If the Binomial Theorem is applied to (x + 2)^4, what is the term containing x^2

If the Binomial Theorem is applied to (x + 2)^4, what is the term containing x^2?

If the Binomial Theorem is applied to (x + 2)^4, what is the term containing x^2?

Step 1: Identify the expression we are working with, which is (x + 2)^4.
Step 2: Recognize that we need to find the term that contains x^2.
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 = 4.
Step 5: We want the term where x has the power of 2, which means we need k = 2 (because x^(n-k) = x^2).
Step 6: Calculate n - k, which is 4 - 2 = 2. This means we will use b^2, which is 2^2.
Step 7: Calculate the binomial coefficient C(4, 2), which is 4! / (2! * (4-2)!) = 6.
Step 8: Now, combine everything: the term is C(4, 2) * x^2 * 2^2 = 6 * x^2 * 4.
Step 9: Simplify the expression: 6 * 4 = 24, so the term is 24x^2.

Binomial Theorem – The Binomial Theorem provides a formula for expanding expressions of the form (a + b)^n, where n is a non-negative integer.
Combination Notation – C(n, k) represents the number of ways to choose k elements from a set of n elements, which is crucial for determining coefficients in the expansion.
Term Extraction – Identifying specific terms in a polynomial expansion, particularly those with a certain power of a variable.