If the Binomial Theorem is applied to (x + 2)^3, what is the coefficient of x^2?

If the Binomial Theorem is applied to (x + 2)^3, what is the coefficient of x^2?

Step 1: Identify the expression we are working with, which is (x + 2)^3.
Step 2: Recognize that we need to find the coefficient of x^2 in the expansion of this expression.
Step 3: Use the Binomial Theorem, which states that (a + b)^n = Σ (C(n, k) * a^(n-k) * b^k) for k = 0 to n.
Step 4: In our case, a = x, b = 2, and n = 3.
Step 5: We want the term where x is raised to the power of 2, which means we need k = 1 (since n - k = 2).
Step 6: Calculate C(3, 2), which is the number of ways to choose 2 items from 3. This equals 3.
Step 7: Calculate 2^(1), which is 2 raised to the power of 1. This equals 2.
Step 8: Multiply the results from Step 6 and Step 7: 3 * 2 = 6.
Step 9: Conclude that the coefficient of x^2 in the expansion of (x + 2)^3 is 6.

Binomial Theorem – The Binomial Theorem provides a formula for expanding expressions of the form (a + b)^n, where the coefficients can be determined using combinations.
Coefficients in Binomial Expansion – In the expansion of (a + b)^n, the coefficient of a^k b^(n-k) is given by C(n, k) * b^(n-k), where C(n, k) is the binomial coefficient.