Algebra
# Polynomial Arithmetic

Let \(P(x) : (x - 1)(x - 2)(x - 3) \dots\ (x - 50)\)

Let \(Q(x) : (x + 1)(x + 2)(x + 3) \dots\ (x + 50)\)

If \[P(x)Q(x) = a_{100}x^{100} + a_{99}x^{99} + \ldots\ + a_{1}x^1 + a_0,\] then compute \(a_{100} -a_{99} - a_{98} - a_{97}.\)

Suppose the polynomials \(P_1(x)\) and \(P_2(x)\) are obtained by expanding and simplifying the following algebraic expressions

\[P_1(1) = (1 + x^2 - x^5)^{2000} \\ P_2(x) = (1 - x^2 + x^5)^{2000}.\]

Let \(C_1\) and \(C_2\) be the coefficient of \(x^{800}\) in the polynomials \(P_1(x)\) and \(P_2(x)\) respectively. Which of the given option is true?

Let \(T(x)=x^{124}+x^{123}+x^{122}+\ldots+x+1\) and \(a_{1},a_{2},a_{3}, \ldots, a_{123},a_{124}\) be the values of \(x\) for which \(T(x) = 0\).

If \(V_{n} = a_{1}^{n} + a_{2}^{n} + a_{3}^{n} + \ldots + a_{124}^{n}\), then find

\[\large \sum_{n=0}^{100000} (-1)^{n}V_{n}\]

What is the minimum value of \(p(2)\) if the following 4 conditions are followed?

\(p(x)\) is a polynomial of degree \(17\).

All roots of \(p(x)\) are real.

All coefficients are positive.

The coefficient of \(x^{17}\) is 1.

The product of roots of \(p(x)\) is -1.

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