So, the expanded form of the function f(x) = (x - 3)^4 is f(x) = x^4 - 12x^3 + 54x^2 - 108x + 81.
To expand the function f(x) = (x - 3)^4, we can use the binomial theorem. The binomial theorem allows us to expand expressions of the form (a + b)^n, where n is a positive integer.
In this case, we have (x - 3)^4. Using the binomial theorem, we can expand this expression as follows:
f(x) = (x - 3)^4
= C(4, 0) * x^4 * (-3)^0 + C(4, 1) * x^3 * (-3)^1 + C(4, 2) * x^2 * (-3)^2 + C(4, 3) * x^1 * (-3)^3 + C(4, 4) * x^0 * (-3)^4
Here, C(n, k) represents the binomial coefficient, which is the number of ways to choose k items from a set of n items. It can be calculated using the formula C(n, k) = n! / (k! * (n - k)!), where ! denotes the factorial of a number.
Let's calculate each term:
C(4, 0) = 4! / (0! * (4 - 0)!) = 1
C(4, 1) = 4! / (1! * (4 - 1)!) = 4
C(4, 2) = 4! / (2! * (4 - 2)!) = 6
C(4, 3) = 4! / (3! * (4 - 3)!) = 4
C(4, 4) = 4! / (4! * (4 - 4)!) = 1
Substituting these values back into the expanded form, we have:
f(x) = 1 * x^4 * (-3)^0 + 4 * x^3 * (-3)^1 + 6 * x^2 * (-3)^2 + 4 * x^1 * (-3)^3 + 1 * x^0 * (-3)^4
Simplifying each term, we get:
f(x) = x^4 + 4 * x^3 * (-3) + 6 * x^2 * 9 + 4 * x * (-27) + 81
Finally, combining like terms, we have the expanded form of f(x):
**f(x) = x^4 - 12x^3 + 54x^2 - 108x + 81**