Calculate the term containing x^3 in the expansion of (x + 2)^7.

Calculate the term containing x^3 in the expansion of (x + 2)^7.

Step 1: Identify the expression to expand, which is (x + 2)^7.
Step 2: 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 3: In our case, a = x, b = 2, and n = 7.
Step 4: We want the term that contains x^3. This means we need to find the term where the exponent of x is 3.
Step 5: If x has an exponent of 3, then 2 must have an exponent of 7 - 3 = 4.
Step 6: The term we are looking for is given by C(7, 3) * (x^3) * (2^4).
Step 7: Calculate C(7, 3), which is the number of combinations of 7 items taken 3 at a time. C(7, 3) = 7! / (3! * (7-3)!) = 35.
Step 8: Calculate 2^4, which is 16.
Step 9: Multiply the results from Step 7 and Step 8: 35 * 16 = 560.
Step 10: The term containing x^3 in the expansion of (x + 2)^7 is 560.

Binomial Expansion – The process of expanding expressions of the form (a + b)^n using the binomial theorem.
Combination Formula – Understanding how to calculate combinations, denoted as C(n, k), which represents the number of ways to choose k elements from a set of n elements.
Term Extraction – Identifying specific terms in a polynomial expansion based on their degree.