Math Problems Vault

Problem 2739

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

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Problem 2740

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

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Problem 2741

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

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Problem 2742

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

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Problem 2743

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

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Problem 2744

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

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Problem 2745

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

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Problem 2746

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

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Problem 2747

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

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Problem 2748

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

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Problem 2749

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

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Problem 2750

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

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Problem 2751

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

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Problem 2752

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

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Problem 2753

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

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Problem 2754

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

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Problem 2755

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

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Problem 2756

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

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Problem 2757

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

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Problem 2758

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

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Problem 2759

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

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Problem 2760

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

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Problem 2761

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

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Problem 2762

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

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Problem 2763

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

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Problem 2764

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

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Problem 2765

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

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Problem 2766

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

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Problem 2767

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

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Problem 2768

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

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Problem 2769

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

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Problem 2770

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

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Problem 2771

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

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Problem 2772

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

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Problem 2773

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

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Problem 2774

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

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Problem 2775

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

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Problem 2776

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

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Problem 2777

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

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Problem 2778

(a)(b)

Construct the Sturm sequence for the equations:

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(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 $x^{5}-5px^{3}+5p^{2}x+2q=0$ .

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Problem 2780

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

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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 $q$ .

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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} .

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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} .

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Problem 2784

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

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Problem 2785

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

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Problem 2786

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

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Problem 2787

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

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Problem 2788

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

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Problem 2789

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

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Problem 2790

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

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Problem 2791

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

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Problem 2792

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

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Problem 2793

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

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Problem 2794

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

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Problem 2795

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

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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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