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Polynomial Arithmetic

Polynomial Arithmetic: Level 4 Challenges


Let P(x):(x1)(x2)(x3) (x50)P(x) : (x - 1)(x - 2)(x - 3) \dots\ (x - 50)

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

If P(x)Q(x)=a100x100+a99x99+ +a1x1+a0,P(x)Q(x) = a_{100}x^{100} + a_{99}x^{99} + \ldots\ + a_{1}x^1 + a_0, then compute a100a99a98a97.a_{100} -a_{99} - a_{98} - a_{97}.

Suppose the polynomials P1(x)P_1(x) and P2(x)P_2(x) are obtained by expanding and simplifying the following algebraic expressions

P1(1)=(1+x2x5)2000P2(x)=(1x2+x5)2000.P_1(1) = (1 + x^2 - x^5)^{2000} \\ P_2(x) = (1 - x^2 + x^5)^{2000}.

Let C1C_1 and C2C_2 be the coefficient of x800x^{800} in the polynomials P1(x)P_1(x) and P2(x)P_2(x) respectively. Which of the given option is true?

Let T(x)=x124+x123+x122++x+1T(x)=x^{124}+x^{123}+x^{122}+\ldots+x+1 and a1,a2,a3,,a123,a124a_{1},a_{2},a_{3}, \ldots, a_{123},a_{124} be the values of xx for which T(x)=0T(x) = 0.

If Vn=a1n+a2n+a3n++a124nV_{n} = a_{1}^{n} + a_{2}^{n} + a_{3}^{n} + \ldots + a_{124}^{n}, then find

n=0100000(1)nVn\large \sum_{n=0}^{100000} (-1)^{n}V_{n}

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

  1. p(x)p(x) is a polynomial of degree 1717.

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

  3. All coefficients are positive.

  4. The coefficient of x17x^{17} is 1.

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

Image Credit: Wikimedia Septic Graph

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


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