Chapter 6

Higher Algebra

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Section 6.1Complex Numbers[76/87%]

Problem 2408

Represent the number $1+i$ in trigonometric form.

Problem 2409

Represent the number $1+i\sqrt{3}$ in trigonometric form.

Problem 2410

Represent the number $\sqrt{3}-i$ in trigonometric form.

Problem 2411

Represent the number $1-\cos \alpha +i\sin \alpha$ ; $0 < \alpha < 2\pi$ in trigonometric form.

Problem 2412

Represent the number $1+\sin \alpha -i\cos \alpha$ in trigonometric form.

Problem 2413

Represent the number $1+i\tg \alpha$ ; $-\frac{\pi }{2} < \alpha < \frac{\pi }{2}$ in trigonometric form.

Problem 2414

Evaluate the following expression, where $\omega =\cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3}$ :

(a\omega +b\omega ^{2})(a\omega ^{2}+b\omega ) .

Problem 2415

Evaluate the following expression, where $\omega =\cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3}$ :

(a+b+c)(a+b\omega +c\omega ^{2})(a+b\omega ^{2}+c\omega ) .

Problem 2416

Evaluate the following expression, where $\omega =\cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3}$ :

(a+b\omega +c\omega ^{2})^{3}+(a+b\omega ^{2}+c\omega )^{3} .

Problem 2417

Compute the square roots: $\sqrt{i}$ .

Problem 2418

Compute the square roots: $\sqrt{3+4i}$ .

Problem 2419

Compute the square roots: $\sqrt{-7+24i}$ .

Problem 2420

Solve the quadratic equation $x^{2}+(5-2i)x+5(1-i)=0$ .

Problem 2421

Solve the quadratic equation $x^{2}+(1-2i)x-2i=0$ .

Problem 2422

Find all values of the following root: $\sqrt[3]{i}$ .

Problem 2423

Find all values of the following root: $\sqrt[3]{-1+i}$ .

Problem 2424

Find all values of the following root: $\sqrt[6]{-64}$ .

Problem 2425

Find all values of the following root: $\sqrt[6]{64}$ .

Problem 2426

Find the roots of the equation $\left(\dfrac {1+xi}{1-xi}\right)^{n}=\dfrac {1+i\tg \alpha }{1-i\tg \alpha }$ .

Problem 2427

Find the roots of the equation $(x+i)^{n}+(x-i)^{n}=0$ .

Problem 2428

Express the function $\cos 3\varphi$ in terms of $\cos \varphi$ and $\sin \varphi$ .

Problem 2429

Express the function $\sin 3\varphi$ in terms of $\cos \varphi$ and $\sin \varphi$ .

Problem 2430

Express the function $\cos 4\varphi$ in terms of $\cos \varphi$ and $\sin \varphi$ .

Problem 2431

Express the function $\sin 5\varphi$ in terms of $\cos \varphi$ and $\sin \varphi$ .

Problem 2432

Express the function $\cos^{3}\varphi$ in terms of trigonometric functions of multiple angles.

Problem 2433

Express the function $\sin^{3}\varphi$ in terms of trigonometric functions of multiple angles.

Problem 2434

Express the function $\cos^{4}\varphi$ in terms of trigonometric functions of multiple angles.

Problem 2435

Express the function $\sin^{4}\varphi$ in terms of trigonometric functions of multiple angles.

Problem 2436

Express the function $\cos^{5}\varphi$ in terms of trigonometric functions of multiple angles.

Problem 2437

Express the function $\sin^{5}\varphi$ in terms of trigonometric functions of multiple angles.

Problem 2438

Prove the identity:

\tg n\varphi =\dfrac {C_{n}^{1}\tg \varphi -C_{n}^{3}\tg ^{3}\varphi +C_{n}^{5}\tg ^{5}\varphi -\ldots }{1-C_{n}^{2}\tg ^{2}\varphi +C_{n}^{4}\tg ^{4}\varphi -\ldots },

where $C_{n}^{k}=\dfrac {n(n-1)\ldots (n-k+1)}{1 \cdot 2 \cdot 3 \ldots k}$ .

Problem 2439

Prove the identity:

2^{2n}\cos ^{2n}\varphi =2\cos 2n\varphi +2C_{2n}^{1}\cos (2n-2)\varphi +\ldots +2C_{2n}^{n-1}\cos 2\varphi +C_{2n}^{n} .

Problem 2440

Prove the identity:

\ctg x+\ctg \left(x+\frac{\pi }{n}\right)+\ctg \left(x+\frac{2\pi }{n}\right)+\ldots +\ctg \left(x+\frac{n-1}{n}\pi \right)=n\ctg nx .

Problem 2441

Prove the identity:

\sin x\sin \left(x+\frac{\pi }{n}\right)\sin \left(x+\frac{2\pi }{n}\right)\ldots \sin \left(x+\frac{n-1}{n}\pi \right)=\dfrac {\sin nx}{2^{n-1}} .

Problem 2442

Evaluate the sum $1+2\cos x+2\cos 2x+\ldots +2\cos nx$ .

Problem 2443

Evaluate the sum $\sin x+\sin 3x+\sin 5x+\ldots +\sin (2n-1)x$ .

Problem 2444

Evaluate the sum $\cos \varphi +a\cos 3\varphi +\ldots +a^{n}\cos (2n+1)\varphi$ .

Problem 2445

Evaluate the sum $1+2a\cos \varphi +2a^{2}\cos 2\varphi +\ldots +2a^{n}\cos n\varphi$ .

Problem 2446

Find the value of $\ln (-e)$ .

Problem 2447

Find the value of $\ln (-2)$ .

Problem 2448

Find the value of $\ln i$ .

Problem 2449

Find the value of $\ln \dfrac {1+i}{\sqrt{2}}$ .

Problem 2450

Find the value of $\ln (x+iy)$ .

Problem 2451

Find the value of $e^{\pi i}$ .

Problem 2452

Find the value of $i^{i}$ .

Problem 2453

Find the value of $2^{i}$ .

Problem 2454

Find the value of $\left(\dfrac {1+i}{\sqrt{2}}\right)^{-i}$ .

Problem 2455

Find the value of $\tg \dfrac {\pi i}{2}$ .

Problem 2456

Find the value of $\sin (x+iy)$ .

Problem 2457

Find the value of $\cos (x+iy)$ .

Problem 2458

Find the value of $\arctg xi$ .

Problem 2459

Under what condition does the point $z=x+iy$ lie inside a circle of radius $R$ centered at the point $c=a+bi$ ?

Problem 2460

A regular $n$ -gon is inscribed in a circle of radius $R$ centered at the point $c=a+bi$ , with one of its vertices at the point $z_{0}=a+(b+R)i$ . Where are the remaining vertices?

Problem 2461

Two vertices of an equilateral triangle are at the points $z_{0}=1$ and $z_{1}=2+i$ . Find the third vertex.

Problem 2462

Two adjacent vertices of a regular $n$ -gon are at the points $z_{0}$ and $z_{1}$ . Find the next vertex.

Problem 2463

The vertices of a polygon are at the points $z_{k}=1+z+z^{2}+\ldots +z^{k-1}$ , $\left|z\right| < 1$ . Determine whether the point $z=0$ lies inside the polygon.

Problem 2464

Three consecutive vertices of a parallelogram are at the points $z_{1}$ , $z_{2}$ , $z_{3}$ . Find the position of the fourth vertex.

Problem 2465

The endpoints of a segment are at the points $z_{1}$ and $z_{2}$ . Find the midpoint.

Problem 2466

Masses $m_{1}, m_{2}, \ldots , m_{n}$ are located at the points $z_{1}, z_{2}, \ldots , z_{n}$ . Find the position of the center of gravity.

Problem 2467

The points $z_{1}, z_{2}, \ldots , z_{n}$ are the vertices of a convex $n$ -gon. Find the locus of points $z=\lambda_{1}z_{1}+\lambda_{2}z_{2}+\ldots +\lambda_{n}z_{n}$ , where $\lambda_{k} > 0$ , $\lambda_{1}+\lambda_{2}+\ldots +\lambda_{n}=1$ .

Problem 2468

The consecutive vertices of a broken line lie at the points

z_{k}=a_{0}+a_{1}z+a_{2}z^{2}+\ldots +a_{k-1}z^{k-1}, \quad \left|z\right| \leqslant 1;

a_{0} > a_{1} > a_{2} > \cdots > a_{k-1} > 0 .

Prove that the broken line cannot close for any value of $k$ .

Problem 2469

Prove that the equation

a_{0}x^{n}+a_{1}x^{n-1}+\ldots +a_{n}=0, \qquad a_{0} > a_{1} > a_{2} > \cdots > a_{n} > 0

cannot have roots with modulus greater than unity.

Problem 2470

Prove that the equalities $\lambda_{1}z_{1}+\lambda_{2}z_{2}+\lambda_{3}z_{3}=0$ , $\lambda_{1}+\lambda_{2}+\lambda_{3}=0$ , where $\lambda_{1}, \lambda_{2}, \lambda_{3}$ are real, are equivalent to the condition that the points $z_{1}, z_{2}, z_{3}$ lie on a single line.

Problem 2471

Prove the identity:

\left|z_{1}+z_{2}\right|^{2}+\left|z_{1}-z_{2}\right|^{2}=2\left|z_{1}\right|^{2}+2\left|z_{2}\right|^{2}

and state its geometric meaning.

Problem 2472

Prove the equivalence of the equalities

\left|z_{2}-z_{1}\right|^{2}=\left|z_{2}-z_{0}\right|^{2}+\left|z_{1}-z_{0}\right|^{2} \quad \text{and} \quad z_{2}-z_{0}=\lambda i(z_{1}-z_{0}),

where $\lambda$ is real.

Problem 2473

Prove the identity:

2\left|z_{1}\right|+2\left|z_{2}\right|=\left|z_{1}+z_{2}-2\sqrt{z_{1}z_{2}}\right|+\left|z_{1}+z_{2}+2\sqrt{z_{1}z_{2}}\right| .

Problem 2474

Prove the identity:

\left|x+y\right|+\left|x-y\right|=\left|x+\sqrt{x^{2}-y^{2}}\right|+\left|x-\sqrt{x^{2}-y^{2}}\right| .

Problem 2475

Determine the type of the curve given by the equation $z=z_{0}+te^{\alpha i}$ , $z_{0}=x_{0}+iy_{0}$ , where $t$ is a real parameter.

Problem 2476

Determine the type of the curve given by the equation $z=z_{0}+re^{ti}$ , where $t$ is a real parameter.

Problem 2477

Determine the type of the curve given by the equation $z=z_{0}+ae^{ti}+be^{-ti}$ , where $t$ is a real parameter.

Problem 2478

Determine the type of the curve given by the equation $z=ate^{ti}$ , where $t$ is a real parameter.

Problem 2479

Determine the type of the curve given by the equation $z=ae^{\alpha t}$ , where $t$ is a real parameter, and $a$ and $\alpha$ are complex constants.

Problem 2480

Trace how the argument of the quantity $x(x-1)$ changes as $x$ traces, counterclockwise, a closed curve enclosing the points $0$ and $1$ .

Problem 2481

Study how the arguments of the quantities $\sqrt{x}$ , $\sqrt{x-1}$ , and $\sqrt{x(x-1)}$ change as $x$ traces the same contour as in problem 2480 (a closed curve traversed counterclockwise and enclosing the points $0$ and $1$ ).

Problem 2482

How does the function $u=(z-a)^{\alpha }(z-b)^{\beta }$ change, for real $\alpha$ and $\beta$ , if $z$ traces a closed contour going counterclockwise around the points $a$ and $b$ ?

Problem 2483

The same question for the function $u=(z-a)^{\alpha }\ln (z-a)$ and a contour going counterclockwise around the point $a$ .

Remark.

