Problems

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 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?

Let $z=x+iy$, $w=u+iv$. Find a) $z+w$; b) $zw$; c) $z/w$.

Prove the equalities:

a) $\overline{z+w}=\bar{z}+\bar{w}$; b) $\overline{zw}=\bar{z}\bar{w}$; c) $\overline{\frac{z}{w}}=\frac{\bar{z}}{\bar{w}}$; d) $|\bar{z}|=|z|$; d) $\overline{\stackrel{―}{z}}=z$.

Prove the equalities:

a) $z+\bar{z}=2\mathrm{Re}z$;

b) $z-\bar{z}=2i\mathrm{Im}z$;

c) $\bar{z}z=|z{|}^2$.

Let $z_1$ and $z_2$ be fixed points of a complex plane. Give a geometric description of the sets of all points $z$ that satisfy the conditions:

a) $\arg \frac{z-z_1}{z-z_2}=0$;

b) $\arg \frac{z_1-z}{z-z_2}=0$.

Prove the formulae: $\arcsin (-x)=-\arcsin x$, $\arccos (-x)=\pi -\arccos x$.

Prove that amongst any 7 different numbers it is always possible to choose two of them, $x$ and $y$, so that the following inequality was true: $$0<\frac{x-y}{1+xy}<\frac{1}{\sqrt{3}}.$$