Math Problems Vault

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).

?

Footnotes

Footnotes

Footnotes