%5E5-binomial-expansion.jpeg "(x+2)^5 Binomial Expansion")
Understanding the Binomial Expansion of (x+2)^5
The binomial theorem provides a powerful tool for expanding expressions of the form (x + y)^n. In this case, we are interested in finding the expansion of (x + 2)^5.
The Binomial Theorem
The binomial theorem states that:
(x + y)^n = ∑_(k=0)^n (n choose k) * x^(n-k) * y^k
Where:
- (n choose k) represents the binomial coefficient, calculated as n! / (k! * (n-k)!). This is often written as nCk or (n_C_k).
- ∑_(k=0)^n denotes the sum from k = 0 to k = n.
Applying the Theorem to (x+2)^5
Let's apply the binomial theorem to our specific case of (x + 2)^5:
- Identify n and y: In our expression, n = 5 and y = 2.
- Calculate the binomial coefficients: We need to calculate (5 choose k) for k = 0, 1, 2, 3, 4, and 5. These are:
- (5 choose 0) = 1
- (5 choose 1) = 5
- (5 choose 2) = 10
- (5 choose 3) = 10
- (5 choose 4) = 5
- (5 choose 5) = 1
- Apply the formula: Substitute the values into the binomial theorem formula:
(x + 2)^5 = (5 choose 0) * x^5 * 2^0 + (5 choose 1) * x^4 * 2^1 + (5 choose 2) * x^3 * 2^2 + (5 choose 3) * x^2 * 2^3 + (5 choose 4) * x^1 * 2^4 + (5 choose 5) * x^0 * 2^5
- Simplify: After simplifying the terms, we get:
(x + 2)^5 = x^5 + 10x^4 + 40x^3 + 80x^2 + 80x + 32
Understanding the Expansion
The expansion of (x + 2)^5 shows the distribution of terms when multiplying (x + 2) by itself five times. Each term represents a different combination of x and 2 chosen from the five factors. The coefficients reflect the number of ways to obtain each combination.
Conclusion
Using the binomial theorem, we have successfully expanded (x + 2)^5 to obtain the expression: x^5 + 10x^4 + 40x^3 + 80x^2 + 80x + 32. This method provides a straightforward approach to expanding binomial expressions to any power.