When water is drained from a pool, the water level \(h\) in it varies depending on the time \(t\) according to the function \(h (t) = at^2 + bt + c\), and at the time \(t_0\) of when the draining is ending, the equalities \(h (t_0) = h' (t_0) = 0\) are satisfied. For how many hours does the pool drain completely, if in the first hour the water level in it is reduced by half?
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)\(P (x) = x^5 + x^4 - 2x^3 - 2x^2 + x + 1\).
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 for \(n> 0\) the polynomial \(nx^{n + 1} - (n + 1) x^n + 1\) is divisible by \((x - 1)^2\).
Prove that for \(n > 0\) the polynomial \[P (x) = n^2x^{n + 2} - (2n^2 + 2n - 1) x^{n + 1} + (n + 1)^2x^n - x - 1\] is divisible by \((x - 1)^3\).
Prove that for \(n> 0\) the polynomial \(x^{2n + 1} - (2n + 1)x^{n + 1} + (2n + 1)x^n - 1\) is divisible by \((x - 1)^3\).