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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38 views

Let $r_0,r_1,…,r_m$ be the real roots of $a_nx^n+a_{n-1}x^{n-1}+…+a_0$.Is there a closed-form expression for $\sum_{i=1}^mr_i -\sum_{i=1}^m1/r_i$?

Let $r_0, r_1, ... ,r_m$ be the real roots of $a_nx^n+a_{n-1}x^{n-1}+...+a_0$, with $a_0\ne0.$ Is there a closed-form expression for $$ \ \ \ \ \ \sum_{i=1}^mr_i - \sum_{i=1}^m \frac{1}{r_i} \ \ \ ...
1
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0answers
31 views

Find the positive root of the equation $ce^{-c}-2(1-e^{-c})^2=0$

Can you help me find a root for $c$ in the equation below? $$ce^{-c}-{10\over5}(1-e^{-c})^2=0$$ By expanding this I got, $$ce^{-c}-2 + 4 e^{-c}-2e^{-2c}=0$$ now grouping, ...
2
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2answers
91 views

How to show the equation $x^4 + 2cx^3 + 6x^2 + 60x =-11$ has exactly two real solutions?

How can we show that $x^4 + 2cx^3 + 6x^2 + 60x =-11$ has exactly two real roots? $c$ is any element in the interval $(-2,2)$.
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2answers
39 views

When looking for zeros of a rational function, why is the numerator equated to zero and not the denominator?

If you have a function $F(x)=\dfrac{a(x)}{b(x)}$ and you are asked to find the zero(s) of the function, why do you set the numerator equal to zero, and not the denominator?
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1answer
30 views

Solving for the roots of a polynomial

Suppose we have a polynomial of the form: $$-x^3+3x^2+9x-27=0$$ Is there an easy way to find the solutions of $x$? I know that they will be factors of $27$, so I begin by factoring $27$ into ...
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3answers
49 views

Finding the roots of a polynomial on a complex plane [duplicate]

I use an online calculator in order to calculate $x^5-1=0$ I get the results x1=1 x2=0.30902+0.95106∗i x3=0.30902−0.95106∗i x4=−0.80902+0.58779∗i x5=−0.80902−0.58779∗i I know that this is the ...
0
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1answer
31 views

Show uniqueness of a point

I use the banana function $F(x_1,x_2)=(1-x_1)^2+100(x_2-x_1^2)^2$ and I found the minimum point X to be (1,1). I need to show the uniqueness of that point. Could you please help me on how to show ...
0
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1answer
30 views

Counting function for the number of zeros of a continuous positive function?

Let $f(x)$ within $x\in[a,b]$ an absolute continuous function with $f(x)\geq0$ $f(x_m)=0$ for all absolute minima $x_m$ no other zeros than at $x_m$ I am trying to define a counting function for ...
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2answers
45 views

Finding zeroes of $x^3-5x^2+11x+17$

I'm trying to find all the zeros of $x^3-5x^2+11x+17$. I figured the possible zeros as being +/- 1, +/- 17$. The book says that -1 is supposed to be a factor, but I tried dividing the polynomial by ...
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0answers
20 views

Polynomial Root Multiplicity Testing.

I would appreciate some help here. Either a reference or a proof or just a statement that helps me to conduct research of my own. Long ago when I was studying polynomials intently I read about a ...
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0answers
27 views

Show that $\sin z$ has only one series expansion

The question goes: An extension of the real function $\sin x$ into a complex analytic function is by defining $\sin z = z- z^3/3! + z^5/5!- \cdots$. Show that this is the only way6 to extend $\sin x$ ...
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1answer
57 views

Estimating the modulus of the roots of $\sinθ_1z^3+\sinθ_2z^2+\sinθ_3z+\sinθ_4=3$

If $θ_1,θ_2,θ_3,θ_4$ are four real numbers, then any root of the equation $$\sinθ_1z^3+\sinθ_2z^2+\sinθ_3z+\sinθ_4=3$$ lying inside the unit circle $\vert z\vert$=1, satisfies which inequality? ...
3
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0answers
115 views

On polynomials over finite fields?

Pick prime $q\in\Bbb Z$ such that $q>B^{3t}>(mB)^{2t}$. Suppose I have a multilinear polynomial $g(x)\in \Bbb Z[x_1,\dots,x_n]$ of degree $t$ with $m^t$ non-zero coefficients that bound by ...
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2answers
45 views

Number of roots of $x^3+2x-1$ in $\mathbb Q$

How many roots does $x^3+2x-1$ have in $\mathbb Q$? I know that it has one real and two complex conjugate roots because the determinant is $-59$.
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1answer
47 views

$P_n(x):=1+ \sum_{m=1}^n\dfrac{x^m}{m!}$ has no real root for even $n$ and exactly one real root for odd $n$

Is it true that $P_n(x):=1+ \sum_{m=1}^n\dfrac{x^m}{m!}$ has no real root for even $n$ and exactly one real root for odd $n$ ? I can only prove that the polynomial cannot have any multiple roots . Am ...
2
votes
3answers
200 views

How to find a solution for this inequation?

