## 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.