%5En-expansion-formula.jpeg "(a+b)^n Expansion Formula")
The Binomial Theorem: Expanding (a + b)^n
The Binomial Theorem is a powerful tool in algebra that allows us to expand expressions of the form (a + b)^n for any positive integer n. This theorem provides a systematic way to determine the coefficients of each term in the expanded form.
Understanding the Theorem
The Binomial Theorem states:
(a + b)^n = a^n + ¹C₁a^(n-1)b + ²C₂a^(n-2)b² + ... + ⁿCₙb^n
Where:
- n is a positive integer representing the power to which the binomial is raised.
- ¹C₁, ²C₂, ... , ⁿCₙ are the binomial coefficients, which represent the number of ways to choose k objects from a set of n objects (also known as combinations).
- a and b are any real numbers.
The Binomial Coefficients
The binomial coefficients are calculated using the following formula:
ⁿCₖ = n! / (k! * (n-k)!)
Where:
- n! represents the factorial of n, which is the product of all positive integers less than or equal to n.
For example: 5! = 5 * 4 * 3 * 2 * 1 = 120
Using the Binomial Theorem
Let's expand (x + y)^4 using the Binomial Theorem:
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Identify n: n = 4
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Calculate the binomial coefficients:
- ⁴C₀ = 4! / (0! * 4!) = 1
- ⁴C₁ = 4! / (1! * 3!) = 4
- ⁴C₂ = 4! / (2! * 2!) = 6
- ⁴C₃ = 4! / (3! * 1!) = 4
- ⁴C₄ = 4! / (4! * 0!) = 1
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Substitute into the formula: (x + y)^4 = 1x⁴ + 4x³y + 6x²y² + 4xy³ + 1*y⁴
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Simplify: (x + y)^4 = x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴
Key Observations
- Pascal's Triangle: The binomial coefficients can be easily visualized using Pascal's Triangle, where each number is the sum of the two numbers directly above it.
- Symmetry: The binomial coefficients are symmetric, meaning ⁿCₖ = ⁿC(n-k).
- Applications: The Binomial Theorem has wide applications in various fields, including probability, statistics, calculus, and computer science.
Conclusion
The Binomial Theorem is a powerful tool for expanding binomials raised to any positive integer power. It provides a systematic way to determine the coefficients of each term, simplifying the process of expanding complex expressions. Understanding this theorem opens up a world of possibilities in algebraic manipulations and various applications in mathematics and beyond.