what's the best way to find a solution for the following inequation: $$ \sqrt{x^2-1}>x $$ The result is as Wolfram says: $$ x \leq-1 $$
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1answer
33 views

complex root finding

I have a problem of root finding. Format of the function can be arbitrary whereas I want to solve the following one: $$f(x)= a+e^{-x^2}(b+cx+dx^2)$$ where a,b,c and d are given parameters, not ...
3
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3answers
557 views

Using Vieta's Formulas to find expression involving polynomial roots

I'm having trouble with this problem. Show that if the roots of $$5x^3-x^2-2x+3=0$$ are $a_1,a_2,a_3$, then $$1/a_1+1/a_2+1/a_3=2/3$$
3
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1answer
24 views

maximal injective neighborhoods centered at the zero of a polynomial

I was working on a particular problem involving the injectivity of a certain polynomial, $p(z) = z^5 + z -1$, $z \in \mathbb{C}$, in which I needed to find a neighborhood around it's real root so that ...
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0answers
57 views

Application of Rouché's theorem to $e^{z-1}=z$

I am reviewing my complex analysis and I got stuck with an exercise about Rouché's theorem. It states: for $0 \leq C \leq \frac{1}{e}$, show that $Ce^z=z$ has exactly one root in the closed unit disc. ...
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1answer
24 views

Find zeroes of trigonometric polynomial

I know this is a rudimentary question but I'm not really sure how to do this. For my homework problem I have to verify some error term of trapazoidal quadrature. I end up with $$f^{(3)} = -8\sin ...
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1answer
26 views

Analyzing Singularities of a Complex Function

I have the function $f(z) = \frac{1}{e^z - 1} - \frac{1}{z}$. I need to determine wither the singularities $z_0$ of the function are removable, a pole, or essential. $f(z) = \frac{1}{e^z - 1} - ...
3
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4answers
80 views

How many zeros does $f(x)= 3x^4 + x + 2 $ have?

How many zeros does this function have? $$f(x)= 3x^4 + x + 2 $$
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6answers
101 views

Finding the roots of $x^2+(3+5i)x+(7+11i)=0$

how can I solve following equation analytically $$x^2+(3+5i)x+(7+11i)=0$$ I need the roots as follow $x=a+bi$
1
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1answer
32 views

Finding all z (complex) that satisfies an equation

I'm having a little trouble with this problem. It's asking to find all $z\in\mathbb C$ that satisfy $z^3 = -2(1+i\sqrt{3})\overline z$, and to keep the answers in standard form. I tried expanding ...
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0answers
41 views

Undergraduate Complex Analysis: Use of Rouche's Theorem

We are asked to prove $ f = z^{3}e^{1-z} = 1 $ has exactly 2 roots inside $|z| = 1$ We've tried creating functions $p$ and $q$ where $p + q = f$, $p$ with 2 roots inside our boundary, and using ...
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1answer
35 views

Build a polynomial

I have $f=x^3 + ax^2 +bx +c \in \mathbb C[x], \alpha_1,\alpha_2,\alpha_3 \in \mathbb C$ are roots of $f$. $\beta_1 = {\alpha_1 \over \alpha_2} + {\alpha_2 \over \alpha_3} + {\alpha_3 \over \alpha_1}, ...
0
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1answer
47 views

Is Rufinni's rule the quickest hand-method to find roots in high order polynomials?

I'm wondering if there's another method where I do not have to "trial-error" with every guess neither using numerical methods. When the searched root is 3*π/5 with Ruffini's rule I cannot find it ...
3
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1answer
60 views

How to show that the polynomilal $\sum_{k=0}^n \dfrac 1{3^{k^2}}x^k$ has $n$ distinct real roots for any positive integer $n$ ?

From this Rational roots of polynomials ; How might we show that the polynomilal $\sum_{k=0}^n \dfrac 1{3^{k^2}}x^k$ has $n$ distinct real roots $\forall n \in \mathbb N$ ?
6
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2answers
279 views

Can we make a sequence of real numbers such that polynomial of any degree with co-efficients of the sequence has all its roots real and distinct ?

Does there exist a sequence of real numbers $(a_n)$ such that $\forall n \in \mathbb N$ , the polynomial $a_nx^n+a_{n-1}x^{n-1}+...+a_o$ has all $n$ real roots ? Can we make a sequence so that all ...
7
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2answers
140 views

Real roots of a polynomial of real co-efficients , with the co-efficients of $x^2 , x$ and the constant term all $1$

Can all the roots of the polynomial equation (with real co-efficients) $a_nx^n+...+a_3x^3+x^2+x+1=0$ be real ? I tried using Vieta's formulae $\prod \alpha=\dfrac {(-1)^n}{a_n}$ , $(\prod \alpha ...
1
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1answer
39 views

Counting number of roots inside a circle, using Rouche's theorem,

Using the Argument Principle and applying Rouche's Theorem, I know that there are 6 zeroes of the polynomial $$z^{10}-6z^6+3z^4-1$$ inside the unit circle $|z|=1$, no zeroes inside $|z|=1/2$, but I'm ...
0
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1answer
13 views

multiplicity and zeros of a function with more on right side

I got to the problem of finding the zeros and their multiplicity for $f(x)=(x^2 -5x + 6)^2$. How do you do it with all that on the right hand side? There isn't an example like that in the book chapter ...
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3answers
62 views

