Question:
The coefficient of \( x^{49} \) in the product \( (x-1)(x-2)(x-3) \dots (x-50) \) is:
The coefficient of \( x^{49} \) in the product \( (x-1)(x-2)(x-3) \dots (x-50) \) is:
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This is a property of Vieta's formulas. For a monic polynomial $x^n + a_{n-1}x^{n-1} + \dots + a_0 = 0$, the sum of roots is $-a_{n-1}$.
Updated On: May 6, 2026
- \( -2250 \)
- \( -1275 \)
- \( 1275 \)
- \( 2250 \)
- \( -49 \)
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The Correct Option is B
Solution and Explanation
Concept:
For a polynomial \( (x-a_1)(x-a_2)\dots(x-a_n) \), the coefficient of \( x^{n-1} \) is the negative sum of the roots:
\[ \text{Coefficient of } x^{n-1} = -(a_1 + a_2 + a_3 + \dots + a_n) \]
Step 1: Identify the roots.
The product is \( (x-1)(x-2)\dots(x-50) \). The roots are \( 1, 2, 3, \dots, 50 \). The total number of terms is \( n = 50 \).
Step 2: Calculate the sum of the roots.
Using the sum formula for first \( n \) natural numbers, \( S = \frac{n(n+1)}{2} \): \[ \text{Sum} = \frac{50(50+1)}{2} = \frac{50 \times 51}{2} \] \[ \text{Sum} = 25 \times 51 = 1275 \]
Step 3: Find the coefficient.
The coefficient of \( x^{49} \) is negative of this sum: \[ \text{Coefficient} = -1275 \]
Step 1: Identify the roots.
The product is \( (x-1)(x-2)\dots(x-50) \). The roots are \( 1, 2, 3, \dots, 50 \). The total number of terms is \( n = 50 \).
Step 2: Calculate the sum of the roots.
Using the sum formula for first \( n \) natural numbers, \( S = \frac{n(n+1)}{2} \): \[ \text{Sum} = \frac{50(50+1)}{2} = \frac{50 \times 51}{2} \] \[ \text{Sum} = 25 \times 51 = 1275 \]
Step 3: Find the coefficient.
The coefficient of \( x^{49} \) is negative of this sum: \[ \text{Coefficient} = -1275 \]
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