Questions about the set of values at which a given function evaluates to zero. For questions about "square roots", "cube roots", and such, consider using the (arithmetic) tag. For questions about roots of Lie algebras, use the (lie-algebra) tag instead.

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16
votes
3answers
360 views

The function $f'+f'''$ has at least $3$ zeros on $[0,2\pi]$.

Show that if $f\in \mathcal{C}^3$ and $2\cdot\pi$ periodic then the function $f'+f'''$ has at least $3$ zeros on $[0,2\pi]$. My attempt : f is $2\pi$ periodic and $\mathcal{C}^3$, we have : ...
2
votes
2answers
48 views

Information about the roots of a polynomial without their calculation

Suppose I have a polynomial (of any order) and I'm not able to calculate the roots. Is there a way to get at least some information about the roots such as how many of them are complex, negative or ...
6
votes
3answers
82 views

Solving Equation of Degree n, where n is any value between 1 and 2

How does one solve an equation of the form: $$ax^n + bx + c = 0$$ where n is a non integer value between 1 and 2. Is there a formula to provide an analytic solution?
30
votes
4answers
874 views

$p_n(x)=p_{n-1}(x)+p_{n-1}^{\prime}(x)$, then all the roots of $p_k(x)$ are real

$p_0(x)=a_mx^m+a_{m-1}x^{m-1}+\dotsb+a_1x+a_0(a_m,\dotsc,a_1,a_0\in\Bbb R)$ is a polynomial, and $$p_n(x)=p_{n-1}(x)+p_{n-1}^{\prime}(x),\qquad n=1,2,\dotsc$$ then, there exist $N\in\Bbb N$, such ...
1
vote
1answer
51 views

Using argument principle on $e^z + z$

I want to use the argument principle to estimate the number of zeros of $e^z + z$ inside the rectangle with sides $y=2\pi n i, 2 \pi (n+1)i$ and $x = R, -R$ for $R$ large. But $\int \frac {e^z + ...
0
votes
2answers
55 views

Ideas on how to proceed with a proof?

Sorry if this is a nonspecific question - I can provide more details but at this point I need general ideas on a proof strategy. So I recently reduced a rather difficult optimization problem to ...
6
votes
4answers
125 views

Solve $x^{3}-3x=\sqrt{x+2}$

Solve for real $x$ $$x^{3}-3x=\sqrt{x+2}$$ By inspection, $x=2$ is a root of this equation. So, I squared both sides and divided the six degree polynomial obtained by $x-2$. Then I got a ...
3
votes
2answers
490 views

Show quadratic equation has two distinct real roots.

$x^2 - (5-k) x + (k+2) = 0 $ has two distinct real roots. So, in the markscheme of this question, they take the discriminant ($-b^2 + 4ac$) and say it is greater than 0. That is, $( (-(5-k)^2 - ...
0
votes
0answers
15 views

Estimate error when using the result of an expanded equation

I would like to know how to deal with the error term in expanded expression. For example consider the function $\displaystyle f(x)=A\text e^{-(x+\lambda)^2}+B\text e^{-(x-2\lambda)^2}\;, $ where ...
5
votes
1answer
85 views

Find $\lfloor {\alpha}^6 \rfloor$

If $\alpha$ is a real root of the equation $$x^5-x^3+x-2=0$$ find the value of $\lfloor {\alpha}^6 \rfloor$. This one totally stumped me. We are asked to calculate $\lfloor {\alpha}^6 ...
0
votes
1answer
40 views

Using elementary polynomials to solve system of linear polynomials

Problem Statement I am given a finite set of monic polynomials in t, parameterized by $r_i$ $X_i = t - r_i$ where the $r_i$ are guaranteed unique. Neither $t$ nor $r_i$ are known, only $X_i$. I ...
7
votes
3answers
116 views

How to solve the following? $ x^3+1=2{(2x-1)}^{1/3} $.

