-%5E3.jpeg "(x+1) ^3")
Expanding (x + 1)³
The expression (x + 1)³ represents the cube of the binomial (x + 1). To understand this, we need to expand the expression, which means multiplying it by itself three times:
(x + 1)³ = (x + 1) * (x + 1) * (x + 1)
Expanding the expression
We can expand this using the distributive property (often called FOIL for first, outer, inner, last) multiple times:
- First Expansion: (x + 1) * (x + 1) = x² + 2x + 1
- Second Expansion: (x² + 2x + 1) * (x + 1) = x³ + 2x² + x + x² + 2x + 1
- Simplifying: x³ + 3x² + 3x + 1
Therefore, the expanded form of (x + 1)³ is x³ + 3x² + 3x + 1.
Understanding the Pattern
The coefficients of the expanded form (1, 3, 3, 1) follow a pattern known as Pascal's Triangle. This triangle is a visual representation of binomial coefficients, where each number is the sum of the two numbers directly above it.
Applications
Understanding the expansion of (x + 1)³ has applications in various fields like:
- Algebra: Solving equations, simplifying expressions, and understanding polynomial behavior.
- Calculus: Finding derivatives and integrals of polynomial functions.
- Probability: Calculating probabilities in binomial distributions.
Conclusion
Expanding (x + 1)³ is a fundamental concept in algebra with various applications. By understanding the expansion process and the pattern of coefficients, we can efficiently work with this expression and its applications.