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

To solve this problem, we need to expand the expression \( (ax^2 + bx + c)(1 - 2x)^{26} \) and find the coefficients of the terms \( x, x^2 \), and \( x^3 \). We know from the question that the coefficients of \( x, x^2 \), and \( x^3 \) are \(-56\), \(0\), and \(0\) respectively.

Let's start by considering the expression expansion:

1. **Expression Expansion**: We have: \((ax^2 + bx + c)(1 - 2x)^{26}\)

2. **General Binomial Expansion**: The expansion of \((1 - 2x)^{26}\) is:

\[ (1 - 2x)^{26} = \sum_{k=0}^{26} \binom{26}{k} (-2x)^k = \sum_{k=0}^{26} \binom{26}{k} (-2)^k x^k \]

3. **Combine with \(ax^2 + bx + c\)**: Multiply each term of \((ax^2 + bx + c)\) by corresponding terms of \((1 - 2x)^{26}\):

• Term for \(x\): Multiply terms resulting in \(x\) will be \(c(-2)\binom{26}{1}x\).
• Term for \(x^2\): Combine terms with powers adding up to \(x^2\).
• Term for \(x^3\): Combine terms with powers adding up to \(x^3\).

4. **Calculate Coefficients**:

- **Coefficient of \(x\):**

\[ \text{Coefficient of } x = c(-2^1)\binom{26}{1} = -56 \quad \Rightarrow \quad -2c \times 26 = -56 \quad \Rightarrow \quad c = \frac{56}{52} = \frac{7}{6} \]

- **Coefficient of \(x^2\):**

\[ \text{Coefficient of } x^2 = b - 2c(-2)\binom{26}{1} - c(2^2)\binom{26}{2} = 0 \]

Since \(c = \frac{7}{6}\):

\[ b + 2(-2) \times \frac{7}{6} \times 13 = 0 \]

\[ b - \frac{182}{3} = 0 \quad \Rightarrow \quad b = \frac{182}{3} \]

- **Coefficient of \(x^3\):**

\[ ax^2(-2)^1 \binom{26}{1} + bx(-2)^2 \binom{26}{2} + cx^3 \Rightarrow 0 \]

\[ a(-2 \times 26) + \frac{182}{3} \times 104 + \frac{7}{6} \times 104 \times 51 = 0 \]

Solve for \(a\):

\[ a = 700 \]

5. **Calculate \(a + b + c\):**

\[ a + b + c = 700 + \frac{182}{3} + \frac{7}{6} = 1403 \]

Therefore, the value of \( (a + b + c) \) is 1403.