Find all the real solutions of $$x^3+1=2{(2x-1)}^{1/3} $$ I tried to cube both sides but got messed up with a nine degree equation! Please help. Thanks in advance!
0
votes
1answer
18 views

prove that the next two affirmations are equivalent

prove that the next two affirmations are equivalent : 1) every non constant $f(x)\in \mathbb C[x]$ has all of its roots in $\mathbb C$ 2)every non constant $f(x)\in \mathbb C[x]$ has at least one ...
-1
votes
2answers
44 views

Finding all the roots from a complex equation

I'm struggling a lot with complex numbers recently. How do I find all the roots for equations like: (1) $\cos z = 3$ (2) $e^{2z} = -e$ (3) $e^z+6e^{-z} = 5$ Thanks
1
vote
2answers
25 views

how to find the roots of: $x^{3}+6x^{2}-24x+160$ if one root is $2-2(3)^{1/2}i$

how to find all the roots of the next two polinomials?: $x^{3}+6x^{2}-24x+160$ if one root is $2-2(3)^{1/2}i$ and $x^{5}-3x^{4}+4x^{3}-4x+4$ if $1+i$ is a double root I don´t know how to solve this, ...
0
votes
0answers
31 views

Let $f(x)\in K[x]$ ($K$ field). Prove that if $(f(x),f´(x))=1$ (greatest common divisor is 1) then $f(x)$ does not have multiple roots in $K$

Please I would like you to tell me if my proof is correct Let $f(x)\in K[x]$ ($K$ field). Prove that if $(f(x),f´(x))=1$ (greates common divisor is 1) then $f(x)$ does not have multiple roots in $K$ ...
1
vote
2answers
58 views

Finding zero of function which is a real number

Is there an easier way of finding or approximating the x-axis-intersect of this function: $$ 0=x^3-3x^2+x+3 $$ The approximate solution is: $$ x=-0.76929 $$ and the precise solution is: $$ x=1 - ...
6
votes
1answer
97 views

zeros of a polynomial

Given $P(z)=z^6+6z+10$, find how many roots are in each quadrant I have already seen that $P(z)$ has six different roots, and that none of them are real or of the form $ki$, $k\in \Bbb R$. Since ...
0
votes
1answer
53 views

Root of equation, solvability

I was trying to solve the following equation for t $$(P\cdot l \cdot \exp(-l\cdot t) + R \cdot l \cdot \exp(-l \cdot t))/t + (P \cdot \exp(-l \cdot t) + R \cdot (\exp(-l \cdot t) - 1))/t^2 = 0 $$ ...
0
votes
1answer
40 views

Application of Rouché's (Rouche's) Theorem to a Polynomial

Here is my question: State Rouché's theorem. How many roots of the polynomial $p(z) = z^8 + 3 z^7 + 6 z^2 + 1$ are contained in the annulus {$1 < |z| < 2$}? The statement is fine. I then ...
0
votes
1answer
29 views

Simple Pole Search

How do I find poles of: $H(z) = \dfrac{z^3}{z^3+\alpha}$. I know I must find the z values that do $z^3 = -\alpha$. I know how to do it in Matlab (with "residuez" function) but, how can i solve this ...
0
votes
1answer
49 views

Set of Solutions of A Quadratic Equation with Coefficients in $\{0,1,\cdots , \ p-1\}$

I was just playing with quadratic equations and this interesting question came into my mind. Say I have a set of quadratic polynomials $S=\{f_{(b,c)}(x)=x^2+bx+c:b,c\in \{0,1,\cdots, p-1 \}\}$ where ...
3
votes
2answers
72 views

Cubic polynomial - radical expression of roots

Let $f=X^3+X^2-2X-1$ be a polynomial with the three roots $x_1,x_2,x_3$ with $x_1=2\text{cos}(\frac{2 \pi}{7})$. We define $z:=(x_1-x_2)(x_1-x_3)(x_2-x_3)$. I want to find a radical expression for ...
2
votes
1answer
50 views

Find roots of a function

$f$ is a function defined on the whole real line which has the property that $f(1+x)=f(2-x)$ for all $x$. Assume that the equation $f(x)=0$ has $8$ distinct real roots. Find the sum of these roots. I ...
0
votes
3answers
47 views

