%5E3-formula-example.jpeg "(a+b)^3 Formula Example")
Understanding the (a + b)³ Formula
The formula (a + b)³ is a fundamental concept in algebra, used to expand the cube of a binomial expression. Understanding this formula is crucial for solving various algebraic equations and simplifying expressions.
The Formula
The formula for (a + b)³ is:
(a + b)³ = a³ + 3a²b + 3ab² + b³
Breaking Down the Formula
This formula expands the expression (a + b)³ into a sum of four terms:
- a³: The cube of the first term 'a'
- 3a²b: Three times the square of the first term 'a' multiplied by the second term 'b'
- 3ab²: Three times the first term 'a' multiplied by the square of the second term 'b'
- b³: The cube of the second term 'b'
Example: Expanding (x + 2)³
Let's apply the formula to expand (x + 2)³:
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Identify 'a' and 'b': In this case, a = x and b = 2.
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Substitute into the formula: (x + 2)³ = x³ + 3(x²)(2) + 3(x)(2²) + 2³
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Simplify: (x + 2)³ = x³ + 6x² + 12x + 8
Therefore, the expanded form of (x + 2)³ is x³ + 6x² + 12x + 8.
Applications of the Formula
The (a + b)³ formula is widely used in various mathematical applications, including:
- Solving equations: This formula can be used to simplify equations involving binomials raised to the power of three.
- Factorization: The formula can be used to factorize expressions of the form a³ + 3a²b + 3ab² + b³.
- Calculus: The formula finds applications in finding derivatives and integrals of expressions involving binomials raised to the power of three.
Understanding and applying the (a + b)³ formula is a fundamental skill in algebra. It simplifies complex expressions, enables solving equations, and serves as a foundation for more advanced mathematical concepts.