(1+3x)^8 Binomial Expansion

3 min read Jun 16, 2024

Binomial Expansion of (1 + 3x)^8

The binomial theorem provides a formula for expanding expressions of the form (a + b)^n. In this case, we want to expand (1 + 3x)^8.

The binomial theorem states:

(a + b)^n = ∑_(k=0)^n (n_C_k) * a^(n-k) * b^k

where:

n is a non-negative integer (the power)
k is an integer from 0 to n
(n_C_k) is the binomial coefficient, calculated as n! / (k! * (n-k)!), which represents the number of ways to choose k items from a set of n items.

Let's apply the binomial theorem to our expression (1 + 3x)^8:

Identify a and b: In our case, a = 1 and b = 3x.
Calculate the binomial coefficients: We need to calculate the binomial coefficients for n = 8 and k ranging from 0 to 8. Here are the values:
- (8_C_0) = 1
- (8_C_1) = 8
- (8_C_2) = 28
- (8_C_3) = 56
- (8_C_4) = 70
- (8_C_5) = 56
- (8_C_6) = 28
- (8_C_7) = 8
- (8_C_8) = 1
Substitute and expand: Now we can substitute these values into the binomial theorem:

(1 + 3x)^8 = (8_C_0) * 1^8 * (3x)^0 + (8_C_1) * 1^7 * (3x)^1 + (8_C_2) * 1^6 * (3x)^2 + ... + (8_C_8) * 1^0 * (3x)^8

Simplifying this, we get:

(1 + 3x)^8 = 1 + 24x + 252x^2 + 1512x^3 + 5670x^4 + 13608x^5 + 20412x^6 + 17496x^7 + 6561x^8

Therefore, the binomial expansion of (1 + 3x)^8 is:

(1 + 3x)^8 = 1 + 24x + 252x^2 + 1512x^3 + 5670x^4 + 13608x^5 + 20412x^6 + 17496x^7 + 6561x^8

This process can be used to expand any binomial expression raised to a non-negative integer power.