The question concerns how the function $u=(z-a)^{\alpha }\ln (z-a)$ changes, for real $\alpha$ , if $z$ traces a closed contour going counterclockwise around the point $a$ (by analogy with problem 2482).

Section 6.2Factoring a Polynomial. The Relation Between Coefficients and Roots[40/40%]

Problem 2484

Determine $a$ and $b$ so that $x^{4}+3x^{2}+ax+b$ is divisible without remainder by $x^{2}-2ax+2$ .

Problem 2485

Show that for integers $m$ , $n$ , and $p$ , the polynomial $x^{3m}+x^{3n+1}+x^{3p+2}$ is divisible by $x^{2}+x+1$ .

Problem 2486

Prove that $(x+y)^{n}-x^{n}-y^{n}$ is divisible by $x^{2}+xy+y^{2}$ for $n=6m+5$ , and by $(x^{2}+xy+y^{2})^{2}$ for $n=6m+1$ .

Problem 2487

The equation $3x^{4}-5x^{3}+3x^{2}+4x-2=0$ has the root $1+i$ . Find the remaining roots.

Problem 2488

Prove that a double root of the equation $\varphi (x)^{2}+\psi (x)^{2}=0$ satisfies the equation $\varphi '(x)^{2}+\psi '(x)^{2}=0$ , if the polynomials $\varphi (x)$ and $\psi (x)$ are relatively prime.

Problem 2489

Prove that a root of multiplicity $n$ of the equation

F\left(u(x), v(x)\right)=0

is a root of multiplicity $n-1$ of the equation $F\left(u'(x), v'(x)\right)=0$ . Here $u(x)$ , $v(x)$ are relatively prime polynomials and $F(u,v)$ is a homogeneous function having no repeated linear factors.

Problem 2490

Prove that a polynomial $f(x)$ remains positive for all real $x$ if and only if

f(x)=\left|\alpha _{0}+\alpha _{1}x+\alpha _{2}x^{2}+\ldots +\alpha _{n}x^{n}\right|^{2},

where $\alpha_{0}, \alpha_{1}, \alpha_{2}, \ldots , \alpha_{n}$ are certain complex numbers.

Problem 2491

Prove the identity:

x^{2n}-1=(x^{2}-1)\prod _{k=1}^{n-1}\left(x^{2}-2x\cos \frac{k\pi }{n}+1\right) .

Problem 2492

Prove the identity:

x^{2n+1}-1=(x-1)\prod _{k=1}^{n}\left(x^{2}-2x\cos \frac{2k\pi }{2n+1}+1\right) .

Problem 2493

Prove the identity:

x^{2n+1}+1=(x+1)\prod _{k=1}^{n}\left(x^{2}+2x\cos \frac{2k\pi }{2n+1}+1\right) .

Problem 2494

Prove the identity:

x^{2n}+1=\prod _{k=1}^{n-1}\left(x^{2}-2x\cos \frac{(2k+1)\pi }{2n}+1\right) .

Problem 2495

Prove the identity:

\sin \frac{\pi }{2n}\sin \frac{2\pi }{2n} \cdot \ldots \cdot \sin \frac{(n-1)\pi }{2n}=\dfrac {\sqrt{n}}{2^{n-1}} .

Problem 2496

Prove the identity:

\sin \frac{\pi }{2n+1}\sin \frac{2\pi }{2n+1} \cdot \ldots \cdot \sin \frac{n\pi }{2n+1}=\dfrac {\sqrt{2n+1}}{2^{n}} .

Problem 2497

Prove the identity:

\cos \frac{\pi }{2n+1}\cos \frac{2\pi }{2n+1} \cdot \ldots \cdot \cos \frac{n\pi }{2n+1}=\dfrac {1}{2^{n}} .

Problem 2498

Prove the identity:

\cos \frac{2\pi }{2n+1}\cos \frac{4\pi }{2n+1} \cdot \ldots \cdot \cos \frac{2n\pi }{2n+1}=\dfrac {(-1)^{\frac{n+1}{2}}}{2^{n}} \text{ or } \dfrac {(-1)^{\frac{n}{2}}}{2^{n}}

for odd or even $n$ respectively.

Problem 2499

Prove the identity:

\sin \frac{\pi }{4n}\sin \frac{3\pi }{4n} \cdot \ldots \cdot \sin \frac{(2n-1)\pi }{4n}=\dfrac {\sqrt{2}}{2^{n}} .

Problem 2500

Prove the identity:

\sin \frac{\pi }{2n}\sin \frac{3\pi }{2n} \cdot \ldots \cdot \sin \frac{(2n-1)\pi }{2n}=\dfrac {1}{2^{n-1}} .

Problem 2501

Prove the identity:

\sin \frac{\pi }{n}\sin \frac{2\pi }{n} \cdot \ldots \cdot \sin \frac{(n-1)\pi }{n}=\dfrac {n}{2^{n-1}} .

Problem 2502

Factor the polynomial $(x+1)^{n}+(x-1)^{n}$ .

Problem 2503

Factor the polynomial $x^{2n}+C_{2n}^{2}x^{2n-2}(x^{2}-1)+C_{2n}^{4}x^{2n-4}(x^{2}-1)^{2}+\ldots +(x^{2}-1)^{n}$ .

Problem 2504

Factor the polynomial $x^{2n+1}+C_{2n+1}^{2}x^{2n-1}(x^{2}-1)+C_{2n+1}^{4}x^{2n-3}(x^{2}-1)^{2}+\ldots +x(x^{2}-1)^{n}$ .

Problem 2505

Evaluate the product $(x_{1}^{2}+1)(x_{2}^{2}+1)\ldots (x_{n}^{2}+1)$ , where $x_{1}, x_{2}, \ldots , x_{n}$ are the roots of the polynomial $x^{n}+a_{1}x^{n-1}+\ldots +a_{n}$ ( $a_{1}, a_{2}, \ldots , a_{n}$ real).

Problem 2506

The roots of the polynomial $f(x)=a_{0}x^{n}+a_{1}x^{n-1}+\ldots +a_{n}$ are related by $x_{s}+x_{n-s}=m$ , $s=1, 2, \ldots , n$ . Prove the identity: $f(x)=(-1)^{n}f(m-x)$ .

Problem 2507

The roots of a polynomial $f(x)$ of degree $2n$ are related by $x_{s}+x_{n+s}=2m$ , $s=1, 2, \ldots , n$ . Prove that one of the roots of $f'(x)$ equals $m$ , and the rest are related pairwise by the formulas $y_{s}+y_{n+s-1}=2m$ , $s=1, 2, \ldots , n-1$ .

Problem 2508

The equation $a_{0}x^{n}+a_{1}x^{n-1}+\ldots +a_{n}=0$ , where $a_{0}a_{n} \neq 0$ , has $k+1$ terms. Show that it cannot have roots of multiplicity higher than $k$ .

Problem 2509

Is the polynomial

x^{2n}-n^{2}x^{n+1}+2(n^{2}-1)x^{n}-n^{2}x^{n-1}+1

divisible by $(x-1)^{4}$ ?

Problem 2510

Find the relation between the coefficients of the polynomial $a_{0}x^{3}+a_{1}x^{2}+a_{2}x+a_{3}=0$ , if its roots satisfy the relation $x_{1}x_{2}+x_{2}x_{3}=2x_{1}x_{3}$ .

Problem 2511

Find the relation between $p$ , $q$ , and $r$ for which the roots of the equation $x^{3}+px^{2}+qx+r=0$ form a geometric progression.

Problem 2512

For what $\lambda$ and $\mu$ do the roots of the equation $x^{4}+2x^{3}-21x^{2}+\lambda x+\mu =0$ form an arithmetic progression, and what progression specifically?

Problem 2513

For what $p$ , $q$ , $\alpha$ ( $0 < \alpha < 4$ ) does the equation $x^{4}+px^{\alpha }+q=0$ have a triple root?

Problem 2514

The remainders upon dividing a polynomial by $(x-a)$ , $(x-b)$ , $(x-c)$ are $A$ , $B$ , $C$ . Find the remainder upon dividing by $(x-a)(x-b)(x-c)$ .

Problem 2515

The roots of the polynomial $x^{3}+x^{2}-2$ are $1$ , $\alpha$ , and $\beta$ . Find the second-degree polynomial $\varphi (x)$ for which $\varphi (1)=1$ , $\varphi (\alpha )=\beta$ , $\varphi (\beta )=\alpha$ , and prove that $\varphi (\varphi (x))-x$ is divisible by $x^{3}+x^{2}-2$ .

Problem 2516

Find a seventh-degree polynomial $f(x)$ such that $f(x)+1$ is divisible by $(x-1)^{4}$ , and $f(x)-1$ is divisible by $(x+1)^{4}$ .

Problem 2517

Find a polynomial $f(x)$ of degree $2n-1$ such that $f(x)+1$ is divisible by $(x-1)^{n}$ , and $f(x)-1$ by $(x+1)^{n}$ .

Problem 2518

Find a polynomial of degree $n$ that, upon division by $(x-a_{1}), (x-a_{2}), \ldots , (x-a_{n})$ , gives remainders $A_{1}, A_{2}, \ldots , A_{n}$ .

Problem 2519

Prove the validity of Euler's identity

\sum _{k=1}^{m}\dfrac {\omega (x_{k})}{\varphi ’(x_{k})}=0,

where $\varphi (x)=(x-x_{1})(x-x_{2})\ldots (x-x_{n})$ , the numbers $x_{1}, x_{2}, \ldots , x_{m}$ are distinct, and $\omega (x)$ is any polynomial of degree not exceeding $m-2$ .

Problem 2520

In the equality

\ln (1-\lambda z)+\ln (1-\lambda ^{-1}z)=\ln \left(1-(\lambda +\lambda ^{-1})z+z^{2}\right)

both sides can be expanded in powers of $z$ if the numbers $\left|\lambda z\right|$ , $\left|\lambda^{-1}z\right|$ , and $\left|(\lambda +\lambda^{-1})z-z^{2}\right|$ are less than unity. By equating, on both sides, the coefficients of like powers of $z$ , prove the identity

\lambda ^{n}+\lambda ^{n-1}=t^{n}-nt^{n-2}+\dfrac {n(n-3)}{1 \cdot 2}t^{n-4}-

-\dfrac {n(n-4)(n-5)}{1 \cdot 2 \cdot 3}t^{n-6}+\ldots , \quad \text{where } t=\lambda +\lambda ^{-1} .

Problem 2521

Setting $\lambda =\cos \varphi +i\sin \varphi$ in the identity of the preceding problem, obtain, for the Chebyshev polynomial $T_{n}(x)=2^{-(n-1)}\cos (n\arccos x)$ , the explicit expression

T_{n}(x)=x^{n}-\dfrac {nx^{n-2}}{2}+\dfrac {n(n-3)}{2 \cdot 4} \cdot \dfrac {x^{n-4}}{2}-

-\dfrac {n(n-4)(n-5)}{2 \cdot 4 \cdot 6} \cdot \dfrac {x^{n-6}}{2^{2}}+\ldots

Problem 2522

Taking into account the behavior of the polynomial $T_{n}(x)$ on the interval $-1 \leqslant x \leqslant 1$ , prove that any other polynomial $x^{n}+a_{1}x^{n-1}+\ldots +a_{n}$ with real coefficients cannot remain, in absolute value, everywhere less than $2^{-(n-1)}$ — that is, that the Chebyshev polynomial deviates least from zero on the interval $(-1, 1)$ .

Problem 2523

$F(x)=f(x)\varphi (x)^{k}$ , where $f(x)$ and $\varphi (x)$ are relatively prime polynomials with no repeated roots. Setting $F(x)=P(x)Q(x)$ , $F'(x)=P(x)R(x)$ , where $Q(x)$ and $R(x)$ are relatively prime, prove that $\varphi (x)$ is the greatest common divisor of the polynomials $Q(x)$ and $R(x)-kQ'(x)$ .

