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What is the coefficient of x^0 in the expansion of (x + 5)^3?

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Question: What is the coefficient of x^0 in the expansion of (x + 5)^3?

Options:

  1. 125
  2. 100
  3. 75
  4. 50

Correct Answer: 125

Solution: 

The coefficient of x^0 is C(3,0) * 5^3 = 1 * 125 = 125.

What is the coefficient of x^0 in the expansion of (x + 5)^3?

Practice Questions

Q1
What is the coefficient of x^0 in the expansion of (x + 5)^3?
  1. 125
  2. 100
  3. 75
  4. 50

Questions & Step-by-Step Solutions

What is the coefficient of x^0 in the expansion of (x + 5)^3?
Correct Answer: 125
  • Step 1: Understand that x^0 means we are looking for the constant term in the expansion.
  • Step 2: Recognize that (x + 5)^3 is a binomial expression that can be expanded using the Binomial Theorem.
  • Step 3: The Binomial Theorem states that (a + b)^n = Σ (C(n, k) * a^(n-k) * b^k) for k = 0 to n, where C(n, k) is the binomial coefficient.
  • Step 4: In our case, a = x, b = 5, and n = 3.
  • Step 5: To find the coefficient of x^0, we need to set a (which is x) to 0. This means we are looking for the term where k = 3 (since 3 - k = 0).
  • Step 6: Calculate the binomial coefficient C(3, 3), which is equal to 1.
  • Step 7: Calculate 5^3, which is 5 * 5 * 5 = 125.
  • Step 8: Multiply the coefficient from Step 6 by the result from Step 7: 1 * 125 = 125.
  • Step 9: Conclude that the coefficient of x^0 in the expansion of (x + 5)^3 is 125.
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