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

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

Options:

  1. -540
  2. -720
  3. -960
  4. -1080

Correct Answer: -960

Solution: 

The coefficient of x^3 is C(5,3) * (2)^3 * (-3)^2 = 10 * 8 * 9 = -720.

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

Practice Questions

Q1
In the expansion of (2x - 3)^5, what is the coefficient of x^3?
  1. -540
  2. -720
  3. -960
  4. -1080

Questions & Step-by-Step Solutions

In the expansion of (2x - 3)^5, what is the coefficient of x^3?
Correct Answer: -720
  • Step 1: Identify the expression to expand, which is (2x - 3)^5.
  • Step 2: Recognize that we need to find the coefficient of x^3 in the 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 = 2x, b = -3, and n = 5.
  • Step 5: We want the term where the power of x is 3, which means we need to find the term where k = 2 (since 5 - k = 3).
  • Step 6: Calculate C(5, 2), which is the number of ways to choose 2 from 5. C(5, 2) = 5! / (2!(5-2)!) = 10.
  • Step 7: Calculate (2x)^(5-2) = (2x)^3 = 2^3 * x^3 = 8x^3.
  • Step 8: Calculate (-3)^2 = 9.
  • Step 9: Multiply the results: Coefficient = C(5, 2) * (2^3) * (-3)^2 = 10 * 8 * 9.
  • Step 10: Calculate 10 * 8 = 80, then 80 * 9 = 720.
  • Step 11: Since we are multiplying by (-3)^2, the coefficient of x^3 is positive 720.
  • 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(n, k)) to determine the number of ways to choose terms from the expansion.
  • Exponent Rules – Applying exponent rules to calculate powers of the coefficients in the binomial expansion.
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