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Questions tagged [roots]

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 (radicals) and the (arithmetic) tag. For questions about roots of Lie algebras, use the (lie-algebra) tag instead.

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20 views

The real solution from a system of equation

I found this question from my friend's math competition, I don't know where I must start it There are 3 couples of real numbers $$(x_1,y_1) (x_2,y_2)$$ and $$(x_3, y_3)$$ that satisfies the system of ...
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1answer
67 views

Closed form of the real zeros of $x^{n+1}-2x^n+1$ for positive integer $n$

I was wondering: What are the real zeros of the function $$x^{n+1}-2x^n+1$$ where $n$ is a positive integer? Obviously, there is a zero at $x=1$. But, if $n$ is even, there are two other ...
2
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2answers
57 views

How many roots does an exponential polynomial have?

Let $s$ be a complex variable and consider two polynomials with real coefficients: $$A(s) = s^n + a_{n-1}s^{n-1}+\ldots+a_1s+a_0,$$ $$B(s) = s^m + b_{m-1}s^{m-1}+\ldots+b_1s+b_0,$$ where $n \ge m$. ...
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3answers
55 views

If one root of the equation $x^3-7x^2+14x-8=0$ be the square of the other [on hold]

If one root of the equation $$x^3-7x^2+14x-8=0$$ be the square of the other then solve the equation by getting the relationship between the coefficients and roots.
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0answers
43 views

An open cover of $\mathbb{R}^n$ and $\mathbb{C}^n$

Consider the following subset of $\mathbb{R}^n$: \begin{eqnarray}V_i:=\{(p_1, \cdots, p_n)\in\mathbb{R}^n|x^n-p_1x^{n-1}+p_2x^{n-2}-\cdots+(-1)^np_n=0\text{ has at least one root with multiplicity at ...
2
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2answers
29 views

Roots in a polynomial ring

If we have a polynomial in a polynomial ring, are its "roots" only valid if the roots are in the ring itself? i.e. For $x^2+1 \in \mathbb{C}[x]$ we have roots $i$ and $-i$. But if we consider it as $...
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1answer
45 views

Analytical approximate solution to a trascendental equation

I have the following equation to solve $$ z+e^{z^2}\operatorname{erfc}(z)=0 $$ being $$ \operatorname{erfc}(z)=1-\frac{2}{\sqrt{\pi}}\int_0^ze^{-s^2}ds. $$ I solved it numerically and appears to have ...
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1answer
21 views

Discussing the convergence of a sequence

If $k, x_1$ are positive, and $x_{n+1}$ = $\sqrt {k + x_n},$ discuss the convergence of the sequence $x_n$, according to whether $x_1$ is less than or greater than $\alpha$, the positive root of $x^2 =...
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2answers
69 views

If If $x^2+ax+b+1=0$ has roots which are positive integers, then $(a^2+b^2)$ can be which of the given choices?

If $x^2+ax+b+1=0$ has roots which are positive integers, then $(a^2+b^2)$ can be (A) 50 (B) 37 (C) 19 (D) 61 My approach: I first took roots $\alpha$, $\beta$ and then applied sum and ...
2
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2answers
30 views

Solving quadratics writing solutions

I am confused on the notation used when writing down the solution of x and y in quadratic equations. For example in $x^2+2x-15=0$, do I write : $x=-5$ AND $x=3$ or is it $x=-5$ OR $x=3$ which ...
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0answers
25 views

Why should we learn all the method for root finding?

if we can find root with Newton-Raphson method very firstly then why should we learn all other method also i.e Bisection, False position,Fixed point, Secant etc. if we have problem with finding ...
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1answer
39 views

Connectedness of the domain in Identity Theorem

Let $f$ be a complex-valued holomorphic function defined on an open set $\Omega\subseteq\mathbb{C}$, $f:\Omega\rightarrow\mathbb{C}$, which is not identically zero. Let $S=\{a\in \Omega :f(a)=0\}$=...
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4answers
57 views

The classification of and description of the root near $1/2$ of $x^d +x^\left({d-1}\right) + x^\left({d-2}\right) + \dots + x^2 + x - 1=0$

