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Welcome to our Math lesson on Cube of a Difference, this is the sixth lesson of our suite of math lessons covering the topic of Special Algebraic Identities Obtained through Expanding, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
Cube of a Difference
The cube of a difference is handled in a similar way to the cube of sum. Just be careful with the signs of each term and ensure they are handled in accordance with the sign rule.
We have
(a - b)3 = (a-b) ∙ (a - b)2
= (a - b) ∙ (a2 - 2ab + b2 )
= a ∙ a2 + a ∙ (-2ab) + a ∙ b2 - b ∙ a2 - b ∙ (-2ab) - b ∙ b2
= a3 - 2a2 b + ab2 - a2 b + 2ab2 - b3
= a3 - 3a2 b + 3ab2 - b3
In this way, we obtained the fifth special algebraic identity
(a - b)3 = a3 - 3a2 b + 3ab2 - b3
Example 5
Expand using the shortest method:
- (3x - 2)3
- (1 - 2x)3
Solution 5
Again, the shortest method is the direct use of the cube of difference formula.
- We have a = 3x and b = 2. Thus,
(3x-2)3 = (3x)3 - 3 ∙ (3x)2 ∙ 2 + 3 ∙ 3x ∙ 22 + 23
= 27x3 - 3 ∙ 9x2 ∙ 2 + 9x ∙ 4 + 8
= 27x3 - 54x2 + 36x + 8
We have a = 1 and b = 2x. Thus, (1 - 2x)3 = 13 - 3 ∙ 12 ∙ 2x + 3 ∙ 1 ∙ (2x)2 + (2x)3
= 1 - 3 ∙ 1 ∙ 2x + 3 ∙ 1 ∙ 4x2 + 8x3
= 1 - 6x + 12x2 + 8x3
More Special Algebraic Identities Obtained through Expanding Lessons and Learning Resources
Expressions Learning Material| Tutorial ID | Math Tutorial Title | Tutorial | Video
Tutorial | Revision
Notes | Revision
Questions |
|---|
| 6.3 | Special Algebraic Identities Obtained through Expanding | | | | |
| Lesson ID | Math Lesson Title | Lesson | Video
Lesson |
|---|
| 6.3.1 | The Meaning of a Binomial | | |
| 6.3.2 | Square of a Sum | | |
| 6.3.3 | Square of a Difference | | |
| 6.3.4 | Product of Conjugates | | |
| 6.3.5 | Cube of a Sum | | |
| 6.3.6 | Cube of a Difference | | |
| 6.3.7 | Expanding the Algebraic Expression of the Form (a - b) · (a2 + ab + b2) | | |
| 6.3.8 | Expanding the Algebraic Expression of the Form (a + b) · (a2 - ab + b2) | | |
| 6.3.9 | Expanding Expressions of the Form (a + b + c)2 | | |
| 6.3.10 | Combining Special Identities | | |
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- Continuing learning expressions - read our next math tutorial: Factorising
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