Will thought of a number: 1, 2 or 3. You can ask him only one question, to which he can answer “yes”, “no” or “I do not know”. Can you guess the number by asking just one question?
Is it possible to draw from some point on a plane \(n\) tangents to a polynomial of \(n\)-th power?
Prove the following formulae are true: \[\begin{aligned} a^{n + 1} - b^{n + 1} &= (a - b) (a^n + a^{n-1}b + \dots + b^n);\\ a^{2n + 1} + b^{2n + 1} &= (a + b) (a^{2n} - a^{2n-1}b + a^{2n-2}b^2 - \dots + b^{2n}). \end{aligned}\]
Prove that if \((p, q) = 1\) and \(p/q\) is a rational root of the polynomial \(P (x) = a_nx^n + \dots + a_1x + a_0\) with integer coefficients, then
a) \(a_0\) is divisible by \(p\);
b) \(a_n\) is divisible by \(q\).
Derive from the theorem in question 61013 that \(\sqrt{17}\) is an irrational number.
Prove that the root a of the polynomial \(P (x)\) has multiplicity greater than 1 if and only if \(P (a) = 0\) and \(P '(a) = 0\).
For a given polynomial \(P (x)\) we describe a method that allows us to construct a polynomial \(R (x)\) that has the same roots as \(P (x)\), but all multiplicities of 1. Set \(Q (x) = (P(x), P'(x))\) and \(R (x) = P (x) Q^{-1} (x)\). Prove that
a) all the roots of the polynomial \(P (x)\) are the roots of \(R (x)\);
b) the polynomial \(R (x)\) has no multiple roots.
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?