Question

If \( A \) denotes the sum of all the coefficients in the expansion of \( (1 - 3x + 10x^2)^n \) and \( B \) denotes the sum of all the coefficients in the expansion of \( (1 + x^2)^n \), then:

• A. \( A = B^3 \)
• B. \( 3A = B \)
• C. \( B = A^3 \)
• D. \( A = 3B \)

Answer 1

The Correct Option is A

To address the problem, we must first determine how to compute sums \( A \) and \( B \). These sums represent the aggregate of all coefficients within their respective expansions. The solution proceeds as follows:

1. Calculating the sum of coefficients in polynomial expansions:
   • For the polynomial \( f(x) = (1 - 3x + 10x^2)^n \), the sum of its coefficients is obtained by evaluating \( f(1) \). Consequently, \( A = f(1) = (1 - 3 \cdot 1 + 10 \cdot 1^2)^n = (1 - 3 + 10)^n = 8^n \).
   • Similarly, for \( g(x) = (1 + x^2)^n \), substituting \( x = 1 \) yields \( B = g(1) = (1 + 1^2)^n = (1 + 1)^n = 2^n \).
2. Establishing the relationship between \( A \) and \( B \):
   • Given \( A = 8^n \) and \( B = 2^n \).
   • Observe that \( 8^n \) can be rewritten as \( (2^3)^n = (2^n)^3 = B^3 \). Therefore, the relationship is \( A = B^3 \).

Following these steps, the relationship between \( A \) and \( B \) is clearly established as \( A = B^3 \). The definitive answer is:

\( A = B^3 \)