In Arturas Dubickas paper "On the number of reducible polynomials of bounded naive height", manuscripta math. 144, 439–456 (2014) he discusses a bounding polynomial $x^d + x^\left({d-1}\right) + x^\...
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1answer
69 views

Solving an interesting polynomial with degree 4? [duplicate]

So the equation is as follows: $$ 6x^2 -\ 25x \ + 12 \ +\ \frac6{x^2}\ + \frac{25}{x} = 0$$ So one thing that is immediately observable is that pairs of roots will be of the from $$x_1=-\frac{1}{...
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1answer
35 views

Positive solution for an exponential equation

Define a function of $t$, $F(t) = e^{X_1t}-e^{X_2t}-e^{X_3t}+e^{X_4t},$ for some fixed real values $X_1, X_2, X_3, X_4 \in \mathbb{R}$ and $X_1<X_2 < X_3 < X_4$. Whether $F(t)$ has at most ...
0
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1answer
39 views

How determine relationship between two $n$ degree polynomials roots

Suppose we have the following polynomial equations: $$a_nx^n+a_{n-1}x^{n-1}+...a_0=0,$$ $$b_nz^n+b_{n-1}z^{n-1}+...b_0=0.$$ I need to analytically determine the relationship between $x^*$ and $z^*$ in ...
5
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1answer
68 views

Zeroes of $\sin(z)+2\sin(8z)$

Clearly the function $f(x)=\sin z+2\sin8z$ has many zeroes on the real line. Does it have any off the real line? I thought of inspecting its real and imaginary parts separately: $$f(x+iy) = (\sin x\...
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1answer
53 views

How many roots are in case of polynomial $f(x).g(x)$

If two real polynomials $f(x)$ and $g(x)$ of degrees $m(\geq 2)$ and $n(\geq 1)$ respectively satisfy $$f(x^2+1)=f(x)g(x)$$ for every $x \in R$ then (A) $f$ has exactly one real root $x_0$ such that $...
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1answer
23 views

How to find number of solutions using the derivative?

I know these are probably well-known results, but I want to find how many roots a function has using its derrivatives. Consider a two times differentiable function $f:\mathbb{R} \to \mathbb{R} $. If $...
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2answers
65 views

Number of roots of $f(x)=x^2-2^{x-\frac{1}{x}}$

How many roots does $f(x)=x^2-2^{x-\frac{1}{x}}$ have in $(0,1]$? Since this function is continuous, I plugged in a couple of values and looked at the signs of the function's values and I concluded ...
1
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1answer
51 views

$\sum_{i=1}^n {(a_i\sqrt{b_i})} \ne 0$

In a surd $a\sqrt{b}$   ($b \in \mathbb{Z^+}$)   the value of $b$ can assumed to be a square-free integer ($b = p_1p_2\dots p_k$, where $p_i$ are distinct primes), since otherwise a ...
1
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1answer
44 views

zeros of convex combination of monic real polynomials with degree $2$

Suppose $f(x) = x^2 + a_1 x + a_0 \in \mathbb R[x]$ and $g(x) = x^2 + b_1 x + b_0 \in \mathbb R[x]$ with zeroes in the open unit disk of $\mathbb C$. Let $h_t(x) = (1-t) f + t g$ where $t \in [0,1]$. ...
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0answers
35 views

Counting the number of roots of a polynomial in each quadrant of the complex plane

I'm looking to answer the question: Given a polynomial of a single variable $x$: $\sum_n a_n x^n = 0 $, how many roots are there in each quadrant of the complex plane, counting positive/...
2
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1answer
31 views

number of roots of a multilinear polynomial over a finite field

I'm trying to upper-bound the number of roots of a degree $k$ multilinear polynomial $p(X_1,\ldots,X_n) \in F_2[X_1,\ldots,X_n]$ (for $1 < k \le n$). Unfortunatelly, the Schwartz–Zippel Lemma ...
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3answers
61 views

Extending Cauchy-Schwarz to any $p \in (1,\infty)$

Given $n$ non-negative real number $\left\lbrace a_i\right\rbrace_{i=1}^n$, we know that using Cauchy-Schwarz, one can prove that $$ \sum\limits_{i=1}^n \sqrt{a_i} \leq \sqrt{n} \cdot \sqrt{\sum\...
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0answers
94 views

