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Welcome to our Math lesson on Cube of a Sum, this is the fifth 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 Sum
Again, we can apply the expanding rules we discussed in the previous tutorial to find an expanded expression of cube of a sum. 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 fourth special algebraic identity
(a + b)3 = a3 + 3a2 b + 3ab2 + b3
Example 4
Expand using the shortest method:
- (2x + 1)3
- (x + 6y)3
Solution 4
Using the shortest method means applying the formula of the cube of a sum, as it gives the result directly without the need for simplifications.
- We have a = 2x and b = 1. Thus,
(2x - 1)3 = (2x)3 + 3 ∙ (2x)2 ∙ 1 + 3 ∙ (2x) ∙ 12 + 13
= 23 ∙ x3 + 3 ∙ 22 ∙ x2 ∙ 1 + 3 ∙ 2x ∙ 12 + 13
= 8x3 + 12x2 + 6x + 1
- We have a = x and b = 6y. Thus,
(x + 6y)3 = x3 + 3 ∙ x2 ∙ (6y) + 3 ∙ x ∙ (6y)2 + (6y)3
= x3 + 3 ∙ x2 ∙ (6y) + 3 ∙ x ∙ 62 ∙ y2 + 63 ∙ y3
= x3 + 18x2 y + 108xy2 + 216y3
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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