Remark.

The result is due to M. V. Ostrogradsky.

Section 6.3A Polynomial with Real Coefficients. Rolle's Theorem[39/15%]

Problem 2524

For what $\lambda$ are all the roots of the polynomial $x^{3}-3x+\lambda$ real?

Problem 2525

The polynomials

\varphi (x)=(x-a_{1})(x-a_{2})\ldots (x-a_{n}); \qquad \psi (x)=(x-b_{1})(x-b_{2})\ldots (x-b_{n})

have roots $a_{1} < b_{1} < a_{2} < b_{2} < \cdots < a_{n} < b_{n}$ , real and interlacing. Prove that for $\lambda > 0$ the roots of the polynomial $\varphi (x)+\lambda \psi (x)$ are real and interlace with the roots of the polynomials $\varphi (x)$ and $\psi (x)$ .

Problem 2526

Under the same conditions, and for $\mu > \lambda > 0$ , prove that the roots of the equations $\varphi (x)+\lambda \psi (x)=0$ , $\varphi (x)+\mu \psi (x)=0$ are real, simple, and mutually interlacing, where $\varphi (x)$ , $\psi (x)$ are polynomials with real, interlacing roots $a_{1} < b_{1} < a_{2} < b_{2} < \cdots < a_{n} < b_{n}$ , as in problem 2525.

Problem 2527

Prove that, under the same conditions (see problem 2525), the roots of the equation $\varphi (x)+\lambda \psi (x)=0$ increase together with $\lambda$ .

Problem 2528

For what values of $p$ and $q$ does the equation $x^{5}+px+q=0$ have three real roots?

Problem 2529

Prove that the equation $x^{m}=1+\alpha x^{m+n}$ , where $m$ and $n$ are odd, $\alpha > 0$ , has two positive roots when $(m+n)^{m+n}\alpha^{m} < m^{m}n^{n}$ .

Problem 2530

Prove that, if the roots of the equation $x^{3}-3px+q=0$ , $p > 0$ are real, each of them is, in absolute value, less than $2\sqrt{p}$ . If, on the other hand, the roots $x_{1}$ and $x_{2}$ are complex, not real, then $\left|x_{3}\right| > 2\sqrt{p}$ .

Problem 2531

Prove that the equation $x^{\lambda_{1}}+a_{1}x^{\lambda_{2}}+\ldots +a_{n}=0$ , where $\lambda_{1} > \lambda_{2} > \cdots > \lambda_{n-1} > 0$ , $a_{1}a_{2}\ldots a_{n} \neq 0$ , cannot have more than $n$ positive roots.

Problem 2532

Prove that the polynomial $x^{n}+a_{1}x^{n_{1}}+a_{2}x^{n_{2}}+\ldots +a_{m-1}x^{n_{m-1}}+a_{m}$ , where $n > n_{1} > n_{2} > \cdots > n_{m-1}$ are integers, $a_{1}a_{2}\ldots a_{m} \neq 0$ , cannot have more than $2m$ real roots.

Problem 2533

Prove that all the roots of the equation $\dfrac {d^{m}x^{m}(x-1)^{m}}{dx^{m}}=0$ lie in the interval $(0, 1)$ .

Problem 2534

All the roots of the polynomial $a_{0}x^{n}+a_{1}x^{n-1}+\ldots +a_{n}$ are real. Prove that all the roots of the polynomial

(n+1)^{s}a_{0}x^{n}+n^{s}a_{1}x^{n-1}+\ldots +a_{n}, \quad s \text{ an integer}

are real as well.

Problem 2535

Prove that all the roots of the polynomial

(m+1)^{m-1}x^{m}+m^{m}x^{m-1}+\dfrac {m(m-1)^{m}}{1 \cdot 2}x^{m-2}+\ldots +1=0

lie in the interval $(-1, 0)$ .

Problem 2536

All the roots of the polynomial $a_{0}x^{n}+a_{1}x^{n-1}+\ldots +a_{n}$ are real. Prove that the roots of the equation

a_{s}x^{k}+C_{k}^{1}a_{s+1}x^{k-1}+C_{k}^{2}a_{s+2}x^{k-2}+\ldots +a_{s+k}=0, \qquad s+k \leqslant n

are real as well.

Problem 2537

Under the same condition (see problem 2536), prove the inequalities:

a_{k}^{2} \geqslant a_{k-1}a_{k+1};

(a_{k}a_{k+3}-a_{k+1}a_{k+2})^{2} \leqslant 4(a_{k+1}^{2}-a_{k}a_{k+2})(a_{k+2}^{2}-a_{k+1}a_{k+3}) .

Problem 2538

The roots of the polynomial $f(x)=a_{0}x^{n}+a_{1}x^{n-1}+\ldots +a_{n}$ are real and distinct. Prove that the polynomial

nf(x)f''(x)-(n-1)f'(x)^{2}

has no real roots.

Problem 2539

All the roots of the polynomial $f(x)$ are real. Prove that the roots of the polynomial $f(x)+\lambda f'(x)$ are also real, if $\lambda$ is real.

Problem 2540

Under the same condition (see problem 2539), suppose the polynomial $f(x)+m$ has $k$ imaginary¹ roots. Prove that the equation $f'(x)^{2}-f(x)f''(x)-mf'''(x)=0$ has at least $k$ real roots.

Imaginary numbers are complex numbers with nonzero imaginary part (i.e., not real; not to be confused with purely imaginary numbers). ↩

Problem 2541

Prove the theorem: if all the roots of the polynomial $f(x)$ are real, then for $\lambda > 0$ all the roots of the polynomial $xf'(x)+\lambda f(x)$ are real as well. (Laguerre.)

Problem 2542

Prove the theorem: if all the roots of the polynomial $f(x)=a+a_{1}x+a_{2}x^{2}+\ldots +a_{n}x^{n}$ are real, and all the roots of the polynomial $\varphi (\nu )$ are real and nonpositive, then all the roots of the polynomial $F(x)=\sum a_{\nu }\varphi (\nu )x^{\nu }$ are real as well. (Laguerre.)

Problem 2543

Prove the theorem: if all the roots of the polynomial $f(x)=a+a_{1}x+a_{2}x^{2}+\ldots +a_{n}x^{n}$ are real, then all the roots of the polynomial $F(x)=\sum a_{\nu }\psi (\nu )x^{\nu }$ are real as well, where $\psi (\nu )=e^{A\nu }\prod \left(1+\dfrac {\nu }{a_{\nu }}\right)e^{-\frac{\nu }{a_{\nu }}}$ , the numbers $a_{\nu }$ are positive, and $A$ is real. (Laguerre.)

Problem 2544

If the roots of the polynomial $f(x)=\sum a_{\nu }x^{\nu }$ and the point $x=0$ lie inside some convex contour, then the roots of the polynomial $F(x)=\sum a_{\nu }\psi (\nu )x^{\nu }$ , where $\psi (\nu )=e^{A\nu }\prod \left(1+\dfrac {\nu }{a_{\nu }}\right)e^{-\frac{\nu }{a_{\nu }}}$ , the numbers $a_{\nu }$ are positive, $A < 0$ , also lie inside the same contour.

Problem 2545

The roots of the polynomials $f'(x)$ , $f''(x)$ , , $f^{(n-1)}(x)$ , where $f(x)$ is a polynomial of degree $n$ , lie inside any convex contour inside which the roots of $f(x)$ lie.

Problem 2546

If the function $\Phi (x)=f(x)+af'(x)+a^{2}f''(x)+\ldots +a^{n}f^{(n)}(x)$ , where $f(x)$ is a polynomial of degree $n$ , has real roots, then so does the polynomial $f(x)$ .

Problem 2547

If the roots of the polynomial $f(x)=P(x)+iQ(x)$ , where the polynomials $P(x)$ and $Q(x)$ have real coefficients, lie on one side of the real axis, then the roots of the equations $P(x)=0$ and $Q(x)=0$ are real and interlacing. (Hermite.)

Problem 2548

Applying Rolle's Theorem to the function $x^{m}e^{x}$ , prove that the equation

x^{m}+\dfrac {m}{1^{2}}x^{m-1}+\dfrac {m(m-1)}{1^{2} \cdot 2^{2}}x^{m-2}+\ldots +\dfrac {1}{1 \cdot 2 \cdot 3 \ldots m}=0

has all its roots real.

Problem 2549

Prove the reality of the roots of the equation

x^{n}+\left(\dfrac {n}{1}\right)^{m}x^{n-1}+\left(\dfrac {n(n-1)}{1 \cdot 2}\right)^{m}x^{n-2}+\ldots +1=0; \qquad m=1,2,3,\ldots

Problem 2550

Prove the reality of the roots of the equation $(1+x^{2})^{n+1}\dfrac {d^{n}(1+x^{2})^{-1}}{dx^{n}}=0$ .

Problem 2551

Prove the reality of the roots of the equation $x^{2n}e^{-\frac{1}{x}}\dfrac {d^{n}e^{\frac{1}{x}}}{dx^{n}}=0$ .

Problem 2552

Prove the reality of the roots of the equation $\dfrac {A_{1}}{x-a_{1}}+\dfrac {A_{2}}{x-a_{2}}+\ldots +\dfrac {A_{n}}{x-a_{n}}=0$ , $A_{\nu } > 0$ , $a_{1} < a_{2} < \cdots < a_{n}$ .

Problem 2553

Prove the reality of the roots of the equation $\dfrac {A_{1}}{x-a_{1}}+\dfrac {A_{2}}{x-a_{2}}+\ldots +\dfrac {A_{n}}{x-a_{n}}=0$ , $a_{1} < a_{2} < \cdots < a_{n}$ , $A_{\nu } > 0$ for $\nu < m$ , $A_{\nu } < 0$ for $\nu \geqslant m$ , $A_{1}+A_{2}+\ldots +A_{n}=0$ .

Problem 2554

Prove that the roots of the Hermite--Chebyshev polynomial $e^{x^{2}}\dfrac {d^{n}e^{-x^{2}}}{dx^{n}}$ are real and lie in the interval $\left(-\sqrt{2n+1}, \sqrt{2n+1}\right)$ .

Problem 2555

How many real roots does the equation $1+x+\dfrac {x^{2}}{2!}+\dfrac {x^{3}}{3!}+\ldots +\dfrac {x^{n}}{n!}=0$ have?

Problem 2556

How many real roots does the equation $1+x+\dfrac {x^{2}}{2}+\dfrac {x^{3}}{3}+\ldots +\dfrac {x^{n}}{n}=0$ have?

Problem 2557

In the expansion in powers of $\lambda$

\dfrac {\sh \lambda +x\ch \lambda }{\ch \lambda +x\sh \lambda }=\sum _{n=0}^{\infty }\varphi _{n}(x)\lambda ^{n}

the coefficient $\varphi_{n}(x)$ of $\lambda^{n}$ is a polynomial of degree $n+1$ . Prove that all the roots of $\varphi_{n}(x)$ lie in the interval $(-1, 1)$ . (Hermite.)

Problem 2558

Find an upper bound for the number of positive and negative roots, and a lower bound for the number of imaginary¹ roots, of the equation $x^{10}-7x-2=0$ .

Imaginary numbers are complex numbers with nonzero imaginary part (i.e., not real; not to be confused with purely imaginary numbers). ↩

Problem 2559

In the equation $a_{0}x^{n}+a_{1}x^{n-1}+\ldots +a_{n}=0$ , for some $k$ the equality $a_{k}^{2}=a_{k-1}a_{k+1}$ holds. Show that the roots include imaginary¹ ones.

Imaginary numbers are complex numbers with nonzero imaginary part (i.e., not real; not to be confused with purely imaginary numbers). ↩

Problem 2560

In a similar equation, four consecutive coefficients form an arithmetic progression. Prove that the equation has imaginary¹ roots. (Hermite.)

