%5E6-binomial-expansion.jpeg "(x+2)^6 Binomial Expansion")
Understanding the Binomial Theorem and Expanding (x+2)^6
The Binomial Theorem provides a formula to expand expressions of the form (x + y)^n, where n is a positive integer. This theorem is fundamental in algebra and has applications in various fields like probability, statistics, and calculus.
The Binomial Theorem
The Binomial Theorem states that:
(x + y)^n = ∑(k=0 to n) [nCk * x^(n-k) * y^k]
where:
- nCk represents the binomial coefficient, calculated as n! / (k! * (n-k)!). It indicates the number of ways to choose k items from a set of n items.
- ∑(k=0 to n) denotes the sum from k = 0 to k = n.
Expanding (x+2)^6
To expand (x+2)^6, we need to apply the Binomial Theorem with n = 6, x = x, and y = 2.
Let's break it down step by step:
-
Identify the binomial coefficients:
- 6C0 = 1
- 6C1 = 6
- 6C2 = 15
- 6C3 = 20
- 6C4 = 15
- 6C5 = 6
- 6C6 = 1
-
Apply the formula:
(x + 2)^6 = 1 * x^6 * 2^0 + 6 * x^5 * 2^1 + 15 * x^4 * 2^2 + 20 * x^3 * 2^3 + 15 * x^2 * 2^4 + 6 * x^1 * 2^5 + 1 * x^0 * 2^6
-
Simplify the terms:
(x + 2)^6 = x^6 + 12x^5 + 60x^4 + 160x^3 + 240x^2 + 192x + 64
Therefore, the expansion of (x + 2)^6 is x^6 + 12x^5 + 60x^4 + 160x^3 + 240x^2 + 192x + 64.
Key Observations
- Symmetry: Notice the symmetry in the coefficients. The first and last coefficients are 1, the second and second-to-last are 6, and so on.
- Increasing and Decreasing Powers: The powers of x decrease from 6 to 0, while the powers of 2 increase from 0 to 6.
Understanding the Binomial Theorem and its applications provides a powerful tool for simplifying complex algebraic expressions and solving various mathematical problems.