightly different: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

Let's see this in action! When factoring $x^3 - 27$ , identify it as $x^3 - 3^3$ . Using the difference of cubes formula, this becomes $(x-3)(x^2+3x+9)$ . 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 $2a^4 + 16a$ , first factor out $2a$ to get $2a(a^3+8)$ . Then the expression inside parentheses is a sum of cubes that factors as $2a(a+2)(a^2-2a+4)$ .

of 2

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 $(a+b)(a-b)$ or as a perfect square minus another term. For example, with $x^2+6x+9-y^2$ , recognize that the first three terms form a perfect square trinomial: $(x+3)^2-y^2$ . This becomes $(x+3+y)(x+3-y)$ .

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

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

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

We thought you’d never ask...

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.

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

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

Algebra 2

Properties of Real Numbers

Notes about the topic

1412

Algebra 2

1.2 - Intro to Sets

Sets, subsets, set builder notation

10th372

Algebra 2

Special Right Triangles

Examples of two of the most common special right triangles.

11th2002

Algebra 1

Linear Function: Solving Linear Equations in 1 Variable

Study note on linear equations with one variable, rules on linear equations, examples on simple linear equations, algebra 2, algebra 1

9th3341

Algebra 2

Midterm Study Guide: Review of the First Half of the Course

Simple review notes and examples for the first half of the algebra 2 course! Not all classes teach the content in the same order, but this study guide should have most of the more basic concepts from algebra 2!

9th1,207120

Algebra 2

Rational Exponents

Writing rational exponents in radical form and exponential form. And even and odd radicals

1535

Algebra 2

math 3 final exam study guide

covers all content learned in tj math 3 (algebra 2)

10th1,660180

Algebra 2

Absolute value functions

Steps to to solve absolute value functions

980

Algebra 2

Transfermations - Quadratic Functions

Learn how to identify the vertex, direction, and maximum or minimum for different parabolas in vertex form.

10th640

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

Algebra 221 views·Updated Jul 8, 2026·2 pages

Understanding the Sum and Difference of Cubes

Chelsea@zuncho

Add to Google sources

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.

of 2

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: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

For the sum of cubes, the pattern is slightly different: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

💡 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!

of 2

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.

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

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

We thought you’d never ask...

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

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

Students love us — and so will you.

4.6/5App Store

4.7/5Google Play

Stefan SiOS user

Samantha KlichAndroid user

AnnaiOS user

Sum and Difference of Cubes

Factoring by Grouping Terms

We thought you’d never ask...

What is the Knowunity AI companion?

Where can I download the Knowunity app?

Is Knowunity really free of charge?

Similar Content

Triangle Congruence

6.5 and 6.6 Proportionality

Proving Triangles Similar: SSS~ and SAS~

Finding Coterminal Angles

Introduction to Conic Sections

Triangle Congruencee by SSS and SAS