Construct the polynomial \(R (x)\) from the problem 61019 if:
a) \(P (x) = x^6 - 6x^4 - 4x^3 + 9x^2 + 12x + 4\);
b)\(P (x) = x^5 + x^4 - 2x^3 - 2x^2 + x + 1\).
Prove that the following polynomial does not have any identical roots: \(P(x) = 1 + x + x^2/2! + \dots + x^n/n!\)
For which \(A\) and \(B\) does the polynomial \(Ax^{n + 1} + Bx^n + 1\) have the number \(x = 1\) at least two times as its root?
Prove that the polynomial \(x^{2n} - nx^{n + 1} + nx^{n - 1} - 1\) for \(n > 1\) has a triple root of \(x = 1\).
Prove that for \(n> 0\) the polynomial \(nx^{n + 1} - (n + 1) x^n + 1\) is divisible by \((x - 1)^2\).
Let \(a, b\) be positive integers and \((a, b) = 1\). Prove that the quantity cannot be a real number except in the following cases \((a, b) = (1, 1)\), \((1,3)\), \((3,1)\).
For what values of \(n\) does the polynomial \((x+1)^n - x^n - 1\) divide by:
a) \(x^2 + x + 1\); b) \((x^2 + x + 1)^2\); c) \((x^2 + x + 1)^3\)?
Find the largest and smallest values of the functions
a) \(f_1 (x) = a \cos x + b \sin x\); b) \(f_2 (x) = a \cos^2x + b \cos x \sin x + c \sin^2x\).
Prove that the function \(\cos \sqrt {x}\) is not periodic.
The Babylonian algorithm for deducing \(\sqrt{2}\). The sequence of numbers \(\{x_n\}\) is given by the following conditions: \(x_1 = 1\), \(x_{n + 1} = \frac 12 (x_n + 2/x_n)\) (\(n \geq 1\)).
Prove that \(\lim\limits_{n\to\infty} x_n = \sqrt{2}\).