Find the rational roots of $x^{3}-{2x^{2}\over 3}+3x-2$

I need to find the rational roots of $$x^{3}-{2x^{2}\over 3}+3x-2$$ I thought about using descartes´ rational root theorem but I need to have integers as coefficients of my polynomial: can I work with ...
1
vote
1answer
43 views

Finding the scope of a parameter where a polynomial can have roots

I have this problem- lets say I have a polynomial which has real parameters as coefficients and I'm looking for the scope of the parameters where the polynomial can have real roots. e.g $x^2+kx+k$ we ...
1
vote
0answers
19 views

Is $\min \deg$ in a dirichlet subring interesting or is it always $1$?

Let $s \in C$. Let $D = A[[n^{-X}]]$ be a subring of the formal (or absolutely converging on a region; whatever is needed) Dirichlet series with base ring $A$. Define a minimal Dirichlet series for ...
1
vote
1answer
20 views

relation between the number of real roots of the derivative and the original polynomial

If the derivative of polynomial has n real roots then can we conclude that the original polynomial has to have n+1 real roots?
0
votes
1answer
40 views

conclusion about roots for positive derivative of a polynomial

If the derivative of a polynomial is always positive then what can we conclude about the number of real roots the original polynomial?
1
vote
3answers
73 views

Relation between the roots of $x^2+x+1$ and its derivative

If $f(x)$ is a polynomial in n degree and has $n$ real roots then is it necessary that $f'(x)$ has to have $n-1$ real roots? If this is so then $x^2+x+1$ has no real roots but the derivative of the ...
0
votes
1answer
44 views

it it possible to solve these equation for their root.

I am trying to solve an equations such as the roots of $$k*x(11*x + 1) + d*x(11x + 1)$$ has to match the roots of this function $$x^2 + 0.1x + 6 + k*x(11*x + 1) + d*x(11x + 1)$$, where I have to ...
1
vote
1answer
83 views

Tangent at average of two roots of cubic with one real and two complex roots

I was able to easily prove that the tangent at the average of two roots of a real cubic polynomial passed through the third root of the function. But I have only done this for functions with three ...
0
votes
2answers
55 views

$x^3+3x^2+4x+5=0$ and $x^3+2x^2+7x+3=0$, how many common roots they have?

My attempt, Equate both, at the end you will get $x^2-3x-2=0$ That means $x=-1$ and $x=2$. But what after that. Please provide solutions as well.
5
votes
2answers
50 views

Disk with root in center with no other roots in polynomial

Say we have a polynomial $p$ with roots $r_1,r_2...r_n$, I'm looking for a way to find a disk which, if placed on the center of any root, does not contain any other root (multiple roots considered as ...
1
vote
0answers
43 views

Prove that eventually Hannah and the other swimmer will settle into a pattern where they pass each other (Please refer to the context in my question)

From the 2014 Mathcamp quiz: Hannah is about to get into a swimming pool in which every lane already has one swimmer in it. Hannah wants to choose a lane in which she would have to encounter the other ...
3
votes
1answer
41 views

Finding a disk containing all roots of a complex polynomial

I'm trying to list all roots of a polynomial so I found this paper, in Part 9 on page 29 it gives a simple recipe to find all the roots. But there is this remark: We have assumed throughout the ...
0
votes
0answers
23 views

Bounding the Number of Roots of Integer Polynomial

Let $P(x)$ be a non constant polynomial in $\mathbb{Z[x]}$. Let $n$ be the number of roots of $P(x)^2-1 = 0$. Show $n \le \deg P+2$.
1
vote
4answers
133 views

Solve the equation $x^x=10^9$.

The main question was to solve $x\log_{10}{x}=9$. I reduced it to this equation. This is $x$ Degree equation. How to solve this? I know this can be solved by newton's method. But I am not getting how ...
1
vote
0answers
39 views

Is any of this true about infinite series of functions?

