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Welcome to our Math lesson on Combining Special Identities, this is the tenth 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.
Combining Special Identities
Certain algebraic expressions may contain more than one special identity, so it is appropriate to know how to write them in the correct way so that simplifications are fast and easy. Let's consider an example where more than one special identity is present.
Example 9
Simplify the expression
(x - 1)(x + 1) - (x - 1)2 - 2x + (x - 2)(x2 + 2x + 4) + 8
Solution 9
In this example there are three special identities involved. From left to right there are the third, the second and the sixth identies respectively in the same algebraic expression. Thus, we have
(x - 1)(x + 1) - (x - 1)2 - 2x + (x - 2)(x2 + 2x + 4) + 8
= (x2 - 12 )-(x2 - 2 ∙ x ∙ 1 + 12 ) - 2x + (x3 - 23 ) + 8
= (x2 - 1) - (x2 - 2x + 1) - 2x + (x3 - 8) + 8
= x2 - 1 - x2 + 2x - 1 - 2x + x3 - 8 + 8
= x3 - 2
A particular Example:
There are situations where the information is partially evident (is given indirectly). In these scenarios we have to combine two or more special identities to find the result. Let's consider an example.
Example 10
The sum of two numbers x and y is 8 and the sum of their squares is 48. What is the product of x and y?
Solution 10
We have
(x + y) = 8
Raising this expression in square, yields
(x + y)2 = 82
x2 + 2xy + y2 = 64
x2 + y2 + 2xy = 64
From the clues, it is known that
x2 + y2 = 48
Hence,
48 + 2xy = 64
2xy = 64 - 48
2xy = 16
xy = 8
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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