In the polynomial (x - 1)(x - 2)(x - 3) dots | vedclass

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(B) The given polynomial is $P(x) = (x - 1)(x - 2)(x - 3) \dots (x - 100)$.This is a product of $100$ linear factors of the form $(x - a_i)$ where $a_

Vedclass · Accepted Answer

$-5050$

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(B) The given polynomial is $P(x) = (x - 1)(x - 2)(x - 3) \dots (x - 100)$.This is a product of $100$ linear factors of the form $(x - a_i)$ where $a_i = i$ for $i = 1, 2, \dots, 100$.When we expand this product,the term $x^{99}$ is obtained by choosing $x$ from $99$ factors and the constant term $-a_i$ from the remaining factor.Thus,the coefficient of $x^{99}$ is the sum of all constant terms: $-(1 + 2 + 3 + \dots + 100)$.Using the formula for the sum of the first $n$ natural numbers,$\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$,we get:Coefficient $= -\frac{100 	imes 101}{2} = -5050$.

In the polynomial $(x - 1)(x - 2)(x - 3) \dots (x - 100)$,the coefficient of $x^{99}$ is

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