What is the coefficient of x^2 in the expansion of (3x + 4)^5?

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Question: What is the coefficient of x^2 in the expansion of (3x + 4)^5?

Options:

Correct Answer: 80

Solution:

The coefficient of x^2 in (3x + 4)^5 is C(5, 2) * (3)^2 * (4)^3 = 10 * 9 * 64 = 5760.

What is the coefficient of x^2 in the expansion of (3x + 4)^5?

What is the coefficient of x^2 in the expansion of (3x + 4)^5?

Step 1: Identify the expression we need to expand, which is (3x + 4)^5.
Step 2: Recognize that we want the coefficient of x^2 in this expansion.
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 = 3x, b = 4, and n = 5.
Step 5: We need to find the term where the power of x is 2. This happens when we choose (3x) twice (which gives us x^2) and (4) three times.
Step 6: Calculate the number of ways to choose 2 from 5, which is C(5, 2). C(5, 2) = 5! / (2!(5-2)!) = 10.
Step 7: Calculate (3)^2, which is 9.
Step 8: Calculate (4)^3, which is 64.
Step 9: Multiply these values together: C(5, 2) * (3)^2 * (4)^3 = 10 * 9 * 64.
Step 10: Perform the multiplication: 10 * 9 = 90, and then 90 * 64 = 5760.
Step 11: Conclude that the coefficient of x^2 in the expansion of (3x + 4)^5 is 5760.

Binomial Expansion – The question tests the understanding of the binomial theorem, specifically how to find coefficients in the expansion of a binomial expression.
Combinatorics – The use of combinations (C(5, 2)) to determine the number of ways to choose terms in the expansion is a key concept.
Exponentiation – Understanding how to correctly apply exponents to the coefficients in the binomial expansion is crucial.