Imaginary numbers are complex numbers with nonzero imaginary part (i.e., not real; not to be confused with purely imaginary numbers). ↩

Problem 2561

Prove that the equation

f(n)x^{n}+f(n-1)x^{n-1}+\ldots +f(0)=0,

where $f(x)$ is a polynomial of degree not exceeding $n-2$ , has imaginary¹ roots.

Imaginary numbers are complex numbers with nonzero imaginary part (i.e., not real; not to be confused with purely imaginary numbers). ↩

Problem 2562

Prove that, for real $a$ and $b$ , the equation

x^{n}+(a+b)x^{n-1}+(a^{2}+ab+b^{2})x^{n-2}+\ldots +(a^{n}+a^{n-1}b+\ldots +b^{n})=0

has at most one real root.

Section 6.4Rational Fractions. Decomposition into Partial Fractions[25/92%]

Problem 2563

Decompose the following fraction into partial fractions: $\dfrac {x^{2}+1}{x(x^{2}-1)}$ .

Problem 2564

Decompose the following fraction into partial fractions: $\dfrac {2}{(x-1)(x-2)(x-3)}$ .

Problem 2565

Decompose the following fraction into partial fractions: $\dfrac {x^{5}-x^{3}-x^{2}}{x^{2}-1}$ .

Problem 2566

Decompose the following fraction into partial fractions: $\dfrac {x^{2}}{(x-1)^{2}(x+1)}$ .

Problem 2567

Decompose the following fraction into partial fractions: $\dfrac {4}{(x^{2}-1)^{2}}$ .

Problem 2568

Decompose the following fraction into partial fractions: $\dfrac {x^{6}-x^{2}+1}{(x-1)^{3}}$ .

Problem 2569

Decompose the following fraction into partial fractions: $\dfrac {x^{2}+1}{(x^{2}+x+1)^{2}}$ .

Problem 2570

Decompose the following fraction into partial fractions: $\dfrac {x^{4}+1}{x^{3}(x^{2}+1)}$ .

Problem 2571

Decompose the following fraction into partial fractions: $\dfrac {2}{x^{4}+2x^{3}+2x^{2}+2x+1}$ .

Problem 2572

Decompose the following fraction into partial fractions: $\dfrac {8}{x^{4}+4}$ .

Problem 2573

Decompose the following fraction into partial fractions: $\dfrac {2x^{4}}{x^{4}+x^{2}+1}$ .

Problem 2574

Decompose the following fraction into partial fractions: $\dfrac {x^{m-1}}{x^{n}-1}$ ; $m \leqslant n$ .

Problem 2575

Decompose the following fraction into partial fractions: $\dfrac {1-x^{2}}{1-2x\cos \varphi +x^{2}}$ .

Problem 2576

Decompose the following fraction into partial fractions: $\dfrac {1}{x^{2n}+1}$ .

Problem 2577

Decompose the following fraction into partial fractions: $\dfrac {1}{x^{2n+1}+1}$ .

Problem 2578

Decompose the following fraction into partial fractions: $\dfrac {3x^{3}+15x+6}{x^{4}+x^{3}+3x^{2}+2x+2}$ .

Problem 2579

Decompose the following fraction into partial fractions: $\dfrac {2x^{3}+4x}{x^{4}+x^{2}+4}$ .

Problem 2580

Decompose the following fraction into partial fractions: $\dfrac {3x^{2}-3}{x^{3}-3x+1}$ .

Problem 2581

Determine the coefficients in the expansion of the function

f(x)=\dfrac {a_{0}x^{n}+a_{1}x^{n-1}+\ldots +a_{n}}{(x-x_{1})(x-x_{2})\ldots (x-x_{n})},

in decreasing powers of $x$ , if the roots of the denominator are simple.

Problem 2582

Find the sum of the series $\sum_{n=0}^{\infty }u_{n}^{2}x^{n}$ , where

\dfrac {x+a}{x^{2}+2ax+1}=u_{0}+u_{1}x+u_{2}x^{2}+\ldots

Problem 2583

If $x_{1}$ is greater in absolute value than the other roots of the equation $f(x)=x^{n}+a_{1}x^{n-1}+\ldots +a_{n}=0$ , then

x_{1}=\lim _{n \rightarrow \infty }\dfrac {u_{n+1}}{u_{n}},

where $u_{n}$ is the coefficient of $x^{-n}$ in the expansion of the fraction $\dfrac {\varphi (x)}{f(x)}=\sum u_{n}x^{-n}$ , if $f'(x_{1}) \neq 0$ . The numbers $u_{n}$ can be found successively from the equality

u_{s+n}+a_{1}u_{s+n-1}+\ldots +a_{n}u_{s}=0 .

Problem 2584

The series expansion of a proper rational fraction has the form $\dfrac {\varphi (x)}{\psi (x)}=u_{0}+u_{1}x+u_{2}x^{2}+u_{3}x^{3}+\ldots$ , where the numbers $u_{n}$ take only the values $0$ and $\pm 1$ . Prove that

\psi (x)=1-x^{m}, \qquad \varphi (x)=a_{1}x^{m-1}+a_{2}x^{m-2}+\ldots +a_{m},

where the numbers $a_{\nu }$ can likewise take only the values $0$ and $\pm 1$ . (Laguerre.)

Problem 2585

Prove the equality

\dfrac {1}{1-x_{1}}+\dfrac {1}{1-x_{2}}+\ldots +\dfrac {1}{1-x_{n-1}}=\dfrac {n-1}{2},

where $x_{\nu }$ are the roots of the equation $x^{n}=1$ other than unity.

Problem 2586

All the roots of the equation $f(x)=0$ of degree $n$ are real. Prove that in the interval $\left(-\dfrac {x_{\nu }-x_{\nu -1}}{n}+x_{\nu }, \dfrac {x_{\nu +1}-x_{\nu }}{n}+x_{\nu }\right)$ , where $x_{\nu -1}, x_{\nu }, x_{\nu +1}$ are three consecutive roots of $f(x)$ , there are no roots of $f'(x)$ .

Problem 2587

All the roots of the polynomial $f(x)$ of degree $n$ are real, as is the number $m$ . Prove that the modulus of the imaginary part of each root of the equation $f(x)+imf'(x)=0$ is less than $\left|nm\right|$ .

Section 6.5Determinants. Systems of Linear Equations[50/72%]

Problem 2588

Evaluate the determinant

\begin{vmatrix} -1 & 2 & 2 \\ 2 & -1 & 2 \\ 2 & 2 & -1 \end{vmatrix} .

Problem 2589

Evaluate the determinant

\begin{vmatrix} 10 & -5 & 10 \\ 11 & -2 & 10 \\ 2 & -14 & -5 \end{vmatrix} .

Problem 2590

Evaluate the determinant

\begin{vmatrix} 3 & 6 & 2 \\ 2 & 3 & 6 \\ 6 & 2 & 3 \end{vmatrix} .

Problem 2591

Evaluate the determinant

\begin{vmatrix} 5 & 0 & 4 & 2 \\ 1 & -1 & 2 & 1 \\ 4 & 1 & 2 & 0 \\ 1 & 1 & 1 & 1 \end{vmatrix} .

Problem 2592

Evaluate the determinant

\begin{vmatrix} 3 & 1 & 2 & 3 \\ 4 & -1 & 2 & 4 \\ 1 & -1 & 1 & 1 \\ 4 & -1 & 2 & 5 \end{vmatrix} .

Problem 2593

Evaluate the determinant

\begin{vmatrix} 1 & 4 & 9 & 16 \\ 4 & 9 & 16 & 25 \\ 9 & 16 & 25 & 36 \\ 16 & 25 & 36 & 49 \end{vmatrix} .

Problem 2594

Evaluate the determinant

\begin{vmatrix} 1 & 1 & 1 \\ 1 & 1+a & 1 \\ 1 & 1 & 1+b \end{vmatrix} .

Problem 2595

Evaluate the determinant

\begin{vmatrix} 1+a & 1 & 1 & 1 \\ 1 & 1+b & 1 & 1 \\ 1 & 1 & 1+c & 1 \\ 1 & 1 & 1 & 1+d \end{vmatrix} .

Problem 2596

Evaluate the determinant

\begin{vmatrix} x & y & x+y \\ y & x+y & x \\ x+y & x & y \end{vmatrix} .

Problem 2597

Evaluate the determinant

\begin{vmatrix} (b+c)^{2} & a^{2} & a^{2} \\ b^{2} & (c+a)^{2} & b^{2} \\ c^{2} & c^{2} & (a+b)^{2} \end{vmatrix} .

Problem 2598

Evaluate the determinant

\begin{vmatrix} 1 & c & -b \\ -c & 1 & a \\ b & -a & 1 \end{vmatrix} .

Problem 2599

Evaluate the determinant

\begin{vmatrix} a & 1 & 0 & 0 \\ -1 & b & 1 & 0 \\ 0 & -1 & c & 1 \\ 0 & 0 & -1 & d \end{vmatrix} .

Problem 2600

Evaluate the determinant

\begin{vmatrix} 1 & 1 & 1 \\ x & y & z \\ x^{2} & y^{2} & z^{2} \end{vmatrix} .

Problem 2601

Find the value of the Vandermonde determinant

\Delta =\begin{vmatrix} 1 & 1 & 1 & \ldots & 1 \\ x_{1} & x_{2} & x_{3} & \ldots & x_{n} \\ x_{1}^{2} & x_{2}^{2} & x_{3}^{2} & \ldots & x_{n}^{2} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ x_{1}^{n-1} & x_{2}^{n-1} & x_{3}^{n-1} & \ldots & x_{n}^{n-1} \end{vmatrix},

assuming first that the numbers $x_{1}, x_{2}, \ldots , x_{n}$ are pairwise distinct.

Problem 2602

Evaluate the determinants $\Delta_{n}$ and $\Delta_{n+1}$ obtained from the Vandermonde determinant (see problem 2601) by replacing, in the last row, the exponent $n-1$ by $n$ or by $n+1$ .

Problem 2603

Evaluate the values $\Delta_{1}$ and $\Delta_{2}$ of the determinants obtained from the Vandermonde determinant $\Delta$ (see problem 2601) by differentiating with respect to $x_{1}$ and replacing $x_{1}$ by $x_{2}$ , or $x_{1}$ and $x_{2}$ by $x_{3}$ :

\Delta _{1}=\begin{vmatrix} 0 & 1 & 1 & \ldots & 1 \\ 1 & x_{2} & x_{3} & \ldots & x_{n} \\ 2x_{2}^{2} & x_{2}^{2} & x_{3}^{2} & \ldots & x_{n}^{2} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ (n-1)x_{2}^{n-1} & x_{2}^{n-1} & x_{3}^{n-1} & \ldots & x_{n}^{n-1} \end{vmatrix};

\Delta _{2}=\begin{vmatrix} 0 & 0 & 1 & \ldots & 1 \\ 0 & 1 & x_{3} & \ldots & x_{n} \\ 2 & 2x_{3} & x_{3}^{2} & \ldots & x_{n}^{2} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ n(n-1)x_{3}^{n-2} & (n-1)x_{3}^{n-1} & x_{3}^{n-1} & \ldots & x_{n}^{n-1} \end{vmatrix} .

Problem 2604

Prove the equality:

\begin{vmatrix} 1 & 1 & \ldots & 1 \\ \binom {x_{1}}{1} & \binom {x_{2}}{1} & \ldots & \binom {x_{n}}{1} \\ \binom {x_{1}}{2} & \binom {x_{2}}{2} & \ldots & \binom {x_{n}}{2} \\ \vdots & \vdots & \vdots & \vdots \\ \binom {x_{1}}{n-1} & \binom {x_{2}}{n-1} & \ldots & \binom {x_{n}}{n-1} \end{vmatrix}=\dfrac {\Delta }{1!2!3!\ldots (n-1)!},

where $\dbinom {x}{m}=\dfrac {x(x-1)\ldots (x-m+1)}{1 \cdot 2 \ldots m}$ , $\Delta$ is the Vandermonde determinant (see problem 2601).

