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 solve the problem, we need to understand how to compute the sums \( A \) and \( B \). These sums represent the total of all coefficients in the respective expansions. Let's break down the solution step-by-step: 

1. Sum of coefficients in any polynomial expansion:
   • The sum of coefficients in the expansion of a polynomial \( f(x) = (1 - 3x + 10x^2)^n \) is found by substituting \( x = 1 \). Thus, \( A = f(1) = (1 - 3 \cdot 1 + 10 \cdot 1^2)^n = (1 - 3 + 10)^n = (8)^n \).
   • Similarly, for \( g(x) = (1 + x^2)^n \), substituting \( x = 1 \) gives \( B = g(1) = (1 + 1^2)^n = (1 + 1)^n = 2^n \).
2. Relation between \( A \) and \( B \):
   • We have \( A = 8^n \) and \( B = 2^n \).
   • Notice that \( 8^n = (2^3)^n = (2^n)^3 = B^3 \), therefore, \( A = B^3 \).

After these steps, it's evident that the relationship between \( A \) and \( B \) is expressed by the formula \( A = B^3 \). Hence, the correct answer is:

\( A = B^3 \)

Answer 2

To find the sums \( A \) and \( B \), we calculate the sum of all coefficients by setting \( x = 1 \) in each expansion.

Step 1. Calculate \( A \)

Substitute \( x = 1 \) in \( (1 - 3x + 10x^2)^n \):\[ A = (1 - 3 \cdot 1 + 10 \cdot 1^2)^n = (1 - 3 + 10)^n = 8^n \]Therefore, \( A = 8^n \).

Step 2. Calculate \( B \)

Substitute \( x = 1 \) in \( (1 + x^2)^n \):\[ B = (1 + 1^2)^n = 2^n \]Thus, \( B = 2^n \).

Step 3. Find the Relationship Between \( A \) and \( B \)

Since \( A = 8^n \) and \( B = 2^n \), we can write:\[ A = (2^n)^3 = B^3 \]Therefore, \( A = B^3 \).