Question

In the binomial expansion of \( (ax^2 + bx + c)(1 - 2x)^{26} \), the coefficients of \( x, x^2 \), and \( x^3 \) are -56, 0, and 0 respectively. Then, the value of \( (a + b + c) \) is

• A. 1500
• B. 1403
• C. 1300
• D. 1483

Hint:

In binomial expansions, carefully match terms in the expansion to the powers of \( x \) and use the given coefficients to form equations that can be solved for the unknowns.

Answer 1

The Correct Option is B

Step 1: Expansion of the binomial expression.
We need to expand \( (ax^2 + bx + c)(1 - 2x)^{26} \) and look at the coefficients of \( x, x^2, \) and \( x^3 \). Use the binomial expansion for \( (1 - 2x)^{26} \) and multiply it by \( ax^2 + bx + c \). Step 2: Find the coefficients of \( x, x^2, x^3 \).
The coefficient of \( x \) in the expansion is obtained by looking at terms that contribute to \( x^1 \), similarly for \( x^2 \) and \( x^3 \). The conditions given are: - Coefficient of \( x \) = -56 - Coefficient of \( x^2 \) = 0 - Coefficient of \( x^3 \) = 0 Step 3: Solve the system of equations.
From the binomial expansion and the given coefficients, we solve the system of equations to find the values of \( a, b, \) and \( c \). After solving, we get \( a + b + c = 1403 \).