Problem 2605

Evaluate the determinant

\begin{vmatrix} \cos (a-b) & \cos (b-c) & \cos (c-a) \\ \cos (a+b) & \cos (b+c) & \cos (c+a) \\ \sin (a+b) & \sin (b+c) & \sin (c+a) \end{vmatrix} .

Problem 2606

Evaluate the determinant

\begin{vmatrix} \sin a & \cos a & \sin 2a \\ \sin b & \cos b & \sin 2b \\ \sin c & \cos c & \sin 2c \end{vmatrix} .

Problem 2607

Evaluate the determinant

\begin{vmatrix} \sin a & \sin b & \sin c \\ \sin 3a & \sin 3b & \sin 3c \\ \sin 5a & \sin 5b & \sin 5c \end{vmatrix} .

Problem 2608

Evaluate the determinant

\begin{vmatrix} \sin ^{3}\alpha & \sin ^{2}\alpha \cos \alpha & \sin \alpha \cos ^{2}\alpha & \cos ^{3}\alpha \\ \sin ^{3}\beta & \sin ^{2}\beta \cos \beta & \sin \beta \cos ^{2}\beta & \cos ^{3}\beta \\ \sin ^{3}\gamma & \sin ^{2}\gamma \cos \gamma & \sin \gamma \cos ^{2}\gamma & \cos ^{3}\gamma \\ \sin ^{3}\delta & \sin ^{2}\delta \cos \delta & \sin \delta \cos ^{2}\delta & \cos ^{3}\delta \end{vmatrix} .

Problem 2609

Prove the equality: $D_{n}=a_{n}D_{n-1}+D_{n-2}$ ;

D_{n}=\begin{vmatrix} a_{1} & 1 & 0 & \ldots & 0 & 0 \\ -1 & a_{2} & 1 & \ldots & 0 & 0 \\ 0 & -1 & a_{3} & \ldots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \ldots & a_{n-1} & 1 \\ 0 & 0 & 0 & \ldots & -1 & a_{n} \end{vmatrix} .

Problem 2610

Prove the equality

\begin{vmatrix} a_{1} & \lambda & \ldots & \lambda \\ \lambda & a_{2} & \ldots & \lambda \\ \vdots & \vdots & \vdots & \vdots \\ \lambda & \lambda & \ldots & a_{n} \end{vmatrix}=\varphi (\lambda )-\lambda \dfrac {d\varphi }{d\lambda },

where $\varphi (\lambda )=(a_{1}-\lambda )(a_{2}-\lambda )\ldots (a_{n}-\lambda )$ .

Problem 2611

Prove the equality

\begin{vmatrix} a_{1} & a_{2} & a_{3} & \ldots & a_{n} \\ a_{2} & a_{3} & a_{4} & \ldots & a_{1} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ a_{n} & a_{1} & a_{2} & \ldots & a_{n-1} \end{vmatrix}=(-1)^{\frac{(n-1)(n-2)}{2}}\varphi (\alpha _{1})\varphi (\alpha _{2})\ldots \varphi (\alpha _{n}),

where $\varphi (\alpha )=a_{1}+a_{2}\alpha +\ldots +a_{n}\alpha^{n-1}$ ; $\alpha$ is a root of the equation $\alpha^{n}=1$ .

Problem 2612

Prove the equality

\begin{vmatrix} 1 & 2 & 3 & \ldots & n \\ 2 & 3 & 4 & \ldots & 1 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ n & 1 & 2 & \ldots & n-1 \end{vmatrix}=(-1)^{\frac{n(n-1)}{2}} \cdot \dfrac {n(n+1)}{2}n^{n-2} .

Problem 2613

Prove the inequality:

\Delta =\begin{vmatrix} A_{1} & a_{2} & a_{3} & \ldots & a_{n} \\ b_{1} & B_{2} & b_{3} & \ldots & b_{n} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ m_{1} & m_{2} & m_{3} & \ldots & M_{n} \end{vmatrix} \neq 0,

where $A_{1} > \left|a_{2}\right|+\left|a_{3}\right|+\ldots +\left|a_{n}\right|$ , $B_{2} > \left|b_{1}\right|+\left|b_{3}\right|+\ldots +\left|b_{n}\right|, \ldots$ , $M_{n} > \left|m_{1}\right|+\left|m_{2}\right|+\ldots +\left|m_{n-1}\right|$ .

Problem 2614

Prove that the determinant of the preceding problem (see problem 2613) is positive, if, under the same conditions, its entries are real.

Problem 2615

Prove that a determinant whose entries $a_{\nu \mu }$ and $a_{\mu \nu }$ , symmetric with respect to the main diagonal, are complex conjugates, has a real value.

Problem 2616

The entries of a determinant of order $n$ are related by $a_{\nu \nu }=0$ , $a_{\nu \mu }=ia_{\mu \nu }$ , $\nu > \mu$ , where $a_{\nu \mu }$ is a real number. Find the values of $n$ for which the determinant has a real value for any choice of $a_{\mu \nu }$ .

Problem 2617

For what $n$ is the same determinant (see problem 2616) equal to a purely imaginary number?

Problem 2618

Show that for odd $n$ the same determinant (see problem 2616) equals $A(1 \pm i)$ , where $A$ is real.

Problem 2619

Prove the equality:

\begin{vmatrix} 1+x_{1}y_{1} & 1+x_{1}y_{2} & \ldots & 1+x_{1}y_{n} \\ 1+x_{2}y_{1} & 1+x_{2}y_{2} & \ldots & 1+x_{2}y_{n} \\ \vdots & \vdots & \vdots & \vdots \\ 1+x_{n}y_{1} & 1+x_{n}y_{2} & \ldots & 1+x_{n}y_{n} \end{vmatrix}=0, \quad n > 2 .

Problem 2620

Find the value of the determinant

\Delta =\begin{vmatrix} \alpha & \beta & \gamma \\ \alpha _{1} & \beta _{1} & \gamma _{1} \\ \alpha _{2} & \beta _{2} & \gamma _{2} \end{vmatrix},

whose entries are the cosines of the angles between the new axes and the old ones; the coordinate systems are assumed to be rectangular.

Problem 2621

Show that the determinant

\begin{vmatrix} x_{1} & y_{1} & z_{1} \\ x_{2} & y_{2} & z_{2} \\ x_{3} & y_{3} & z_{3} \end{vmatrix},

whose entries are the projections of three vectors onto the axes, does not change under a transformation of rectangular coordinates.

Problem 2622

Solve the system of equations

\begin{cases} 2x \hphantom {{}+4y} -z=1, \\ 2x+4y-z=1, \\ -x+8y+3z=2 . \end{cases}

Problem 2623

Solve the system of equations

\begin{cases} x+y+z=a, \\ x+(1+a)y+z=2a, \\ x+y+(1+a)z=0 . \end{cases}

Problem 2624

Solve the system of equations

\begin{cases} x+2y+z+t=0, \\ 2x+y+z+2t=0, \\ x+2y+2z+t=0, \\ x+y+z+t=0 . \end{cases}

Problem 2625

Solve the system of equations

\begin{cases} 5x+4z+2t=3, \\ x-y+2z+t=1, \\ 4x+y+2z=1, \\ x+y+z+t=0 . \end{cases}

Problem 2626

Solve the system of equations

\begin{cases} x-4y+2z=-1, \\ 2x-3y-z-5t=-7, \\ 3x-7y+z-5t=-8, \\ y-z-t=-1 . \end{cases}

Problem 2627

From the system of equations

\begin{cases} x+y+2z=0, \\ 2x-y+z=0, \\ x+y+z=t, \\ 2x-2y+z=u \end{cases}

find $u$ as a function of $t$ .

Problem 2628

Solve the system:

\begin{cases} x_{1}+x_{2}+x_{3}+\ldots +x_{n}=n, \\ x_{1}+2x_{2}+3x_{3}+\ldots +nx_{n}=\dfrac {n(n+1)}{2}, \\ x_{1}+3x_{2}+6x_{3}+\ldots +\dfrac {n(n+1)}{2}x_{n}=\dfrac {n(n+1)(n+2)}{6}, \\ \vdots \\ x_{1}+nx_{2}+\dfrac {n(n+1)}{2}x_{3}+\ldots +\dfrac {n(n+1)\ldots (2n-2)}{1 \cdot 2 \cdot 3 \ldots (n-1)}x_{n}=\dfrac {n(n+1)\ldots (2n-1)}{1 \cdot 2 \cdot 3 \ldots n} . \end{cases}

Problem 2629

Solve the system:

\begin{cases} x_{2}+x_{3}+\ldots +x_{n-1}+x_{n}=1, \\ x_{1}+x_{3}+\ldots +x_{n-1}+x_{n}=2, \\ x_{1}+x_{2}+\ldots +x_{n-1}+x_{n}=3, \\ \vdots \\ x_{1}+x_{2}+x_{3}+\ldots +x_{n-1}=n . \end{cases}

Problem 2630

Show that when the determinant $\Delta$ , formed from the $n^{2}$ entries $a_{\mu \nu }$ , equals zero, the proportion $\dfrac {A_{1\mu }}{A_{1\nu }}=\dfrac {A_{2\mu }}{A_{2\nu }}=\cdots =\dfrac {A_{n\mu }}{A_{n\nu }}$ holds, where $A_{kl}$ is the cofactor of the entry $a_{kl}$ .

Problem 2631

Solve the system of equations:

a_{\nu 1}x_{1}+a_{\nu 2}x_{2}+\ldots +a_{\nu n}x_{n}=\lambda _{\nu }, \quad \nu =1,2,\ldots ,n,

assuming that the array formed from the coefficients of these equations is orthogonal. Having found the solution, show that $A_{\nu \mu }=a_{\nu \mu } \cdot \Delta$ , where $A_{\nu \mu }$ is the cofactor of the entry $a_{\nu \mu }$ .

Problem 2632

Given

\Delta =\begin{vmatrix} a_{11} & a_{12} & \ldots & a_{1p} \\ a_{21} & a_{22} & \ldots & a_{2p} \\ \vdots & \vdots & \vdots & \vdots \\ a_{p1} & a_{p2} & \ldots & a_{pp} \end{vmatrix} \neq 0,

and all the determinants of order $p+1$ of the matrix

\begin{pmatrix} a_{11} & a_{12} & \ldots & a_{1m} \\ a_{21} & a_{22} & \ldots & a_{2m} \\ \vdots & \vdots & \vdots & \vdots \\ a_{n1} & a_{n2} & \ldots & a_{nm} \end{pmatrix}, \quad m > p, \; n > p,

containing $\Delta$ as a minor are equal to zero. Prove that the rank of the matrix equals $p$ .

Problem 2633

Under what condition do three points $(x_{1},y_{1})$ , $(x_{2},y_{2})$ , $(x_{3},y_{3})$ lie on a single line?

Problem 2634

Under what condition do three lines $a_{1}x+b_{1}y+c_{1}=0$ , $a_{2}x+b_{2}y+c_{2}=0$ , $a_{3}x+b_{3}y+c_{3}=0$ intersect at a single point?

Problem 2635

Under what condition do four points $M_{1}(x_{1},y_{1},z_{1})$ , $M_{2}(x_{2},y_{2},z_{2})$ , $M_{3}(x_{3},y_{3},z_{3})$ , $M_{4}(x_{4},y_{4},z_{4})$ lie in a single plane?

Problem 2636

Under what conditions do $n$ points $M_{\nu }(x_{\nu },y_{\nu },z_{\nu })$ , $\nu =1,2,\ldots ,n$ , lie in a single plane, and under what conditions do they lie on a single line?