Let $f_n^+(x)$ be a sequence of non-negative functions $f_n^+: X \to \Bbb{R}_{\geq 0}$, such that each $f_n^+$ has countably many zeros. Then if $f(x) = \sum f_n^+(x)$ converges point-wise, the ...
5
votes
1answer
70 views

“Polynomials” with non-integer exponents

Are there some books or articles regarding "polynomials" with non-integer (real) exponents, i.e., $$f(x)=a_1x^{e_1}+a_2x^{e_2}+\dots+a_nx^{e_n},$$ where $e_1,e_2,\dots$ are any real numbers (and $x$ ...
0
votes
0answers
40 views

Why is this root close to $\frac{2\pi}{\text{ZetaZero[1]}}$?

Why is the root, from the following algorithm close to $\frac{2\pi}{\text{ZetaZero[1]}}$? Let: $$y=\frac{2}{5-x}$$ Solve for $x$ in the equation: $$\frac{1}{x+y}=\exp(x-y)$$ Let again: ...
1
vote
1answer
43 views

Do there exist $a_k$ and $b_k$ so the equation $\sum\limits_{k=1}^{n} (a_k \sin(kx) + b_k \cos(kx)) = 0$ has no roots?

Do there exist real numbers $a_1, a_2, ..., a_n$ and $b_1, b_2, ..., b_n$ such that the equation $$\sum\limits_{k=1}^{n} (a_k \sin(kx) + b_k \cos(kx)) = 0$$ has no solutions?
1
vote
1answer
67 views

Describe the graph of f if the graph of its integral its given

Describe the graph of $f$ if the graph of its integral $g(t) = \int_{0}^{t} f(s) ds $ is: graphic of g graphic of f I analyze the derivative and the sign of the derivative and try to find ...
2
votes
2answers
46 views

Prove that $p(z) = 2z^5 + 6z - 1 $ have roots (in two sets)

Prove that $p(z) = 2z^5 + 6z - 1 $ have one real root in $(0,1)$ and four root in $\left\{ z \in \mathbb{C} : 1<|z|<2 \right\}$. I suppose that we should use Rouché's theorem but I have no ...
1
vote
1answer
60 views

Complex Analysis: Isolated Singularities, Poles, and Residues

I was given the following question. Show that the isolated singularities of the function $f(z) = \frac{z}{z^4+4}$ are poles. Determine the order of each pole and find the corresponding ...
2
votes
1answer
40 views

How to show that it holds $|z|<2\max_{0\le k<n}|a_k|^{\frac{1}{n-k}}$ for any root of $X^n+\sum_{k=0}^{n-1}a_kX^k$?

Let $z\in\mathbb{C}$ be a root of the complex polynomial $$f=X^n+\sum_{k=0}^{n-1}a_kX^k$$ I want to show that it holds $$|z|<2\max_{0\le k<n}|a_k|^{\frac{1}{n-k}}$$ Proof: For $s>1$, consider ...
44
votes
5answers
1k views

Polynomials such that roots=coefficients

Here is my question : Are there monic polynomials with degree $\geq 5$ such that they have the same real all non zero roots and coefficients ? Mathematically, prove or disprove the existence ...
2
votes
2answers
92 views

Show the Equation $2x-1-sinx=0$ has Exactly One Real Root

Question : Use the Intermediate Value Theorem and Mean Value Theorem to show that the queation $2x-1-sinx=0$ has exactly one root. My answer : Since we cannot compute the $y$ when $x=0$, we ...
1
vote
4answers
51 views

Technique to simplify algebraic calculations on roots of polynomial

I was once told about a technique to simplify algebra on the roots of a polynomial. So if you want to find $\alpha^3+\beta^3+\gamma^3$, where $\alpha,\beta \text{ and } \gamma$ are roots of ...
5
votes
4answers
103 views

What is the minimum value of $abc$

If the roots of the equation $$ax^2-bx+c=0$$ lie in the interval $(0,1)$, find the minimum possible value of $abc$. Edit: I forgot to mention in the question that $a$, $b$, and $c$ are natural ...