Math Problems Vault

Problem 2771 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 6.9 "Isolation and Computation of Roots"

Isolate, by Sturm's method, the roots of the equation $x^{3}+2x-7=0$ .

?

Problem 2772 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 6.9 "Isolation and Computation of Roots"

Isolate, by Sturm's method, the roots of the equation $x^{3}-21x+7=0$ .

?

Problem 2773 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 6.9 "Isolation and Computation of Roots"

Isolate, by Sturm's method, the roots of the equation $x^{4}-6x^{3}+x^{2}-1=0$ .

?

Problem 2774 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 6.9 "Isolation and Computation of Roots"

Isolate, by Sturm's method, the roots of the equation $x^{4}+x^{3}-4x^{2}-4x+1=0$ .

?

Problem 2775 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 6.9 "Isolation and Computation of Roots"

Isolate, by Sturm's method, the roots of the equation $x^{4}-12x^{3}-55x^{2}+96=0$ .

?

Problem 2776 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 6.9 "Isolation and Computation of Roots"

Isolate, by Sturm's method, the roots of the equation $x^{5}+5x^{3}-7x+2=0$ .

?

Problem 2777 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 6.9 "Isolation and Computation of Roots"

Isolate, by Sturm's method, the roots of the equation $x^{5}+7x^{3}-5x+11=0$ .

?

Problem 2778 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 6.9 "Isolation and Computation of Roots"

(a)(b)

Construct the Sturm sequence for the equations:

?

(a)

$x^{2}+px+q=0$ ;

(b)

$x^{3}+px+q=0$ .

Problem 2779 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 6.9 "Isolation and Computation of Roots"

Using Sturm's method, find the condition for the reality of the roots of the equation $x^{5}-5px^{3}+5p^{2}x+2q=0$ .

?

Problem 2783 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 6.9 "Isolation and Computation of Roots"

Determine the number of real roots of the equation

f(x)=1+ax+\dfrac {a(a+1)}{1 \cdot 2}x^{2}+\ldots +\dfrac {a(a+1)\ldots (a+n-1)}{1 \cdot 2 \ldots n}x^{n}=0,

using Sturm's theorem and the equality

(1-x)f'(x)=af(x)-\dfrac {a(a+1)\ldots (a+n-1)(a+n)}{1 \cdot 2 \ldots n}x^{n} .

?

Problem 33.1 from Кострикин А.И. "Сборник задач по алгебре" [Кос15]More problems in Section 33 "Isolation of roots"

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 from Кострикин А.И. "Сборник задач по алгебре" [Кос15]More problems in Section 33 "Isolation of roots"

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 from Кострикин А.И. "Сборник задач по алгебре" [Кос15]More problems in Section 33 "Isolation of roots"

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 from Кострикин А.И. "Сборник задач по алгебре" [Кос15]More problems in Section 33 "Isolation of roots"

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 from Кострикин А.И. "Сборник задач по алгебре" [Кос15]More problems in Section 33 "Isolation of roots"

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

?

Problem 33.18 from Кострикин А.И. "Сборник задач по алгебре" [Кос15]More problems in Section 33 "Isolation of roots"

(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 from Кострикин А.И. "Сборник задач по алгебре" [Кос15]More problems in Section 33 "Isolation of roots"

(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 I.6.4.14 from Кострикин А.И. "Основы алгебры" [Кос04]More problems in Section I.6.4 "Polynomials with real coefficients"

(a)(b)(c)

The Legendre polynomials $P_{0}(X)=1, P_{1}(X)=X, \ldots , P_{n}(X), \ldots$ are defined by the recurrence formula

m P_{m}(X)-(2 m-1) X P_{m-1}(X)+(m-1) P_{m-2}(X)=0

Show that:

?

(a)

$P_{n}(1)=1, P_{n}(-1)=(-1)^{n}$ ;

(b)

$\left\{ P_{n}, P_{n-1}, \ldots , P_{0}\right\}$ is a Sturm system for $P_{n}(X)$ on the interval $[-1,1]$ ;

(c)

$P_{n}(X)$ has $n$ distinct roots in the interval $]-1,1[$ .

References