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.

learn more… | top users | synonyms (1)

2
votes
3answers
69 views

why if x in 1/n power >(<) y in 1/m power then x in c/n power >(<) y in c/m power?

As you might guess this is one more stupid question from non-matematician, and you are right. I found this exercise in "Algebra and trigonometry book": $7^{1/2}$ or $4^{1/4}$. After some googling I ...
2
votes
1answer
125 views

Location of Complex Roots

Here is a problem I think dealing with Rouche's theorem: How many roots does the equation $$ \frac{1}{2}e^z+z^4+1=0 $$ have in the left half plane $Re(z)<0$ I see that in order to have a root in ...
2
votes
0answers
192 views

Can a linear combination of even Legendre polynomials have common real root(s) with a linear combination of odd Legendre polynomials?

I am using the following definition of Legendre Polynomials: $P_0(x)=1$, $P_1(x)=x$ and $$P_{k+1}(x)=\left(\frac{2k+1}{k+1}\right)xP_k(x)−\left(\frac{k}{k+1}\right)P_{k−1}(x)$$ Let ...
0
votes
2answers
106 views

Solve an equation of 3rd order [duplicate]

What is the simplest method to solve an equation of 3rd degree. For example: $$-x^{3} + x^{2} + x - 1 = 0$$ Please I don't want the resolution of this equation I just want the simplest method to use ...
4
votes
1answer
240 views

How to solve a polynomial with power fractions like $a-ax+x^{0.8}-x^{0.2}=0$

I have something like $a-ax+x^{0.8}-x^{0.2}=0$ with parameter a>0 and variable x>0. I know by trial and error that the equation has three real roots for parameter a greater than certain value, ...
3
votes
2answers
946 views

Finding the Number of Zeros of a Function in a Given Annulus

Consider $z^6 - 6z^2 + 10z + 2$ on the annulus $1<|z|<2$. By Rouche's Theorem $|f(z) + g(z)| < |f(z)|$ implies that both sides of the inequality have the same number of zeros. I understand ...
2
votes
2answers
215 views

sum of squares of the roots of equation

The equation is $$x^2-7[x]+5=0.$$ Here $[x]$ the greatest integer less than or equal to $x$. Some other method other than brute forcing. I tried a method of putting $[x]=q$ and $x=q+r$ which gives an ...
9
votes
1answer
315 views

Existence of real roots of a quartic polynomial

Question What is the minimum possible value of $a^{2}+b^{2}$ so that the polynomial $x^{4}+ax^{3}+bx^{2}+ax+1=0$ has at least 1 root? Attempt I divided by $x^{2}$ and got ...
3
votes
4answers
254 views

Solving this 3-degree polynomial

I'm trying to factor the following polynomial by hand: $-x^3 + 9x^2 - 24x + 20 = 0$ The simplest I could get is: $-x^2(x-9) - 4(5x+5) = 0$ Any ideas on how I could go ahead and solve this by hand? ...
4
votes
3answers
131 views

Prove a polynomial has all roots different

I need to prove that $P(x)=x^4+\zeta x+1$ where $\zeta\in\mathbb{R}$ and $\zeta\neq0$ has four different roots. I have tried with the rule of signs of Decartes but it does not give enough information. ...
1
vote
3answers
279 views

Analytic Function Root Finding - Rouche's Theorem

Please help determine the number of roots of $$ z^7+2z^3+1 $$ in the region $1/2\leq|z|<1$. It seems like everything I do with Rouche's theorem does not give a strict inequality for when ...
6
votes
1answer
216 views

Location of zeros of a sum of exponentials

Describe the approximate locations of the zeros of the function $$ f(z) = e^{iz}+e^{-iz}+e^z $$ lying outside the circle $|z|=R >>1$. Another prelim problem. For Rouche's theorem we need to ...
3
votes
2answers
91 views

Convergence of Roots for an analytic function

Show that the roots of $$ f(z) = z^n+z^3+z+2 =0 $$ converge to the circle $|z|=1$ as $n \to \infty$.
2
votes
1answer
235 views

Roots of a finite Fourier series?

In general, are there any clever tricks to help find the roots of a finite Fourier series? Presumably there aren't analytic methods, but can we use the fact that our function is a finite Fourier ...
1
vote
1answer
109 views

Skecth the root locus with respect to K for the characteristic equation.

$$s^4+12s^3+22s^2+(20+K)s+2K=0$$ I don't understand, how can I sketch the root locus? Can anybody help me to understand?
2
votes
4answers
70 views

Show that $1$ and $2$ are zeros of the following polynomial

Show that $1$ and $2$ are zeros of the polynomial $P(x)=x^4-2x^3+5x^2-16x+12$ and hence that $(x-1)(x-2)$ is a factor of $P(x)$
0
votes
1answer
42 views

Complex solutions to $a = (z+b)^n$

I have tried the whole afternoon trying to figure out how to approach an equation of the form $a = (z+b)^n$, more specifically the equation: $1 = (z+1)^4$. Is there a general approach to equations of ...
5
votes
4answers
358 views

Prove $\sqrt[3]{3} \notin \mathbb{Q}(\sqrt[3]{2})$

I've tried solving $\sqrt[3]{3} =a + b* \sqrt[3]{2}+c* \sqrt[3]{4}$, but there is no obvious contradiction, even when taking the norms/traces of both sides. I can't think of another approach. This is ...
0
votes
2answers
40 views

Get polynom from polynom, roots of second one are multiplication of first one.

