In the expansion of (2x + 5)^3, what is the coefficient of x^2?

₹0.0

Question: In the expansion of (2x + 5)^3, what is the coefficient of x^2?

Options:

Correct Answer: 60

Solution:

The coefficient of x^2 in (2x + 5)^3 is given by 3C2 * (2x)^2 * 5^1 = 3 * 4 * 5 = 60.

In the expansion of (2x + 5)^3, what is the coefficient of x^2?

In the expansion of (2x + 5)^3, what is the coefficient of x^2?

Step 1: Identify the expression we are expanding, which is (2x + 5)^3.
Step 2: Recognize that we need to find the coefficient of x^2 in this expansion.
Step 3: Use the Binomial Theorem, which states that (a + b)^n = Σ (nCk * a^(n-k) * b^k) for k = 0 to n.
Step 4: In our case, a = 2x, b = 5, and n = 3.
Step 5: We want the term where x has the power of 2, which means we need to find the term where (2x) is raised to the power of 2.
Step 6: This corresponds to k = 1 in the Binomial Theorem because n - k = 2 (3 - 1 = 2).
Step 7: Calculate the binomial coefficient for k = 1, which is 3C1 = 3.
Step 8: Calculate (2x)^2, which is 4x^2.
Step 9: Calculate 5^1, which is 5.
Step 10: Multiply these values together: 3 (from 3C1) * 4 (from (2x)^2) * 5 (from 5^1) = 3 * 4 * 5.
Step 11: Perform the multiplication: 3 * 4 = 12, and then 12 * 5 = 60.
Step 12: Conclude that the coefficient of x^2 in the expansion of (2x + 5)^3 is 60.

Binomial Expansion – The question tests the understanding of the binomial theorem, specifically how to expand a binomial expression and find specific coefficients.
Combinatorics – The use of binomial coefficients (nCr) to determine the number of ways to choose terms in the expansion.
Algebraic Manipulation – The ability to correctly manipulate and simplify expressions involving powers and coefficients.