Intereting Posts

Ergodicity of tent map
Defining a manifold without reference to the reals
Functional equation $f(x+y)-f(x)-f(y)=\alpha(f(xy)-f(x)f(y))$ is solvable without regularity conditions
$A_n$ is the only subgroup of $S_n$ of index $2$.
partial derivative of cosine similarity
Quick method for finding eigenvalues and eigenvectors in a symmetric $5 \times 5$ matrix?
Prove that $\int\limits_0^1 x^a(1-x)^{-1}\ln x \,dx = -\sum\limits_{n=1}^\infty \frac{1}{(n+a)^2}$
Is the empty set an open ball in a metric space?
If $G/Z(G)$ is cyclic, then $G$ is abelian
rational functions on projective n space
Geometrico-Harmonic Progression
Let $G$ be a Lie group. Show that there is a diffeomorphism $TG \cong G \times T_e G$.
What is the cross product in spherical coordinates?
Change of measure of conditional expectation
If $H$ is a proper subgroup of a $p$-group $G$, then $H$ is proper in $N_G(H)$.

I want to prove that the polynomial

$$

f_p(x) = x^{2p+2} – abx^{2p} – 2x^{p+1} +1

$$

has distinct roots. Here $a$, $b$ are positive real numbers and $p>0$ is an odd integer. How can I prove that this polynomial has distinct roots for any arbitrary $a$,$b$ and $p$.

- Prob. 26, Chap. 5 in Baby Rudin: If $\left| f^\prime(x) \right| \leq A \left| f(x) \right|$ on $$, then $f = 0$
- $\lim\limits_{n\to\infty} \frac{n}{\sqrt{n!}} =e$
- T-invariant sub-sigma algebra
- Evaluating sums using residues $(-1)^n/n^2$
- Improper rational/trig integral comes out to $\pi/e$
- Rudin assumes $(x^a)^b=x^{ab}$(for real $a$ and $b$) without proof?

Thanks in advance.

- how to find the branch points and cut
- Existence and value of $\lim_{n\to\infty} (\ln\frac{x}{n}+\sum_{k=1}^n \frac{1}{k+x})$ for $x>0$
- Is $\sin^2(z) + \cos^2(z)=1$ still true for $z \in \Bbb{C}$?
- Prove that $\lim_{n \to \infty}\bigg^{1/n}=M$
- Why is there no continuous log function on $\mathbb{C}\setminus\{0\}$?
- Mapping circles using Möbius transformations.
- How to prove that $\frac{\zeta(2) }{2}+\frac{\zeta (4)}{2^3}+\frac{\zeta (6)}{2^5}+\frac{\zeta (8)}{2^7}+\cdots=1$?
- Calculating a real integral using complex integration
- On the Riemann mapping theorem
- Why: A holomorphic function with constant magnitude must be constant.

Let $c$ denote $ab$. Note that

\begin{equation*}

f(x) = (x^{p+1}-1)^2 – cx^{2p}

\end{equation*}

and

\begin{equation*}

f'(x) = 2(p+1)x^p(x^{p+1}-1) – 2pcx^{2p-1}

\end{equation*}

Then, $f(x) = 0 \iff$

\begin{equation*}

c = \dfrac{(x^{p+1}-1)^2}{x^{2p}} = \varphi(x)\ (\text{say})

\end{equation*}

and $f'(x) = 0 \iff$

\begin{align*}

c & = \dfrac{(p+1)x^p(x^{p+1}-1)}{px^{2p-1}} = \dfrac{(p+1)x}{p}\dfrac{x^{p+1}-1}{x^p} \iff\\

c & = \left(\dfrac{p+1}{p}\right)x \sqrt{\varphi(x)}.

\end{align*}

Thus, $f(x)$ and $f'(x)$ vanish for the same $x$ if and only if for some root $x$ of $f(x)$,

\begin{align*}

c & = \left(\dfrac{p+1}{p}\right)x \sqrt c \iff\\

x & = \dfrac{p\sqrt c}{p+1}.

\end{align*}

Thus, for every $p$ and $c = ab$, if such an $x$ is a root, it is a multiple root.

Now, when does such a root $x$ exist? Let $x = t$ be one such. Then $c = \left(\dfrac{p+1}{p} \right)^2 t^2$. Then, since $f(t) = 0$, we have (from the original form of the equation):

\begin{align*}

t^{2p+2} – \left(\dfrac{p+1}{p}\right)^2 t^{2p + 2} – 2t^{p+1} + 1 = 0\\

-\dfrac{(2p + 1)}{p^2}t^{2p + 2} – 2t^{p+1} + 1 = 0\\

(2p + 1)t^{2(p + 1)} + 2p^2 t^{p + 1} – p^2 = 0.

