(x+2)^4 Binomial Expansion

3 min read Jun 16, 2024

Understanding the Binomial Expansion of (x + 2)^4

The binomial theorem provides a powerful tool to expand expressions of the form (x + y)^n. In this case, we'll explore the expansion of (x + 2)^4.

The binomial theorem states:

(x + y)^n = ∑_(k=0)^n (n_C_k) x^(n-k) y^k

Where:

n is a non-negative integer (the power)
k is an integer ranging from 0 to n
n_C_k represents the binomial coefficient, calculated as n! / (k! * (n-k)!), also known as "n choose k"

Let's apply the theorem to expand (x + 2)^4:

Identify n: In this case, n = 4.
Expand the summation: We need to expand the sum for k = 0, 1, 2, 3, and 4.
Calculate binomial coefficients:
- 4_C_0 = 4! / (0! * 4!) = 1
- 4_C_1 = 4! / (1! * 3!) = 4
- 4_C_2 = 4! / (2! * 2!) = 6
- 4_C_3 = 4! / (3! * 1!) = 4
- 4_C_4 = 4! / (4! * 0!) = 1
Substitute and simplify:
- (x + 2)^4 = (4_C_0) x^(4-0) 2^0 + (4_C_1) x^(4-1) 2^1 + (4_C_2) x^(4-2) 2^2 + (4_C_3) x^(4-3) 2^3 + (4_C_4) x^(4-4) 2^4
- (x + 2)^4 = 1x^4 + 4x^3 * 2 + 6x^2 * 4 + 4x * 8 + 1 * 16
Final Expansion:
- (x + 2)^4 = x^4 + 8x^3 + 24x^2 + 32x + 16

The binomial theorem provides a systematic way to expand expressions of the form (x + y)^n.
The coefficients in the expansion are represented by binomial coefficients.
The expansion of (x + 2)^4 results in a polynomial with five terms, each with a distinct power of x.

By understanding the binomial theorem and its application, you can efficiently expand any binomial expression raised to a positive integer power.