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Problem #PRU-110157

Problems Geometry Plane geometry Plane transformations Similarity Homothetic transformations, rotations of homothetic transformations Circles Connection between the arc length, the angle, and the chord Calculus Derivative Derivative and multiple roots

14–16 5.0

The circles $σ_{1}$ and $σ_{2}$ intersect at points $A$ and $B$ . At the point $A$ to $σ_{1}$ and $σ_{2}$ , respectively, the tangents $l_{1}$ and $l_{2}$ are drawn. The points $T_{1}$ and $T_{2}$ are chosen respectively on the circles $σ_{1}$ and $σ_{2}$ so that the angular measures of the arcs $T_{1} A$ and $A T_{2}$ are equal (the arc value of the circle is considered in the clockwise direction). The tangent $t_{1}$ at the point $T_{1}$ to the circle $σ_{1}$ intersects $l_{2}$ at the point $M_{1}$ . Similarly, the tangent $t_{2}$ at the point $T_{2}$ to the circle $σ_{2}$ intersects $l_{1}$ at the point $M_{2}$ . Prove that the midpoints of the segments $M_{1} M_{2}$ are on the same line, independent of the positions of the points $T_{1}, T_{2}$ .

Problem #PRU-61018

Problems Calculus Derivative Derivative and multiple roots Algebra and arithmetic Polynomials Polynomial remainder theorem (little Bezout's theorem). Factorisation.

15–16 3.0

Prove that the root a of the polynomial $P (x)$ has multiplicity greater than 1 if and only if $P (a) = 0$ and $P^{'} (a) = 0$ .

Problem #PRU-61021

Problems Calculus Derivative Derivative and multiple roots

15–16 4.0

Prove that the following polynomial does not have any identical roots: $P (x) = 1 + x + x^{2} / 2! + \dots + x^{n} / n!$

Problem #PRU-61022

Problems Calculus Derivative Derivative and multiple roots Algebra and arithmetic Polynomials Division of polynomials with remainders. The greatest common divisor (GCD) and the least common multiple (LCM) for polynomilas.

14–16 3.0

For which $A$ and $B$ does the polynomial $A x^{n + 1} + B x^{n} + 1$ have the number $x = 1$ at least two times as its root?

Problem #PRU-61023

Problems Calculus Derivative Derivative and multiple roots

15–16 4.0

Prove that the polynomial $x^{2 n} - n x^{n + 1} + n x^{n - 1} - 1$ for $n > 1$ has a triple root of $x = 1$ .

Problem #PRU-64412

Problems Calculus Derivative Derivative and multiple roots

15–16 4.0

Prove that the polynomial $P (x) = a_{0} + a_{1} x + \dots + a_{n} x^{n}$ has a number $- 1$ which is a root of multiplicity $m + 1$ if and only if the following conditions are satisfied: $\begin{aligned} a_{0} - a_{1} + a_{2} - a_{3} + \dots + (- 1)^{n} a_{n} & = 0, \\ - a_{1} + 2 a_{2} - 3 a_{3} + \dots + (- 1)^{n} n a_{n} & = 0, \\ \dots \\ - a_{1} + 2^{m} a_{2} - 3^{m} a_{3} + \dots + (- 1)^{n} n^{m} a_{n} & = 0. \end{aligned}$

Problem #PRU-61019

Problems Calculus Derivative Derivative and multiple roots Algebra and arithmetic Polynomials Division of polynomials with remainders. The greatest common divisor (GCD) and the least common multiple (LCM) for polynomilas.

14–16 3.0

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.

Problem #PRU-61020

Problems Calculus Derivative Derivative and multiple roots Algebra and arithmetic Polynomials Division of polynomials with remainders. The greatest common divisor (GCD) and the least common multiple (LCM) for polynomilas.

14–16 3.0

Construct the polynomial $R (x)$ from the problem 61019 if:

a) $P (x) = x^{6} - 6 x^{4} - 4 x^{3} + 9 x^{2} + 12 x + 4$ ;

b) $P (x) = x^{5} + x^{4} - 2 x^{3} - 2 x^{2} + x + 1$ .

Problem #PRU-61025

Problems Calculus Derivative Derivative and multiple roots

13–16 3.0

Prove that for $n > 0$ the polynomial $n x^{n + 1} - (n + 1) x^{n} + 1$ is divisible by $(x - 1)^{2}$ .

Problem #PRU-64410

Problems Calculus Derivative Derivative and multiple roots

15–16 3.0

Prove that for $n > 0$ the polynomial $P (x) = n^{2} x^{n + 2} - (2 n^{2} + 2 n - 1) x^{n + 1} + (n + 1)^{2} x^{n} - x - 1$ is divisible by $(x - 1)^{3}$ .