How do you expand (x + 1)^3 using binomial expansion?

To expand the expression (x + 1)^3 using binomial expansion, we can use the binomial theorem, which states that:

(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

In our case, a = x, b = 1, and n = 3. Let’s break this down step by step:

1. Identify the components: a = x, b = 1, n = 3.
2. Calculate the binomial coefficients for n = 3:
   \(\binom{3}{0} = 1\), \(\binom{3}{1} = 3\), \(\binom{3}{2} = 3\), and \(\binom{3}{3} = 1\).
3. Write the expanded form using the coefficients calculated:

(x + 1)^3 = \binom{3}{0} x^3 (1)^0 + \binom{3}{1} x^2 (1)^1 + \binom{3}{2} x^1 (1)^2 + \binom{3}{3} x^0 (1)^3

Substituting the coefficients and simplifying:

1. For \(k = 0\): 1 * x^3 * 1 = x^3
2. For \(k = 1\): 3 * x^2 * 1 = 3x^2
3. For \(k = 2\): 3 * x^1 * 1^2 = 3x
4. For \(k = 3\): 1 * x^0 * 1^3 = 1

Now, we can add all these terms together:

(x + 1)^3 = x^3 + 3x^2 + 3x + 1

So the final expanded form of (x + 1)^3 is:

x^3 + 3x^2 + 3x + 1