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## Polynomial Arithmetic

From profit/cost models to the flight of a baseball to approximating the motion of a wave, polynomials are a mathematical way to represent values which are sums of powers of variables.

# Level 4

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?

1. $$p(x)$$ is a polynomial of degree $$17$$.

2. All roots of $$p(x)$$ are real.

3. All coefficients are positive.

4. The coefficient of $$x^{17}$$ is 1 and the product of roots of $$p(x)$$ is 1.

##### Image Credit: Wikimedia Septic Graph

$$P(x)$$ is a polynomial with integral coefficients such that the absolute value of the constant term of $$P(x)$$ is smaller than 1000. Given further that $$P(19)=P(94)=1994$$, find the constant term of the polynomial $$P(x)$$.

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