Math Problems Vault

Problem 33.1

Construct the Sturm sequence and separate the roots of the polynomials:

?

(a)

$x^{3}-3 x-1$ ;

(b)

$x^{3}+x^{2}-2 x-1$ ;

(c)

$x^{3}-7 x+7$ ;

(d)

$x^{3}-x+5$ ;

(e)

$x^{3}+3 x-5$ ;

(f)

$x^{4}-12 x^{2}-16 x-4$ ;

(g)

$x^{4}-x-1$ ;

(h)

$2 x^{4}-8 x^{3}+8 x^{2}-1$ ;

(i)

$x^{4}+x^{2}-1$ ;

(j)

$x^{4}+4 x^{3}-12 x+9$ .

Problem 33.2

Construct the Sturm sequence for the real polynomial $x^{5}-5 a x^{3}+5 a^{2} x+2 b$ . Depending on the sign of the number $a^{5}-b^{2}$ , find the number of real roots of the polynomial.

?

Problem 33.3

Construct the Sturm sequence for the real polynomial $x^{n}+p x+q$ . Depending on the parity and sign of the number $d=-(n-1)^{n-1} \times p^{n}-n^{n} q^{n-1}$ , find the number of real roots of the polynomial.

?

Problem 33.4

Construct the Sturm sequence and find the number of real roots of the polynomial

E_{n}(x)=1+\frac{x}{1!}+\frac{x^{2}}{2!}+\ldots +\frac{x^{n}}{n!}.

?

Problem 33.5

Prove that the polynomial $t^{3}-3 t+r$ cannot have more than one real root in the interval $[0,1]$ .

?

Problem 33.6

(a)(b)(c)

Suppose that all roots of the polynomial $f(x) \in \mathbb {R}\left[x\right]$ are real, i.e.,

f(x)=a\left(x-a_{1}\right)^{k_{1}} \ldots \left(x-a_{m}\right)^{k_{m}}, \quad a \neq 0

where $a_{1}<a_{2}<\ldots <a_{m}$ . Prove that:

?

(a)

$f^{\prime }(x)=n a\left(x-a_{1}\right)^{k_{1}-1} \ldots \left(x-a_{m}\right)^{k_{m}-1}\left(x-b_{1}\right) \ldots \left(x-b_{m-1}\right)$ , where $a_{1}<b_{1}<a_{2}<b_{2}<\ldots <a_{m-1}<b_{m-1}<a_{m}$ ;

(b)

if the number $k$ does not exceed the degree of the polynomial $f(x)$ , then the multiple roots of the $k$ -th derivative $f^{(k)}(x)$ are exactly the numbers $a_{i}$ with $k_{i} \geqslant k+2$ , and only these;

(c)

if $f(x)=c_{n} x^{n}+c_{n-1} x^{n-1}+\ldots +c_{0}$ , where $c_{n} \neq 0$ and $c_{k}=c_{k+1}=0$ for some $k=0, \ldots , n-2$ , then $c_{0}=c_{1}=\ldots =c_{k}=c_{k+1}=0$ .

Problem 33.7

Let $g(x)=b_{n} x^{n}+\ldots +b_{0}$ be a real polynomial, with $b_{n}, b_{0} \neq 0$ and $b_{k}=b_{k+1}=0$ for some $k=1, \ldots , n-2$ . Then not all roots of $g(x)$ are real.

?

Problem 33.8

Prove that for the real polynomial

a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots +a_{3} x^{3}+x^{2}+x+1, \quad a_{n} \neq 0

not all roots are real.

?

Problem 33.9

Prove that all complex roots $z$ of the polynomial $n x^{n}-x^{n-1}-\ldots -1$ satisfy $\left|z\right| \leqslant 1$ .

?

Problem 33.10

Prove that all positive roots of the polynomial $f(x)=x(x+1)(x+2) \ldots (x+n)-1$ are less than $1 / n!$ .

?

Problem 33.11

Prove that the polynomial $x^{4}-5 x^{3}-4 x^{2}-7 x+4$ has no negative roots.

?

Problem 33.12

How many roots of the polynomial $x^{6}+6 x+10$ lie in each quadrant of the complex plane?

?

Problem 33.13

Let $n_{1}<\ldots <n_{k}$ be natural numbers. Prove that the polynomial $1+x^{n_{1}}+\ldots +x^{n_{k}}$ has no complex roots $z$ satisfying $\left|z\right|<\frac{\sqrt{5}-1}{2}$ .

?

Problem 33.14

Prove that all complex roots of the polynomial $x^{n+1}-a x^{n}+a x-1$ , where $a$ is a real number, have modulus 1.

?

Problem 33.15

Let $k$ be a natural number and $\left|a_{i}\right|<k$ for $i=1, \ldots , n$ . Prove that then for any root $z$ of the polynomial $a_{n} x^{n}+\ldots +a_{1} x+1$ we have $\left|z\right| \geqslant \frac{1}{k+1}$ .

?

Problem 33.16

Prove that if all roots of the polynomial $f(x) \in \mathbb {C}\left[x\right]$ lie in the upper half-plane, then all roots of $f^{\prime }(x)$ lie in the same half-plane.

?

Problem 33.17

Prove that if $D$ is a convex region of the complex plane containing all roots of the polynomial $f(x) \in \mathbb {C}\left[x\right]$ , then all roots of $f^{\prime }(x)$ lie in $D$ .

?

Problem 33.18

(a)(b)

Let a sequence of real polynomials $f_{0}, f_{1}, \ldots , f_{n}$ with positive leading coefficients be given, such that: the degree of $f_{k}$ equals $k=0, \ldots , n$ ; $f_{k}=a_{k} f_{k-1}-c_{k} f_{k-2}$ , where $a_{k}, c_{k}$ are real polynomials, with $c_{k}(r)>0$ for all $r \in \mathbb {R}$ when $k \geqslant 2$ .

Prove that:

?

(a)

the roots of all the polynomials $f_{k}$ are real;

(b)

between any two roots of the polynomial $f_{k}$ there is a root of the polynomial $f_{k-1}$ .

Problem 33.19

(a)(b)

Determine the number of real roots:

?

(a)

of the Hermite polynomial $(-1)^{n} e^{x^{2} / 2} \frac{d^{n}}{d x^{n}} e^{-x^{2} / 2}$ ;

(b)

of the Laguerre polynomial $(-1)^{n} e^{x} \frac{d^{n}}{d x^{n}}\left(x^{n} e^{-x}\right)$ .

Problem 33.20

Determine all polynomials with coefficients $\pm 1$ that have only real roots.

?