Proving electrostatic analogy for root locus

My teacher told us that there is a provable mathematical analogy between root locus and the lines of force generated by electric charges, where every pole can be associated to a positive charge and ...
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3answers
47 views

Roots of polynomials and their formulae relating to coefficients

Write down the cubic equation given that $\alpha + \beta + \gamma = 4$, $\alpha^2 + \beta^2 + \gamma^2 = 66$, and $\alpha^3 + \beta^3 + \gamma^3 = 280$ Ok so, the sum of roots is given and I'm able ...
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2answers
52 views

Finding number of real roots of an equation

Equation is--> $$ x^{13} + x - 1/e^x - \sin(x) =0 $$ To find number of real roots of the equation. Context--> I am solving previous years questions of IIT Jam Mathematical Statistics (MS entrance ...
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1answer
122 views

For $c\in\mathbb{F}_p^*$, the cubic $t^3-3ct^2-3t+c$ has exactly one root $r\in\mathbb{F}_p$. Express $r$ in terms of $c$ without cubic roots.

For some $c \in \mathbb{F}_p^*$ consider the polynomial $$ f(t) = t^3 - 3ct^2 - 3t + c $$ for $p \equiv 1$ (mod $3$) and $p \equiv 3$ (mod $4$). In this case $3$ is a quadratic non-residue modulo $p$ ...
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0answers
15 views

Complex roots using Muller Method

Please explain what should be $d$ here? How to calculate that? $x_3= x_2 - \displaystyle\frac{2c}{b} + d$ where $c = f(x_2)$ and $b= a h_1+\sigma_1$ This is Muller's method.
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2answers
36 views

For which values of $a$ we will get two different roots?

In given the following system of equations: $$ |x-1| > 2x+2 $$ $$ x^2 + ax + a -1 = 0 $$ For which values of $a$ we will get two different roots?
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0answers
34 views

Complex Roots question.

How many complex solutions does 2^π have? Obviously, something like 2^(7/3) has 3 complex solutions, ie. 3 complex numbers c such that 2∈ c^(3/7). What is the set of c such that 2∈c^(1/π)? Is it ...
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1answer
30 views

Questions about some special classes of polynomials [closed]

Consider natural numbers as coefficients $a_i\in \mathbb{N}_0$. Consider the polynomials $\sum_{i=0}^{m-1}a_iX^i + X^m$. Note that $a_i\in \mathbb{N}_0$. Question 1: What are classical results ...
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0answers
37 views

Algorithm for regular continued fraction of a square root

Say I have a number $n$, and want to find the expression of $\sqrt{n}$ as a regular continued fraction. How would I do such a thing systematically? A naive computer algorithm wouldn't work, due to ...
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3answers
52 views

Finding real $a$, $b$, $c$ such that $x^5 - 2x^4 + ax^3 + bx^2 - 2x + c$ has $1+i$ as a zero, and one negative integer as a zero with multiplicity $2$

Find $a, b, c \in \mathbb{R}$, if one zero of $p$ is $1+i$ and $p$ has one negative integer zero with multiplicity $2$, where: $$p(x) = x^5 - 2x^4 + ax^3 + bx^2 - 2x + c$$ I know that one zero of $...
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2answers
51 views

Solving a quadratic equation using the “splitting the middle term” method.

Use splitting the middle term method to solve the below equation. Is there a limitation to this method? $$5b^2-16b+4=0$$
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0answers
44 views

Roots of equation involving both polynomials and trigonometric functions

I have to determine for which values of $x$ that the velocity vector is orthogonal to the acceleration vector, the position is given by: $(3 \cos(t), - \sin(3t), 2t^3 - t^2)$, I then use that $u \...
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2answers
35 views

Showing that, for $|a|>3$ and $n\geq1$, the function $f(z)=e^z-az^n$ has exactly $n$ roots (all simple and different) in the open unit disc [duplicate]

Let $|a|>3$, $n\geq1$, $n\in\mathbb{N}$, then the function $$f(z)=e^z-az^n$$ has exactly $n$ roots in the disc $\{z\mid|z|<1\}$, and that they are all simple. Hint: look at $f(z)-f'(z)...
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1answer
30 views