I have a polynomial $P$, with unknown roots $r_1,r_2, ... ,r_n$. My goal is to find a polynomial $X$ with roots $s_1,s_2, ... ,s_n$, where each $s_i = 2r_i$ I shall get $X$ with no need to find the ...
2
votes
1answer
93 views

Prove that $f: (a,b)→ℂ$ cannot have infinitely many zeros in $(a,b)$

I have the following nonzero analytic function: $f:ℂ→ℂ$. We will consider only the restriction $f: (a,b)→ℂ$, $a,b∈ℝ$ and $a<b$. My question is: Prove that $f: (a,b)→ℂ$ cannot have infinitely many ...
1
vote
1answer
105 views

Assume that the set of values where $f^{(k)}≠0$ is finite

Let $f:ℝ→ℝ$ be a real analytic function. Let $f^{(k)}$ be the $k$th derivative of $f$. Assume that the set of values where $f^{(k)}≠0$ is finite, then what we can say about the function $f$.
1
vote
1answer
66 views

Prove that $D$ is bijective with the integers set $ℤ$

Let $f:ℝ→ℝ$ be a real analytic function. Assume that $f$ has infinitely many zeros. Let $D$ be the set of those zeros. Prove that $D$ is bijective with the integers set $ℤ$.
0
votes
4answers
119 views

Could someone explain the solution to the problem in the screenshot?

This is from a past-years'-questions PDF for an Indian secondary school olympiad. Could someone explain the answer to question no. 6 shown in the picture?
2
votes
3answers
1k views

$x^4 + 4x^3 - 2x^2 - 12x + k$ has 4 real roots. Find the condition on k.

The question is: $f(x) = x^4 + 4x^3 - 2x^2 - 12x + k$ has 4 real roots. What values can k take? Please drop a hint!
4
votes
2answers
119 views

$x^4 + 4rx + 3s = 0$ has no real roots. Relate $r, s$.

It is given that $x^4 + 4rx + 3s = 0$ has no real roots. What can be said about r and s? a) $r^2 < s^3$ b) $r^2 > s^3$ c) $r^4 < s^3$ d) $r^4 > s^3$ How to even begin??
4
votes
2answers
90 views

Finding root using Hensel's Lemma

Hensel's Lemma calculates root of a polynomial $\in \mathbb{Z}_p[X]$ but is there any other significance to other branches of mathematics or outside mathematics? Why is finding root of ...
3
votes
5answers
457 views

Proof By Contradiction, Rational Roots

This was an exam question that I got totally wrong and am a bit question. Prove $x^3 + x + 1 = 0$ has no solutions. Prove by contradiction. Assume: $x^3 +x +1 =0$ has at least one rational root. ...
0
votes
0answers
36 views

Unicity of solutions in several dimensions

Let $h:ℝ^{r+1}→ℝ^{r+1}$ be a real function. If $r=0$ and $h$ is bijective then we know that the equation $h(x)=y$ has a unique solution. My question is: How about the case where $r>0$? I know that ...
2
votes
1answer
93 views

congruence modulo infinity

Going through Hensel's Lemma, I feel I read somewhere that the limit of sequence of integers $a_0,a_1,a_2,...$=$ a$ is root of the $f(X)\in\mathbb{Z}_p[X]$, where, ...
2
votes
1answer
326 views

Zeros set of analytic functions over complex plane with several variables

I know that the zeros of analytic function (with one variable) over complex plane are isolated. However, I am not aware about the structure of the zeros set of analytic functions over complex plane ...
0
votes
4answers
111 views

Square and square root and negative numbers [duplicate]

Are they equal? -5 = $\sqrt{(-5)^2}$
3
votes
2answers
210 views

Show $\;f(x) = x^{20}-70x^3+1\;$ has zero in $\;[0, 1]$

How can we use the Intermediate Value Theorem to show that the function $$\;f(x) = x^{20}-70x^3+1\;$$ has a zero in the interval $\;[0, 1]\,$? (To use the theorem I need to show that the function ...
5
votes
2answers
144 views

Roots of a Polynomial of degree 3

Is there any condition based on the coefficients of terms that guarantees all real solutions to a general cubic polynomial? e.g. $$ax^3+bx^2+cx+d=0\, ?$$ If not, are there methods rather than ...
1
vote
2answers
69 views

Larger Theory for root formula

Consider the quadratic equation: $$ax^2 + bx + c = 0$$ and the linear equation: $$bx + c = 0$$. We note the solution of the linear equation is $$x = -\frac{c}{b}.$$ We note the solution of the ...
3
votes
2answers
129 views

How to find the roots of $f(x)=x^{2}+2x+2$ in $\mathbb{Z}_{3}$ ? in $\mathbb{Z}_{5}$ ? in $\mathbb{R}$?

