Find the term containing x^3 in the expansion of (x + 5)^6.

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Question: Find the term containing x^3 in the expansion of (x + 5)^6.

Options:

Correct Answer: 150

Solution:

The term containing x^3 is C(6,3) * (5)^3 = 20 * 125 = 250.

Find the term containing x^3 in the expansion of (x + 5)^6.

Find the term containing x^3 in the expansion of (x + 5)^6.

Step 1: Identify the expression to expand, which is (x + 5)^6.
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 = 5, and n = 6.
Step 4: We want the term that contains x^3, which means we need to find the term where the exponent of x is 3.
Step 5: If x has an exponent of 3, then 5 must have an exponent of (6 - 3) = 3.
Step 6: The term we are looking for is given by C(6, 3) * (x^3) * (5^3).
Step 7: Calculate C(6, 3), which is the number of combinations of 6 items taken 3 at a time. C(6, 3) = 6! / (3! * (6-3)!) = 20.
Step 8: Calculate 5^3, which is 5 * 5 * 5 = 125.
Step 9: Multiply the results from Step 7 and Step 8: 20 * 125 = 250.
Step 10: The term containing x^3 in the expansion of (x + 5)^6 is 250.

Binomial Expansion – The expansion of expressions in the form (a + b)^n using the binomial theorem, which involves combinations and powers.
Combinations – The use of binomial coefficients C(n, k) to determine 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 or power.