\end{align*}

When treated as a quadratic equation in $t^{p+1}$, the discriminant is

\begin{equation*}

4p^4 + 4p^2(2p + 1) = 4p^2(p + 1)^2,

\end{equation*}

and therefore, the solutions are

\begin{equation*}

t^{p+1} = -p, \dfrac{p}{2p + 1}.

\end{equation*}

That is,

\begin{equation*}

t = (-p)^{\frac 1 {p + 1}}, \left(\dfrac p {2p + 1} \right)^{\frac 1 {p + 1}}.

\end{equation*}

But substituting the same $c$ in $f'(t) = 0$, we get

\begin{align*}

& 2(p+1)t^p(t^{p+1}-1)-2p\left(\dfrac{p+1}{p}\right)^2t^{2p+1}=0\\

& p(t^{p+1}-1)-(p+1)t^{p+1}=0 \implies\\

& t = (-p)^{\frac{1}{p+1}}

\end{align*}

Thus, only the first of the previous two solutions satisfies both equations.

Then, $c = \left( \dfrac{p+1}{p} \right)^2 t^2$ gives us

\begin{equation*}

\boxed{c= \dfrac{(p+1)^2}{(-p)^{\frac{2p}{p+1}}}}.

\end{equation*}

Thus, the equation has multiple roots exactly when $c$ and $p$ are related as above.

Note that for odd values of $p$, $c$ will be a real number if and only if $p$ is of the form $4k + 1$, and then, $c < 0$. If, as stated in the question, $c = ab$ is a positive real number, the equation will have **distinct roots**.

**Example**

For $p = 1$, $f(x) = x^4 – cx^2 – 2x^2 + 1$ and $f'(x) = 4x^3 – 2cx – 4x$.

Then, $f(x) = 0$ and $f'(x) = 0$ imply that $c = \left(\dfrac{x^2 – 1}{x}\right)^2$ and $c = 2x\left(\dfrac{x^2-1}{x}\right)$ respectively. Thus, if $x$ is a multiple root, then $x = \dfrac{\sqrt c}{2}$.

Taking $t$ to be such a root, so that $c = 4t^2$, and substituting in $f(t) = 0$, we get

\begin{align*}

t^4 – 4t^4 – 2t^2 + 1 = 0\\

3t^4 + 2t^2 – 1 = 0.

\end{align*}

Thus, $t^2 = -1, \dfrac 1 3$, of which only the first one satisfies $f'(t)=0$. Thus, $c = -4$.

For $c = 4$, $x^4 + 2x^2 + 1 = 0$ has roots $\pm i, \pm i$.

The claim is false. Set $ab:=(\frac{729}{16})^\frac{1}{3}$ and $p:=2$

Then $f(x)=x^6-abx^4-2x^3+1$ has a double root at $x=-2^{\frac{1}{3}}$

With the help of *Mathematica* the discriminant is given by:

$$\Delta =

\left \{

\begin{array}{cc}

(4c)^{p+1}

\left[ p^{p} c^{(p+1)/2}+(p+1)^{p+1} \right]^{2} &

\text{odd } p \\[5pt]

(4c)^{p+1} [(p+1)^{2(p+1)}-p^{2p}c^{p+1}] &

\text{even } p

\end{array}

\right. \\$$

$\Delta \neq 0 \,$ if $\,

\left \{

\begin{array}{ll}

\text{odd } p=4n-1 , & c\neq 0 \\[5pt]

\text{odd } p=4n-3, & c^{(p+1)/2} \neq 0,

\displaystyle -\frac{(p+1)^{p+1}}{p^{p}} \\[5pt]

\text{even } p, & c^{p+1} \neq 0,

\displaystyle \frac{(p+1)^{2(p+1)}}{p^{2p}}

\end{array}

\right.$

- For what $n$ is it true that $(1+\sum_{k=0}^{\infty}x^{2^k})^n+(\sum_{k=0}^{\infty}x^{2^k})^n\equiv1\mod2$
- Why the interest in locally Euclidean spaces?
- Cauchy sequence is convergent iff it has a convergent subsequence
- Multinomial Theorem Example Questions
- Tightness condition in the case of normally distributed random variables
- Why do we need to learn Set Theory?
- Is the sequence $\sqrt{p}-\lfloor\sqrt{p}\rfloor$ , $p$ running over the primes , dense in $$?
- No radical in the denominator — why?
- The graph of a measurable function
- Is 'clamp' a formally recognized mathematical function?
- Proving that the sum and difference of two squares (not equal to zero) can't both be squares.
- Why the matrix of $dG_0$ is $I_l$.
- If series $\sum a_n$ is convergent with positive terms does $\sum \sin a_n$ also converge?
- How can you prove $\frac{n(n+1)(2n+1)}{6}+(n+1)^2= \frac{(n+1)(n+2)(2n+3)}{6}$ without much effort?
- Axiom of Choice: Where does my argument for proving the axiom of choice fail? Help me understand why this is an axiom, and not a theorem.