A Polynomial with Real Coefficients. Rolle's Theorem

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

The polynomials

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 $\psi (x)$.

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.

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

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

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

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 $\left|x_{3}\right| > 2\sqrt{p}$.

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.

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.

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

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

are real as well.

Prove that all the roots of the polynomial

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

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

are real as well.

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

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

has no real roots.

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.

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.

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.)

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.)

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.)

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.

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.

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 $f(x)$.

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.)

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

has all its roots real.

Prove the reality of the roots of the equation

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

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

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

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 $\nu \geqslant m$, $A_{1}+A_{2}+\ldots +A_{n}=0$.

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 $\left(-\sqrt{2n+1}, \sqrt{2n+1}\right)$.

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

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

In the expansion in powers of $\lambda$

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.)

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 $x^{10}-7x-2=0$.

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.

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

Prove that the equation

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

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

has at most one real root.