Understanding the Binomial Expansion of (x+3)^4
The binomial theorem provides a powerful tool for expanding expressions like (x+3)^4 without directly multiplying the terms multiple times. It states that for any positive integer n:
(x + y)^n = ∑_(k=0)^n (n choose k) x^(nk) y^k
where (n choose k) represents the binomial coefficient, calculated as n!/(k!(nk)!).
Let's apply this to our specific case, (x+3)^4:
Expanding (x+3)^4

Identify n: In this case, n = 4.

Calculate binomial coefficients: We need to calculate the binomial coefficients for k = 0, 1, 2, 3, and 4.
 (4 choose 0) = 4!/(0!4!) = 1
 (4 choose 1) = 4!/(1!3!) = 4
 (4 choose 2) = 4!/(2!2!) = 6
 (4 choose 3) = 4!/(3!1!) = 4
 (4 choose 4) = 4!/(4!0!) = 1

Apply the formula: Now, we can substitute the values into the binomial theorem:
(x + 3)^4 = (4 choose 0) x^4 3^0 + (4 choose 1) x^3 3^1 + (4 choose 2) x^2 3^2 + (4 choose 3) x^1 3^3 + (4 choose 4) x^0 3^4
Simplifying:
(x + 3)^4 = x^4 + 12x^3 + 54x^2 + 108x + 81
Understanding the Result
The expanded form of (x+3)^4 is a polynomial with five terms, each representing a different combination of powers of x and 3. The coefficients are the binomial coefficients calculated earlier, indicating the number of ways to choose k items from a set of n items.
This expansion is particularly useful for simplifying expressions, solving equations, and exploring the properties of polynomials.
Remember, applying the binomial theorem is more efficient than directly multiplying out (x+3) four times. It allows for quick and accurate expansion of complex binomial expressions.