%5E3-formula-examples.jpeg "(a+b)^3 Formula Examples")
Understanding the (a + b)³ Formula
The formula (a + b)³ = a³ + 3a²b + 3ab² + b³ is a fundamental concept in algebra, particularly when working with binomial expansions. This formula helps us to expand the cube of a binomial expression (a + b) without needing to perform the multiplication directly.
How to Derive the Formula
The formula can be derived by expanding the expression (a + b)³ using the distributive property:
(a + b)³ = (a + b)(a + b)(a + b)
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Expand the first two brackets: (a + b)(a + b) = a² + 2ab + b²
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Multiply the result by (a + b): (a² + 2ab + b²)(a + b) = a³ + 2a²b + ab² + a²b + 2ab² + b³
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Combine like terms: a³ + 3a²b + 3ab² + b³
Examples of Using the (a + b)³ Formula
Let's look at some examples of how to use the formula:
Example 1:
(x + 2)³
Using the formula, we get:
x³ + 3(x²)(2) + 3(x)(2²) + 2³ = x³ + 6x² + 12x + 8
Example 2:
(2y - 3)³
We can apply the formula by treating (2y) as 'a' and -3 as 'b':
(2y)³ + 3(2y)²(-3) + 3(2y)(-3)² + (-3)³ = 8y³ - 36y² + 54y - 27
Example 3:
(a + 2b)³
Following the same pattern:
a³ + 3(a²)(2b) + 3(a)(2b)² + (2b)³ = a³ + 6a²b + 12ab² + 8b³
Key Points to Remember
- The formula applies to binomials (expressions with two terms) only.
- The coefficients in the expanded expression follow Pascal's Triangle.
- The formula can be used to simplify complex expressions and solve problems involving cubic equations.
By understanding and applying the (a + b)³ formula, you can easily expand cubic expressions, simplify algebraic expressions, and develop a deeper understanding of algebraic concepts.