Problems

Prove that if $(p,q)=1$ and $p/q$ is a rational root of the polynomial $P(x)=a_n{x}^n+\cdots +a_{1}x+a_0$ 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 $P^{\prime }(a)=0$.

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 $Q(x)=(P(x),P^{\prime }(x))$ 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: $P(x)=1+x+x^2/2!+\cdots +x^n/n!$

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?

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

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

Let it be known that all the roots of some equation $x^3+p{x}^2+qx+r=0$ are positive. What additional condition must be satisfied by its coefficients $p,q$ and $r$ in order for it to be possible to form a triangle from segments whose lengths are equal to these roots?