Finding remainder when unknown $f(x)$ is divided to $g(x)$

When $f(x)$ is divided by $x - 2$ and $x + 3$, the remainders are 5 and -1, respectively. Find the remainder when $f(x)$ is divided by $x^2 + x - 6$ My method: Since $x - 2$ and $x + 3$ are linears, ...
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2answers
90 views

Prove all roots of $p_n(x)-x$ are real and distinct

Given a polynomial series $\{p_n(x)\}_{n=1}^{\infty}$ in $\mathbb{R}[X]$ with initial value $p_1(x)=x^2-2$. And $p_k(x)=p_1(p_{k-1}(x))=p_{k-1}(x)^2-2,\;k=2,3,\cdots$. Prove that for each integer ...
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3answers
60 views

Solve $x^2 - \frac{1}{\ln(x) - 1} + \frac{1}{x^2(\ln(x)-1)} = 0$ [closed]

How would I find the roots of this function? \begin{equation} x^2 - \frac{1}{\ln(x) - 1} + \frac{1}{x^2(\ln(x)-1)} = 0 \end{equation} I don't really know how i would tackle this problem because we'...
5
votes
2answers
73 views

Is there a simpler method of calculating $\sqrt[n]{x}$?

I've began to reteach myself Algebra and am brushing up on my roots (pun intended). I've been following this website (which makes it simple to refresh your memory), and as I reviewed the root examples ...
2
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1answer
63 views

How is this step acheived? $h(x)= 27x^6+26x^3-1 $

Find all real roots of $h(x)$. Solution I have solved the question by letting $u = x^3$ and then using the quadratic formula to solve $27u^2+26u-1 = 0$. However I don't have a clue as to how they'...
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votes
3answers
32 views

I have the following equation to solve analytically: $\ 0.5q+\sqrt q-2=0$

I sincerely apologize in advance for the simple question I am asking but I really cannot live with this doubt in my mind. I have the following equation to solve analytically: $$\ 0.5q+\sqrt q -2=0$$...
3
votes
2answers
41 views

Why are polynomials tangential to the $x$ axis at real double roots?

If $(x-a)^2$ is a root of a polynomial, then the graph will be tangent to the $x$ axis at $x=a$ but why? I know this is always the case for real double roots however I do not know the explanation for ...
1
vote
1answer
41 views

Can $\frac{xp(x)-ap(a)}{x-a}$ have an inside root if $p(x)$ does not?

Let $p(x)$ be a polynomial that has all of its roots outside of the unit circle and let $a$ be some real number that is not a root of $p(x)$. Can the polynomial $$ \frac{xp(x)-ap(a)}{x-a}$$ have a ...
2
votes
3answers
77 views

Irrationality of $(a_1+\sqrt{b_1})(a_2+\sqrt{b_2})$

Sorry, for a rather silly question. Suppose $a_1$, $b_1$, $a_2$, $b_2$ are integers, all different from zero, while $b_1$ and $b_2$ are co-prime positive integers, neither being a complete square. ...
0
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2answers
21 views

Roots of polynomial irreducible over the rationals

If a polynomial is "irreducible over the rationals", does it mean that it has no rational roots? I would say yes because otherwise I could divide out the linear factors (i.e. rational roots) but ...
1
vote
4answers
61 views

Solving $z^3=1+i \sqrt3$

How can I solve the complex equation $$z^3=1+i \sqrt3$$ Splitting $z$ into $z=a+bi$ gives me the mess $$a^3+3a^2b^2i-3ab^2-b^3i-i \sqrt3-1=0$$ where I dont know how to continue. I have never really ...
3
votes
3answers
40 views

Find the number of roots of $F(x)= \int_0^x e^t(t^2-3t-5)\mathrm dt , x>0$ in the interval $(0,4)$

Let $$F(x)= \int_0^x e^t(t^2-3t-5)\mathrm dt , x>0$$ Find the number of roots of $F(x)=0$ in the interval $(0,4)$. My attempt: I simply integrated it and got $F(x)=e^x(x(x-5))$ which has roots $...