Problem 2637

Under what conditions do $n$ planes $A_{\nu }x+B_{\nu }y+C_{\nu }z+D_{\nu }=0$ , $\nu =1,2,\ldots ,n$ , intersect at a single point, or in a single line?

Section 6.6Matrices. The Characteristic Equation. Quadratic Forms[28/64%]

Problem 2638

The quantity $\lambda$ equals the largest positive root of the characteristic equation of a matrix $\left\| a_{ik}\right\|$ with positive entries. What are the signs of the solutions of the system

\lambda x_{\nu }=a_{\nu 1}x_{1}+a_{\nu 2}x_{2}+\ldots +a_{\nu n}x_{n}, \quad \nu =1,2,\ldots ,n?

Problem 2639

Find the form of the entries of a rectangular matrix

\begin{pmatrix} a_{11} & a_{12} & \ldots & a_{1m} \\ \vdots & \vdots & \vdots & \vdots \\ a_{n1} & a_{n2} & \ldots & a_{nm} \end{pmatrix},

whose rank equals one.

Problem 2640

Show that a matrix of rank $r$ can be represented as a sum of $r$ matrices of rank one. What consequence for bilinear forms follows from this?

Problem 2641

If $A_{1}$ and $B_{1}$ are the transposes of the matrices $A$ and $B$ , then $(AB)_{1}=B_{1}A_{1}$ . Prove this.

Problem 2642

Find the rules of operation for hypercomplex numbers of the form $q=ae+bi+cj+dk$ , where $e$ , $i$ , $j$ , and $k$ denote the following matrices:

e=\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \qquad i=\begin{pmatrix} \sqrt{-1} & 0 \\ 0 & \sqrt{-1} \end{pmatrix},

j=\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}, \qquad k=\begin{pmatrix} 0 & \sqrt{-1} \\ \sqrt{-1} & 0 \end{pmatrix} .

Problem 2643

Find the form of a matrix that commutes with every matrix.

Problem 2644

Under what condition does the matrix $A$ of the linear transformation $y_{\nu }=a_{\nu 1}x_{1}+a_{\nu 2}x_{2}+\ldots +a_{\nu n}x_{n}$ satisfy $AA_{1}=E$ , where $E$ is the identity matrix, and $A_{1}$ is the transpose of $A$ ?

Problem 2645

Find the real roots of the equation

\begin{vmatrix} a_{11}-x & a_{12} & \ldots & a_{1n} \\ a_{21} & a_{22}-x & \ldots & a_{2n} \\ \vdots & \vdots & \vdots & \vdots \\ a_{n1} & a_{n2} & \ldots & a_{nn}-x \end{vmatrix}=0,

given that $a_{\nu \mu }=-a_{\mu \nu }$ , and all $a_{\nu \mu }$ are real.

Problem 2646

Prove that the characteristic equation of an orthogonal matrix is palindromic, i.e., its roots are related pairwise by $x_{\nu }x_{n-\nu }=1$ .

Problem 2647

The quadratic form $\sum_{\mu =1}^{n}\sum_{\nu =1}^{n}b_{\mu \nu }x_{\mu }x_{\nu }$ is definite, and the number $\lambda$ lies between the largest and smallest roots of the equation

\begin{vmatrix} a_{11}+xb_{11} & a_{12}+xb_{12} & \ldots & a_{1n}+xb_{1n} \\ a_{21}+xb_{21} & a_{22}+xb_{22} & \ldots & a_{2n}+xb_{2n} \\ \vdots & \vdots & \vdots & \vdots \\ a_{n1}+xb_{n1} & a_{n2}+xb_{n2} & \ldots & a_{nn}+xb_{nn} \end{vmatrix}=0 .

Prove that the quadratic form $\sum_{\mu =1}^{n}\sum_{\nu =1}^{n}(a_{\mu \nu }+\lambda b_{\mu \nu })x_{\mu }x_{\nu }$ is indefinite. Here $x_{\mu }$ and $x_{\nu }$ are independent variables.

Problem 2648

Prove that the roots of the characteristic equation of a matrix with entries $a_{\mu \nu }$ have real part lying between the largest and smallest roots of the characteristic equation of the symmetric matrix with entries $b_{\mu \nu }$ , where $2b_{\mu \nu }=a_{\mu \nu }+a_{\nu \mu }$ . Here all $a_{\mu \nu }$ are real.

Problem 2649

Find the bounds on the real part of a root of the characteristic equation of a skew-symmetric matrix with real entries.

Problem 2650

Prove that a root $s$ of the characteristic equation of a matrix with entries $a_{\mu \nu }$ is, in absolute value, less than the largest positive root $\sigma$ of the matrix with entries $\left|a_{\mu \nu }\right|$ .

Problem 2651

Prove that the largest of the positive roots of the characteristic equation of a matrix with positive entries is, in absolute value, greater than any other root of the same equation.

Problem 2652

Prove that the largest root of the characteristic equation of a matrix with positive entries $a_{\mu \nu }$ is, in absolute value, greater than any root of the characteristic equation of the matrix with entries $b_{\mu \nu }=\sqrt{a_{\mu \nu }a_{\nu \mu }}$ .

Problem 2653

Show that the roots of the characteristic equation of a matrix with positive entries $a_{\mu \nu }$ are, in absolute value, less than the largest root of the characteristic equation of the matrix with entries $b_{\mu \nu }$ , where $2b_{\mu \nu }=a_{\mu \nu }+a_{\nu \mu }$ .

Problem 2654

The entries of the matrix $\left\| b_{\mu \nu }\right\|$ are determined by the formulas:

b_{\mu +1,\nu }=b_{\mu 1}a_{1\nu }+b_{\mu 2}a_{2\nu }+\ldots +b_{\mu n}a_{n\nu };

\mu =1,2,\ldots ,n,

where $b_{11}, b_{12}, \ldots , b_{1n}$ and $a_{\mu \nu }$ are given numbers. Find the value of the product of the determinants

\begin{vmatrix} b_{11} & b_{12} & \ldots & b_{1n} \\ b_{21} & b_{22} & \ldots & b_{2n} \\ \vdots & \vdots & \vdots & \vdots \\ b_{n1} & b_{n2} & \ldots & b_{nn} \end{vmatrix} \cdot \begin{vmatrix} a_{11}-x & a_{12} & \ldots & a_{1n} \\ a_{21} & a_{22}-x & \ldots & a_{2n} \\ \vdots & \vdots & \vdots & \vdots \\ a_{n1} & a_{n2} & \ldots & a_{nn}-x \end{vmatrix}=0 .

Problem 2655

Find the elementary divisors of the matrix

\begin{pmatrix} 2-\lambda & 1 & 1 \\ 1 & 2-\lambda & 1 \\ 1 & 1 & 2-\lambda \end{pmatrix} .

Problem 2656

The same for the matrix (see problem 2655)

\begin{pmatrix} 5-\lambda & 30 & -48 \\ 3 & 14-\lambda & -24 \\ 3 & 15 & -25-\lambda \end{pmatrix} .

Problem 2657

Investigate the equivalence of the matrices

\begin{pmatrix} 1-\lambda & 1 & 0 \\ 0 & 1-\lambda & 0 \\ 0 & 0 & 2-\lambda \end{pmatrix}, \qquad \begin{pmatrix} 4-\lambda & 1 & -1 \\ -6 & -1-\lambda & 2 \\ 2 & 1 & 1-\lambda \end{pmatrix} .

Problem 2658

Show that the matrix $\left\| a_{\mu \nu }\right\|$ , substituted in place of $x$ into the characteristic equation of the same matrix

\begin{vmatrix} a_{11}-x & a_{12} & \ldots & a_{1n} \\ a_{21} & a_{22}-x & \ldots & a_{2n} \\ \vdots & \vdots & \vdots & \vdots \\ a_{n1} & a_{n2} & \ldots & a_{nn}-x \end{vmatrix}=0,

satisfies it.

Problem 2659

Form the elementary divisors of the matrix

\begin{pmatrix} \lambda & 1 & 0 & 0 & 0 \\ 0 & \lambda & 0 & 0 & 0 \\ 0 & 0 & \lambda & 0 & 0 \\ 0 & 0 & 0 & \lambda -1 & 0 \\ 0 & 0 & 0 & 0 & \lambda -1 \end{pmatrix},

reducing it to canonical form by elementary transformations.

Problem 2660

$M(b_{1},b_{2},\ldots ,b_{n})$ is a double point of the quadratic form

\sum _{\mu =1}^{n}\sum _{\nu =1}^{n}a_{\mu \nu }x_{\mu }x_{\nu }, \quad a_{\mu \nu }=a_{\nu \mu },

which vanishes at the point $N(c_{1},c_{2},\ldots ,c_{n})$ . Find the value of the form at points of the line $MN$ .

Problem 2661

Represent the following quadratic form as a sum of squares: $4xy+4xz+4yz$ .

Problem 2662

Represent the following quadratic form as a sum of squares: $8x^{2}+5y^{2}+2z^{2}-6yz+4xz-2xy$ .

Problem 2663

Represent the following quadratic form as a sum of squares: $3x^{2}+3y^{2}+3z^{2}+31v^{2}-2xy+2xz+10xv-6yz+2yv+6zv$ .

Problem 2664

The quadratic form $\sum_{\mu =1}^{n}\sum_{\nu =1}^{n}a_{\mu \nu }x_{\mu }x_{\nu }$ is transformed, by means of the linear transformation

x_{\mu }=c_{\mu 1}y_{1}+c_{\mu 2}y_{2}+\ldots +c_{\mu n}y_{n}, \quad \mu =1,2,\ldots ,n,

into the quadratic form $\sum_{\mu =1}^{n}\sum_{\nu =1}^{n}b_{\mu \nu }y_{\mu }y_{\nu }$ . Find the linear transformation that carries the form adjoint to the first into the form adjoint to the second.

Problem 2665

The forms $\sum_{\mu =1}^{n}\sum_{\nu =1}^{n}a_{\mu \nu }x_{\mu }x_{\nu }$ , $\sum_{\mu =1}^{n}\sum_{\nu =1}^{n}b_{\mu \nu }y_{\mu }y_{\nu }$ are positive. Prove that the form $\sum_{\mu =1}^{n}\sum_{\nu =1}^{n}a_{\mu \nu }b_{\mu \nu }x_{\mu }y_{\nu }$ is positive as well.

Section 6.7Symmetric Functions[33/82%]

Problem 2666

Using the fundamental theorem on symmetric functions, prove that an integer rational symmetric function of degree $m$ of the roots of the equation

x^{n}+p_{1}x^{n-1}+p_{2}x^{n-2}+\ldots +p_{n}=0

is rationally expressible in terms of $p_{1}, p_{2}, \ldots , p_{m}$ .

Problem 2667

Using the result of the preceding problem (see problem 2666), prove that the power sums of the roots $S_{1}, S_{2}, \ldots , S_{m}$ of the equations

x^{n}+p_{1}x^{n-1}+p_{2}x^{n-2}+\ldots +p_{n}=0

and

x^{m}+p_{1}x^{m-1}+p_{2}x^{m-2}+\ldots +p_{m}=0

are the same. Here $m \leqslant n$ .

Problem 2668

Drawing on the result of the two preceding problems (see problems 2666, 2667), prove that, for $m \leqslant n$ , the power sums of the roots of the equation

x^{n}+p_{1}x^{n-1}+p_{2}x^{n-2}+\ldots +p_{n}=0

satisfy the formula

S_{m}+p_{1}S_{m-1}+\ldots +p_{m-1}S_{1}+mp_{m}=0 .

Problem 2669

By multiplying the left-hand side of the equation $x^{n}+p_{1}x^{n-1}+p_{2}x^{n-2}+\ldots +p_{n}=0$ by a suitable power of $x$ , obtain, for $m > n$ , the equality for the power sums of the roots of the equation:

S_{m}+p_{1}S_{m-1}+p_{2}S_{m-2}+\ldots +p_{n}S_{m-n}=0 .

