Question

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

• A. \(1500\)
• B. \(1403\)
• C. \(1300\)
• D. \(1483\)

Hint:

For coefficient-comparison problems:
Expand only up to required powers
Match coefficients systematically
Use given zero-coefficient conditions to form equations

Answer 1

The Correct Option is B

(ax² + 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:

Step 1: Expand the expression using the binomial theorem

The given expression is:

(ax² + bx + c)(1 - 2x)^{26}.

Using the binomial expansion for \((1 - 2x)^{26}\), we have:

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

Expanding this:

\((1 - 2x)^{26} = 1 - 52x + 1320x² - 20800x³ + \cdots.\)

Step 2: Multiply with the terms in \( ax^2 + bx + c \)

Now, expand the product:

\((ax² + bx + c)(1 - 52x + 1320x² - 20800x³ + \cdots).\)

This gives:

\(ax²(1 - 52x + 1320x² - 20800x³ + \cdots) + bx(1 - 52x + 1320x² - 20800x³ + \cdots) + c(1 - 52x + 1320x² - 20800x³ + \cdots).\)

Expanding each term:

\(= a(x² - 52x³ + 1320x⁴ - 20800x⁵ + \cdots) + b(x - 52x² + 1320x³ - 20800x⁴ + \cdots) + c(1 - 52x + 1320x² - 20800x³ + \cdots).\)

Step 3: Identify the coefficients of \(x\), \(x²\), and \(x³\)

We are given that the coefficients of \(x\), \(x²\), and \(x³\) are -56, 0, and 0, respectively. Now, extract the terms for \(x\), \(x²\), and \(x³\) from the expanded product: - Coefficient of \(x\) from the product of \(bx\) and \(1\) term: \[ b \cdot 1 = b. \] So, \(b = -56\). - Coefficient of \(x^2\) from the product of \(ax²\) and \(1\) term, and \(bx\) and \(-52x\): \[ a \cdot 1 + b \cdot (-52) = 0. \] Substituting \(b = -56\): \[ a - 56 \cdot 52 = 0. \] \[ a - 2912 = 0 \quad \Rightarrow \quad a = 2912. \] - Coefficient of \(x^3\) from the product of \(ax²\) and \(-52x\), \(bx\) and \(1320x²\), and \(c\) and \(-52x\): \[ a(-52) + b(1320) + c(-52) = 0. \] Substituting \(a = 2912\) and \(b = -56\): \[ -2912 \cdot 52 + (-56 \cdot 1320) - 52c = 0. \] After simplifying the equation, solve for \(c\).

Step 4: Calculate \( (a + b + c) \)

After finding \(a\), \(b\), and \(c\), sum the values: \[ a + b + c. \] Substituting the values: \[ 2912 - 56 + c. \] Solve for \(c\) and find the final result.