Normally I just guess a root and then smash one out in high degree functions, or complete squares or any other number of mathemagical tricks, but my textbook has decided to break numbers on me and I ...
2
votes
1answer
214 views

Error in Proof of Residues?

I wanted to prove that the function $$F(z) = \frac{z-\sum_{j =2}^{n-1} z^j}{1-\sum_{k=1}^{n} z^k} $$ will only contain simple poles. Is the following proof correct? Which implies that $z_o$ ...
0
votes
1answer
213 views

Number theory - Primitive root of $338$ [closed]

Im having problem $338$ root. I know it has a root because $13^2\times2=338$ but what is the correct way to find it??
1
vote
2answers
141 views

Easy way to simplify this expression?

I'm teaching myself algebra 2 and I'm at this expression (I'm trying to find the roots): $$ x=\frac{-1-2\sqrt{5}\pm\sqrt{21-4\sqrt{5}}}{4} $$ My calculator gives $ -\sqrt{5} $ and $ -\frac12 $ and ...
1
vote
0answers
44 views

root of binary matrix

There is a square matrix A defined over the field GF(2). It means there are zeros and ones in its cells, xor stands for element summation, logical and - for multiplication. Is there any way to find ...
2
votes
3answers
101 views

Solving $\sqrt{x}-2\sqrt[4]{x}-8 = 0$

How can I do this question? $$\sqrt{x}-2\sqrt[4]{x}-8 = 0$$ Can I solve this? I tried to multiply everything by $x^4$, and got $$8x^4+x^3 -2x = 0$$ I don't know how to proceed from here.
0
votes
1answer
38 views

Square root entries of matrices

How would you simplify something like this? $((\xi'\omega \xi)^{-1})^{0.5}$ where $\xi$ is a $k \times 1$ matrix, $\omega$ is a $k\times k$ square matrix. Thank you very much! Edit: Yes, though ...
3
votes
2answers
461 views

A Question On Euler's Proof Of the Basel Problem

I've studied the proof that Euler gave for the famous Basel Problem, and it would seem that while it is technically correct, he does not justify all of his steps properly. Namely, he assumes that ...
9
votes
4answers
986 views

Prove $\sqrt{3a + b^3} + \sqrt{3b + c^3} + \sqrt{3c + a^3} \ge 6$

If $a,b,c$ are non-negative numbers and $a+b+c=3$, prove that: $$\sqrt{3a + b^3} + \sqrt{3b + c^3} + \sqrt{3c + a^3} \ge 6$$ Here's what I've tried: Using Cauchy-Schawrz I proved that: $$(3a + ...
15
votes
4answers
594 views

Prove $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$

Let $a,b,c$ are non-negative numbers, such that $a+b+c = 3$. Prove that $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$ Here's my idea: $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$ ...
10
votes
5answers
363 views

Finding real roots of $ P(x)=x^8 - x^7 +x^2 -x +15$

Let $ P(x)=x^8 - x^7 +x^2 -x +15 $, Descartes' Rule of Signs tells us that the polynomial has 4 positive real roots , but if we group the terms as $$ P(x)= x(x-1)(x^6+1) +15 $$ we find that $ P(x) $ ...
22
votes
2answers
512 views

How to show that a root of the equation $x (x+1)(x+2) … (x+2009) = c $ can have multiplicity at most 2?

How to show that a root of the equation $$x (x+1)(x+2) ....... (x+2009) = c $$ can have multiplicity at most 2 , and to find the value of $ c $ for which this is possible. I proceeded by using the ...
1
vote
2answers
74 views

Indeterminate Limit (Finding the Remainder to a Root)

So i was working on this: $$ \lim\limits_{x\to1} \frac{x + \sqrt{x} - 2}{x - 1} $$ and I thought to simpify my top by multiplying by a conjugate, taking everything other than the $x$ to be the $b$ ...
1
vote
2answers
153 views

Comparing square roots of negative numbers

If we have for instance $\sqrt{-25}$, that is, a square root of $-25$, I know the answer can be $5i$ (Is $-5i$ also correct? Sorry not professional in mathematics). My main question here is how to ...
0
votes
1answer
123 views

Polynomials with integer coefficients, with value close to $0$, in the interval $[-1,1]$

Are there some interesting properties of polynomials with integer coefficients of degree $2^n$ which satisfy $\mid P(x) \mid \le \frac{1}{2^k}$ ? I know that their coefficients are bounded and the ...
0
votes
1answer
274 views

Finding roots of $e^x\sin^2(x)-\cos(x)=0$ using Mathematica

$$e^x\sin^2(x)-\cos(x)=0$$ I'm trying to find 5 roots of this equation but mathematica keeps giving me back an error saying $\sin^2$ is not a well-formed equation. Thanks, Joe