Find the four complex zeros without given root. [closed]

$$f(x) = 3x^4-x^3+2x^2-x+3$$ Hint: set $= 0$ and divide each side by $x^2$, use identities equation. Please show me the work.
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2answers
40 views

How to use binary search to find a function

I am reading somewhere that $$(\phi'(y))^{-1}=y^{-c_1}+y^{-c_2},$$ $c_1,c_2$ are some numbers, can be solved for $\phi$ using binary search. I am surprised because binary search binary search is used ...
1
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1answer
165 views

How to solve: $x^4+x^2=1$

I solved $x^4+x^2+1=0$. But, the above one is hard. The equation is too hard for me to understand. Can anyone solve it? Please help.
1
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1answer
117 views

Derivation of Cubic Formula

How do we derive the so called cubic formula without using Cardano's method or substitution? I would like to see a step by step proof of where wolfram alpha derives this answer. And also explain where ...
3
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2answers
52 views

Show that no linear polynomial divides $x^k + x^{k-1} + \cdots + 1$ with $k\ge 2$ even

Let $f(x) = x^k + x^{k-1} + \cdots+ 1 \in \mathbb{Q}[x]$, $k\ge 2$ and even. Show there's no linear polynomial which divides $f(x)$. A start: Lets assume by contradiction there's a linear ...
3
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4answers
57 views

How do I find the sum of the cubes of the roots in a cubic polynomial?

I have an equation, $x^3-x^2+x-2$, with three distinct roots, $p$, $q$ and $r$. What is the value of $p^3+q^3+r^3$? I'm not sure how to do this. Using Vieta's formula, we know that: $pqr= 2$ ...
3
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3answers
136 views

Finding the zeroes in a function

I have come across a problem on my trigonometry homework where we need to find the zeros of a function without the use of a calculator. The Equation: Given one of the zeros is $x=5$ $f(x) = ...
1
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1answer
33 views

Finding the argument of a complex number,

I'm trying to locate my four zeroes of a complex-valued function, in order to apply the Residue Theorem. After using the quadratic formula, I am left with $$z^2 = [-3 \pm i\sqrt7] / 2$$ writing the ...
2
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3answers
72 views

By using the Fixed-Point Iteration, I have to find the roots of $f(x)=x^2-x-1=0$

By using the Fixed-Point Iteration, I have to find the roots of $f(x)=x^2-x-1=0$ First I write it in terms $x=f(x)$ $$x^2=x+1$$ $$x=1+\frac{1}{x}$$ Then I make a sequence ...
2
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2answers
198 views

how to find the number of real root

Prove $\;ax^3+bx+c=0\;$, with $\;a,b>0\;$, has at most one real root. Should I use Rolle's theory for this? But I use it, what is the boundary of this function? I can only work out x must not be ...
0
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1answer
24 views

Steffensen's method in Numerical Analysis

In some sources, Steffensen's method is the development of Newton's method to avoid computing the derivative, http://bit.do/Um6M http://cims.nyu.edu/~donev/Teaching/NMI-Fall2010/Homework4.pdf ...
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0answers
35 views

How to find complex roots using Müller's Method?

I have $f(x) = 2x^5 - 2x^4 + 6x^3 - 6x^2 + 8x - 8$ and $x_0 = 0.4$, $x_1 = 0.6$, $x_2 = 0.5$ with a tolerance Es = 10^-4. I solved it using Müller's method in 4 ...
2
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1answer
28 views

How to find initial estimate of roots from graphs?

I have f1(x,y) = x^2 + 3y^2 - 1 = 0 and f2(x,y) = (x-2)^2 + (y-1)^2 - 4 = 0 I am suppossed to find the roots of these nonlinear ...
0
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0answers
19 views

Efficient method to calculate passes (rises and sets) for satellites

There is a function describing the characterisic elevation of ISS seen from an observers horizon. Calculating of an elevation at one time is pretty expensive. So I wanna try to avoid naive iterating ...
2
votes
6answers
242 views

Disproving existence of real root in some interval for a quintic equation

Disprove the statement: There is a real root of equation $\frac{1}{5}x^5+\frac{2}{3}x^3+2x=0$ on the interval (1,2). I am not sure whether to prove by counter-example or by assuming the statement is ...
0
votes
0answers
45 views

What is the condition for the first root of a cubic function to be positive?

Is there any way to determine if the "first root" of a cubic equation is positive, assuming that it's real, given coefficients $a,b,c,$ and $d$? I tried following along with Wikipedia's explaination ...
4
votes
0answers
101 views

$15a+6b+4c+8d=0$ implies $ax^3+bx^2+cx+d$ has a positive root

Let $a,b,c,d$ be real numbers such that $15a+6b+4c+8d=0$. Show that $f(x)=ax^3+bx^2+cx+d$ has a positive root. I want to try to use the intermediate value theorem, showing that $f(s)<0$ and ...