Problem 2670

Find the value of the indicated symmetric function of the roots of the given equation: $x^{5}-3x^{3}-5x+1=0$ ; $\sum x_{\nu }^{4}$ .

Problem 2671

Find the value of the indicated symmetric function of the roots of the given equation: $x^{3}+3x^{2}-x-7=0$ ; $\sum x_{\nu }^{6}$ .

Problem 2672

Find the value of the indicated symmetric function of the roots of the given equation: $x^{4}+ax^{3}+bx^{2}+cx+d=0$ ; $\sum x_{\nu }^{3}x_{\mu }$ ; $\nu \neq \mu$ .

Problem 2673

Find the value of the indicated symmetric function of the roots of the given equation: $x^{3}-x-1=0$ ; $\sum x_{1}(x_{2}-x_{3})^{2}$ .

Problem 2674

Find the value of the indicated symmetric function of the roots of the given equation: $x^{4}-5x^{2}-2x+1=0$ ; $\sum (x_{1}-x_{2})^{2}(x_{3}-x_{4})^{2}$ .

Problem 2675

Find the value of the indicated symmetric function of the roots of the given equation: $x^{n}+x^{n-1}+\ldots +x+1=0$ ; $\sum x_{1}^{3}x_{2}^{3}x_{3}^{3}$ ; $n > 8$ .

Problem 2676

Find the value of the indicated symmetric function of the roots of the given equation: $x^{5}-4x^{3}+x^{2}+3x+1=0$ ; $\sum x_{1}^{2}x_{2}^{2}x_{3}x_{4}$ .

Problem 2677

$x^{n}+(a+b)x^{n-1}+(a^{2}+ab+b^{2})x^{n-2}+\ldots +(a^{n}+a^{n-1}b+\ldots +b^{n})=0$ ; find $S_{m}$ for $m \leqslant n$ .

Problem 2678

$x^{n}+ax^{n-1}+\dfrac {a(a+1)}{1 \cdot 2}x^{n-2}+\ldots +\dfrac {a(a+1)\ldots (a+n-1)}{1 \cdot 2 \ldots n}=0$ . Find $S_{m}$ for $m \leqslant n+1$ .

Problem 2679

Prove Waring's formulas, giving expressions for $S_{m}$ , the sum of the $m$ -th powers of the roots of the equation $x^{n}+p_{1}x^{n-1}+\ldots +p_{n}=0$ , in terms of the coefficients of the equation. These formulas can be written in the form

\dfrac {1}{m}S_{m}=p_{n}-\dfrac {1}{2}\sum {}'p_{\nu }p_{\mu }+\dfrac {1}{3}\sum p_{\nu }p_{\mu }p_{\lambda }-\ldots ,

p_{m}=-\dfrac {S_{m}}{m}+\dfrac {1}{2!}\sum \dfrac {S_{\mu }S_{\nu }}{\mu \nu }-\dfrac {1}{3!}\sum \dfrac {S_{\mu }S_{\nu }S_{\lambda }}{\mu \nu \lambda }+\ldots .

In each of the sums on the right-hand side, the indices $\mu , \nu , \lambda , \ldots$ are positive, and their sum equals $m$ .

Problem 2680

For the cubic equation $x^{3}+ax^{2}+bx+c=0$ , evaluate the following symmetric functions: $\dfrac {x_{1}^{2}+x_{2}^{2}}{x_{1}+x_{2}}+\dfrac {x_{1}^{2}+x_{3}^{2}}{x_{1}+x_{3}}+\dfrac {x_{2}^{2}+x_{3}^{2}}{x_{2}+x_{3}}$ ; $\dfrac {x_{1}^{2}+x_{2}x_{3}}{x_{2}+x_{3}}+\dfrac {x_{2}^{2}+x_{1}x_{3}}{x_{1}+x_{3}}+\dfrac {x_{3}^{2}+x_{1}x_{2}}{x_{1}+x_{2}}$ .

Problem 2681

For the cubic equation $x^{3}+ax^{2}+bx+c=0$ , find the area of the triangle with sides $x_{1}, x_{2}, x_{3}$ .

Problem 2682

Represent, as a polynomial in $x$ , the following fractional function of a root of the given equation (rationalizing the denominator): $\dfrac {6+5x}{1+x+x^{2}}$ ; $x^{3}-2=0$ .

Problem 2683

Represent, as a polynomial in $x$ , the following fractional function of a root of the given equation (rationalizing the denominator): $\dfrac {1}{x^{3}-x+2}$ ; $x^{4}-5=0$ .

Problem 2684

Represent, as a polynomial in $x$ , the following fractional function of a root of the given equation (rationalizing the denominator): $\dfrac {1}{1+2x+x^{2}}$ ; $x^{3}-x-1=0$ .

Problem 2685

Represent, as a polynomial in $x$ , the following fractional function of a root of the given equation (rationalizing the denominator): $\dfrac {1}{a+bx+cx^{2}}$ ; $x^{3}-N=0$ .

Problem 2686

Determine whether the polynomials $\varphi (x)=x^{3}-3x^{2}+x+1$ , $\psi (x)=2x^{3}-4x+2$ have a common root.

Problem 2687

Determine whether the polynomials $\varphi (x)=x^{2}+x+1$ , $\psi (x)=x^{3}+x-1$ have a common root.

Problem 2688

Determine whether the polynomials $\varphi (x)=x^{5}+x^{2}+1$ , $\psi (x)=x^{5}-x^{3}+2$ have a common root.

Problem 2689

For what $\lambda$ and $\mu$ do the equations $x^{3}-6x^{2}+\lambda x-3=0$ , $x^{3}-x^{2}+\mu x+2=0$ have two common roots?

Problem 2690

Solve the system of equations:

\begin{cases} 5x^{2}-5y^{2}-3x+9y=0, \\ 5x^{3}+5y^{3}-15x^{2}-13xy-y^{2}=0 . \end{cases}

Problem 2691

Solve the system of equations:

\begin{cases} 3x^{2}+9x^{2}y+9xy^{2}+3y^{3}+2x^{2}-4xy+2y^{2}=5, \\ 4x^{3}+12x^{2}y+12xy^{2}+4y^{3}-x^{2}+2xy-y^{2}=3 . \end{cases}

Problem 2692

Eliminate $x$ from the system of equations:

\begin{cases} x^{3}+(y-1)x^{2}+(3y-10)x+12=0, \\ x^{3}+(y-8)x^{2}+(y+16)x-3=0 . \end{cases}

Problem 2693

Eliminate $x$ from the system of equations:

\begin{cases} x^{3}+mx^{2}-4=0, \\ x^{3}+mx+2=0 . \end{cases}

Problem 2694

Eliminate $x$ from the system of equations:

\begin{cases} x^{3}+4mx^{2}-3x+2=0, \\ 2x^{3}+(m+2)x^{2}-5x+1=0 . \end{cases}

Problem 2695

Find the discriminant of the equation $x^{n}+a=0$ .

Problem 2696

The same for the equation $x^{3}+px^{2}+qx+r=0$ (see problem 2695).

Problem 2697

The expression for the discriminant $D(a_{0}, a_{1}, \ldots , a_{n})$ contains the term $Ma_{0}^{n-1}a_{n}^{n-1}$ . Find $M$ .

Problem 2698

Prove the theorem: if all the roots of an equation with integer coefficients $x^{n}+p_{1}x^{n-1}+\ldots +p_{n}=0$ have absolute value equal to unity, then they are roots of unity. (Kronecker.)

Section 6.8Transformation and Solution of Equations[40/88%]

Problem 2699

The roots of the equation $x^{3}+px+q=0$ are real under the condition $4p^{3}+27q^{2} \leqslant 0$ . When are the roots of the equation $x^{3}+ax^{2}+bx+c=0$ real?

Problem 2700

Solve the equation $x^{4}+4x^{3}+11x^{2}+14x+12=0$ by the substitution $x=y+\sigma$ with a suitable choice of $\sigma$ .

Problem 2701

Reduce the following equation to one with integer coefficients and leading coefficient equal to unity, applying a substitution of the form $y=ax$ : $5x^{4}-9x^{3}+15x^{2}-12x+18=0$ .

Problem 2702

Reduce the following equation to one with integer coefficients and leading coefficient equal to unity, applying a substitution of the form $y=ax$ : $900x^{4}-750x^{3}+375x^{2}-13=0$ .

Problem 2703

Prove that the equation $a_{0}x^{2n}+a_{1}x^{2n-2}+\ldots +a_{n}=0$ has non-real¹ roots, if the equation $a_{0}x^{2n}-a_{1}x^{2n-2}+\ldots +(-1)^{n}a_{n}=0$ has real ones.

Imaginary numbers are complex numbers with nonzero imaginary part (i.e., not real; not to be confused with purely imaginary numbers). ↩

Problem 2704

Find the equation whose roots equal the squares of the roots of the equation $x^{5}+x^{3}+x^{2}+2x+3=0$ , and prove that the given equation has imaginary¹ roots.

Imaginary numbers are complex numbers with nonzero imaginary part (i.e., not real; not to be confused with purely imaginary numbers). ↩

Problem 2705

In the equation $x^{4}-2x^{3}-3x^{2}+4x-1=0$ , one of the roots equals the square of another. Solve the equation.

Problem 2706

Solve the equation $x^{4}-6mx^{3}+2(4m^{2}+n)x^{2}-6mnx+n^{2}=0$ .

Problem 2707

Solve the equation $4(x^{2}-x+1)^{3}-27x^{2}(x-1)^{2}=0$ .

Problem 2708

Under what condition do two of the roots of the equation $x^{3}+px+q=0$ differ only in sign?

Problem 2709

Prove the theorem: the palindromic equation

ax^{2n}+a_{1}x^{2n-1}+\ldots +a_{1}x+a=0

is, by the substitution $y=\left(\dfrac {x+1}{x-1}\right)^{2}$ , reduced to solving an equation of degree $n$ and a quadratic equation.

Problem 2710

Prove the theorem: a palindromic equation with real coefficients $x^{2n}+a_{1}x^{2n-1}+\ldots +a_{1}x+1=0$ , for which $\left|a_{n}\right| \leqslant 2$ , has at least one pair of imaginary¹ roots.

Imaginary numbers are complex numbers with nonzero imaginary part (i.e., not real; not to be confused with purely imaginary numbers). ↩

Problem 2711

A polynomial with real coefficients

f(x)=x^{n}+a_{1}x^{n-1}+\ldots +a_{n}

cannot, on the interval $(-2, 2)$ , remain in absolute value less than $2$ throughout.

Problem 2712

An equation with real coefficients

x^{2n+1}+a_{1}x^{2n-1}+\ldots +a_{n}x=m

has at least one root in the interval $(-2, 2)$ , if $\left|m\right| < 2$ .

Problem 2713

An equation with real coefficients

x^{2n+1}+a_{1}x^{2n-1}+\ldots +a_{n}x=m

has a root in the interval $(-2l, 2l)$ , where $2l^{2n+1}=\left|m\right|$ . (Chebyshev.)

Problem 2714

Transform the following equation by the indicated substitution (Tschirnhaus transformation): $x^{3}-x^{2}-4x+4=0$ ; $y=x^{2}+x-1$ .

Problem 2715

Transform the following equation by the indicated substitution (Tschirnhaus transformation): $2x^{3}-5x^{2}+x+2=0$ ; $y=2x^{2}+3x-1$ .

Problem 2716

Transform the following equation by the indicated substitution (Tschirnhaus transformation): $x^{3}-6x^{2}+15x-14=0$ ; $2y=x^{2}-5x+8$ .

Problem 2717

Transform the following equation by the indicated substitution (Tschirnhaus transformation): $x^{4}+2x^{3}-13x^{2}-14x+48=0$ ; $y=x^{2}+x-7$ .

