## Expanding (a + 3b)^4 using the Binomial Theorem

The binomial theorem is a powerful tool for expanding expressions of the form (x + y)^n. It states:

**(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)!).

Let's apply this theorem to expand (a + 3b)^4:

**1. Identify x, y, and n:**

- x = a
- y = 3b
- n = 4

**2. Apply the binomial theorem:**

(a + 3b)^4 = ∑_(k=0)^4 (4 choose k) a^(4-k) (3b)^k

**3. Expand the sum:**

This means we need to calculate the terms for k = 0, 1, 2, 3, and 4:

**k = 0:**(4 choose 0) a^4 (3b)^0 = 1 * a^4 * 1 =**a^4****k = 1:**(4 choose 1) a^3 (3b)^1 = 4 * a^3 * 3b =**12a^3b****k = 2:**(4 choose 2) a^2 (3b)^2 = 6 * a^2 * 9b^2 =**54a^2b^2****k = 3:**(4 choose 3) a^1 (3b)^3 = 4 * a * 27b^3 =**108ab^3****k = 4:**(4 choose 4) a^0 (3b)^4 = 1 * 1 * 81b^4 =**81b^4**

**4. Combine the terms:**

Therefore, the expanded form of (a + 3b)^4 is:

**(a + 3b)^4 = a^4 + 12a^3b + 54a^2b^2 + 108ab^3 + 81b^4**

**In conclusion, by applying the binomial theorem, we successfully expanded (a + 3b)^4 into a polynomial with five terms.**