Question

Use the binomial theorem to determine the expanded form of the binomial expression

(x + 4)^4. Select the correct expression.

Answer 1

The binomial theorem states that the expansion of the binomial expression (a + b)^n is given by:

(a + b)^n = C(n,0)a^n b^0 + C(n,1)a^(n-1) b^1 + C(n,2)a^(n-2) b^2 + ... + C(n,n-1)a^1 b^(n-1) + C(n,n)a^0 b^n

where C(n,k) is the binomial coefficient, given by:

C(n,k) = n! / (k! (n-k)!)

Using this formula, we can expand (x + 4)^4 as follows:

(x + 4)^4 = C(4,0) x^4 4^0 + C(4,1) x^3 4^1 + C(4,2) x^2 4^2 + C(4,3) x^1 4^3 + C(4,4) x^0 4^4

Simplifying the terms and coefficients, we get:

(x + 4)^4 = 1x^4 + 4x^3(4) + 6x^2(16) + 4x(64) + 1(256)

Expanding the products and combining like terms, we obtain:

(x + 4)^4 = x^4 + 16x^3 + 96x^2 + 256x + 256

Therefore, the correct expression for the expanded form of (x + 4)^4 is:

x^4 + 16x^3 + 96x^2 + 256x + 256