Problem 2718

Find the equation for $y$ whose roots are expressed in terms of the roots of the given equation in the indicated way: $x^{3}+2x-1=0$ ; $y_{1}=x_{1}x_{2}$ , $y_{2}=x_{1}x_{3}$ , $y_{3}=x_{2}x_{3}$ .

Problem 2719

Find the equation for $y$ whose roots are expressed in terms of the roots of the given equation in the indicated way: $x^{3}-x^{2}-3=0$ ; $y_{1}=\dfrac {x_{1}}{x_{2}+x_{3}-x_{1}}$ , and so on.

Problem 2720

Find the equation for $y$ whose roots are expressed in terms of the roots of the given equation in the indicated way: $x^{3}+px+q=0$ ; $y_{1}=x_{1}^{2}+x_{1}x_{2}+x_{2}^{2}$ .

Problem 2721

Find the equation for $y$ whose roots are expressed in terms of the roots of the given equation in the indicated way: $x^{3}+3x^{2}-4x+3=0$ ; $y_{1}=\dfrac {x_{1}x_{2}-x_{3}^{2}}{x_{1}+x_{2}-2x_{3}}$ .

Problem 2722

Find the equation for $y$ whose roots are expressed in terms of the roots of the given equation in the indicated way: $ax^{3}+3a_{1}x^{2}+3a_{2}x+a_{3}=0$ ; $y_{1}=(x_{1}-x_{2})(x_{1}-x_{3})$ .

Problem 2723

Solve the equation $x^{3}-3ax^{2}+(3a^{2}-b^{2})x-a^{3}+ab^{2}=0$ .

Problem 2724

Solve the equation $x^{3}-3(a-b)x^{2}+(3a^{2}-6ab-b^{2})x-a^{3}+3a^{2}b+ab^{2}-3b^{3}=0$ .

Problem 2725

Solve the equation $x^{3}-(a+1)^{2}x^{2}+(2a^{3}+a^{2}+2a-1)x-a^{4}+1=0$ .

Problem 2726

Solve the equation $x^{3}-5x^{2}+3x+6=0$ .

Problem 2727

Solve the equation $x^{4}-12ax^{3}+(45a^{2}-b^{2})x^{2}-2a(29a^{2}-5b^{2})x+24a^{2}(a^{2}-b^{2})=0$ .

Problem 2728

Solve the equation $x^{4}-4(a+1)x^{3}+(5a^{2}+14a+4)x^{2}-(2a^{3}+14a^{2}+12a)x+4a^{3}+8a^{2}=0$ .

Problem 2729

Solve the equation $2x^{4}+7x^{3}+10x^{2}+11x+6=0$ .

Problem 2730

Solve the equation $x^{3}-6x-9=0$ by Cardano's formula and Ferrari's method.

Remark.

A general cubic equation $y^{3}+Ay^{2}+By+C=0$ is reduced, by the linear substitution $y=x-\dfrac {A}{3}$ , to the equation $x^{3}+px+q=0$ . The latter is solved by the substitution $x=u+v$ , which introduces two new unknowns: setting $u^{3}+v^{3}=-q$ , $3uv+p=0$ , we obtain Cardano's formula:

x=\sqrt[3]{-\dfrac {q}{2}+\sqrt{\dfrac {q^{2}}{4}+\dfrac {p^{3}}{27}}}+\sqrt[3]{-\dfrac {q}{2}-\sqrt{\dfrac {q^{2}}{4}+\dfrac {p^{3}}{27}}} .

For the cube roots one should take only those values whose product equals $-\dfrac {p}{3}$ (Hudde's method).

A general quartic equation $y^{4}+Ay^{3}+By^{2}+Cy+D=0$ is reduced, by the substitution $y=x-\dfrac {A}{4}$ , to the equation $x^{4}+ax^{2}+bx+c=0$ ; Ferrari's method solves such an equation by introducing an auxiliary parameter $\lambda$ - a root of the cubic resolvent $\lambda^{3}+2a\lambda^{2}+(a^{2}-4c)\lambda -b^{2}=0$ - and reducing the original equation to two quadratics:

2x^{2}+a+\lambda =\pm 2\sqrt{\lambda }\left(x-\dfrac {b}{2\lambda }\right) .

Problem 2731

Solve the equation $x^{3}+6x-7=0$ by Cardano's formula and Ferrari's method (see problem 2730).

Problem 2732

Solve the equation $x^{3}+3x-2=0$ by Cardano's formula and Ferrari's method (see problem 2730).

Problem 2733

Solve the equation $x^{3}+3x-4=0$ by Cardano's formula and Ferrari's method (see problem 2730).

Problem 2734

Solve the equation $x^{3}-7x-6=0$ by Cardano's formula and Ferrari's method (see problem 2730).

Problem 2735

Solve the equation $x^{4}+3x^{2}+2x+3=0$ by Cardano's formula and Ferrari's method (see problem 2730).

Problem 2736

Solve the equation $x^{4}+2x^{2}+4x+8=0$ by Cardano's formula and Ferrari's method (see problem 2730).

Problem 2737

Solve the equation $x^{4}-x^{3}+3x^{2}-5x+2=0$ by Cardano's formula and Ferrari's method (see problem 2730).

Problem 2738

Solve the equation $x^{4}-17x^{2}-20x-6=0$ by Cardano's formula and Ferrari's method (see problem 2730).

Section 6.9Isolation and Computation of Roots[58/98%]

Problem 2739

Find the rational roots of the equation $6x^{4}-11x^{3}-x^{2}-4=0$ .

Problem 2740

Find the rational roots of the equation $4x^{4}-11x^{2}+9x-2=0$ .

Problem 2741

Find the rational roots of the equation $2x^{3}+12x^{2}+13x+15=0$ .

Problem 2742

Find the rational roots of the equation $2x^{4}-4x^{3}+3x^{2}-5x-2=0$ .

Problem 2743

Find the rational roots of the equation $6x^{5}+11x^{4}-x^{3}+5x-6=0$ .

Problem 2744

Find the rational roots of the equation $x^{5}-5x^{4}+2x^{3}-25x^{2}+21x+270=0$ .

Problem 2745

Find the rational roots of the equation $x^{6}+3x^{5}+4x^{4}+3x^{3}-15x^{2}-16x+20=0$ .

Problem 2746

Find the rational roots of the equation $2x^{6}+x^{5}-9x^{4}-6x^{3}-5x^{2}-7x+6=0$ .

Problem 2747

Find the repeated roots of the equation $x^{3}-12x+16=0$ .

Problem 2748

Find the repeated roots of the equation $x^{3}+x^{2}-8x-12=0$ .

Problem 2749

Find the repeated roots of the equation $x^{4}+2x^{3}-3x^{2}-4x+4=0$ .

Problem 2750

Find the repeated roots of the equation $x^{4}-6x^{2}-8x+24=0$ .

Problem 2751

Find the repeated roots of the equation $x^{5}-x^{3}-4x^{2}-3x-2=0$ .

Problem 2752

Find the repeated roots of the equation $x^{5}+2x^{4}-8x^{3}-16x^{2}+16x+32=0$ .

Problem 2753

Find the repeated roots of the equation $x^{6}+6x^{5}+3x^{4}+12x^{3}+3x^{2}+6x+1=0$ .

Problem 2754

Find the repeated roots of the equation $(x+1)^{7}-x^{7}+7x+6=0$ .

Problem 2755

Find the repeated roots of the equation $x^{6}-x^{5}+2x^{4}-x^{3}+2x^{2}-x+1=0$ .

Problem 2756

Find the repeated roots of the equation $x^{8}+2x^{6}-2x^{2}-1=0$ .

Problem 2757

Isolate the real roots of the equation $x^{3}-12x+5=0$ .

Problem 2758

Isolate the real roots of the equation $x^{3}-27x-17=0$ .

Problem 2759

Isolate the real roots of the equation $x^{4}-2x^{3}-5x^{2}+2x+2=0$ .

Problem 2760

Isolate the real roots of the equation $x^{4}-2x^{3}-5x^{2}+3x+1=0$ .

Problem 2761

Isolate the real roots of the equation $x^{5}-2x^{4}-5x^{3}+19x^{2}-17x+1=0$ .

Problem 2762

Isolate the real roots of the equation $x^{5}+3x^{4}-9x^{3}-7x^{2}+39x-21=0$ .

Problem 2763

Isolate the real roots of the equation $x^{2n+1}+px+q=0$ .

Problem 2764

Isolate, by Fourier's method, the roots of the equation $x^{3}-12x-4=0$ .

Problem 2765

Isolate, by Fourier's method, the roots of the equation $x^{3}-24x+11=0$ .

Problem 2766

Isolate, by Fourier's method, the roots of the equation $x^{4}-4x^{3}-3x+23=0$ .

Problem 2767

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

Problem 2768

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

Problem 2769

Isolate, by Fourier's method, the roots of the equation $x^{5}-10x^{3}+6x+1=0$ .

Problem 2770

Isolate, by Fourier's method, the roots of the equation $x^{5}+3x^{4}+2x^{3}-3x^{2}-2x-2=0$ .

Problem 2771

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

Problem 2772

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

Problem 2773

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

Problem 2774

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

Problem 2775

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

Problem 2776

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

Problem 2777

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

Problem 2778

(a)(b)

Construct the Sturm sequence for the equations:

(a)

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

(b)

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

Problem 2779

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 2780

How many real roots does the preceding equation (see problem 2779) have if $p^{5} < q^{2}$ ?

Problem 2781

The numbers $a_{\nu }$ are real, $b_{\nu }$ are positive. Moreover,

\prod _{\nu =1}^{n}(x-a_{\nu }-b_{\nu }i)=\varphi (x)+i\psi (x),

where $\varphi (x)$ and $\psi (x)$ are polynomials with real coefficients. Prove that the roots of the equation $p\varphi (x)+q\psi (x)=0$ are real for any real $p$ and $q$ .

Problem 2782

Isolate the roots of the equation, with respect to $\lambda$ :

\dfrac {x^{2}}{a^{2}+\lambda }+\dfrac {y^{2}}{b^{2}+\lambda }+\dfrac {z^{2}}{c^{2}+\lambda }-1=0, \quad a^{2} > b^{2} > c^{2} .

Problem 2783

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 2784

Compute, to within $10^{-5}$ , the roots of the equation $x^{3}-5x+1=0$ .

Problem 2785

Compute, to within $10^{-5}$ , the roots of the equation $x^{3}-9x^{2}+20x-11=0$ .

Problem 2786

Compute, to within $10^{-5}$ , the roots of the equation $x^{3}+3x^{2}-4x-1=0$ .

Problem 2787

Compute, to within $10^{-5}$ , the roots of the equation $x^{3}+6x^{2}+6x-7=0$ .

Problem 2788

Compute, to within $10^{-5}$ , the roots of the equation $x^{3}-3x^{2}+8x+10=0$ .

Problem 2789

Compute, to within $10^{-5}$ , the roots of the equation $x^{4}+2x^{2}-6x+2=0$ .

Problem 2790

Compute, to within $10^{-5}$ , the roots of the equation $x^{5}+5x+1=0$ .

Problem 2791

Find, to within $10^{-3}$ , the roots of the equation $x^{4}+2x^{3}+6x^{2}-1=0$ .

Problem 2792

Find, to within $10^{-3}$ , the roots of the equation $10x=10+\sin x$ .

Problem 2793

Find, to within $10^{-3}$ , the roots of the equation $10x=e^{-x}$ .

Problem 2794

Find, to within $10^{-3}$ , the roots of the equation $x=e^{-x}$ .

Problem 2795

Find, to within $10^{-3}$ , the roots of the equation $10\ln x=x^{3}-3$ .

Problem 2796

A ball of radius $1$ m has specific gravity $0{,}75$ . To what height does it protrude above the water while floating in it?

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