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.
12
votes
3answers
103 views
$\sum_i x_i^n = 0$ for all $n$ implies $x_i = 0$
Here is a statement that seems prima facie obvious, but when I try to prove it, I am lost.
Let $x_1 , x_2 \dots x_k$ be complex numbers satisfying:
$$x_1 + x_2 \dots + x_k = 0$$
$$x_1^2 + x_2^2 ...
1
vote
0answers
43 views
A question about cubic equation.
I'd like to share my doubt on cubic equation.
Step 1: $ax^3+bx^2+cx+d=0$,
Step 2: We can substitute $x=y-\frac b {3a}$ to get $y^3+py+q=0$ where $p,q$ are something.
Step 3: By Vieta's ...
3
votes
5answers
101 views
Polynomials - The sum of two roots
If the sum of two roots of
$$x^4 + 2x^3 - 8x^2 - 18x - 9 = 0$$ is
$0$, find
the roots of the equation
7
votes
2answers
81 views
If $\mathbb f$ is analytic and bounded on the unit disc with zeros $a_n$ then $\sum_{n=1}^\infty \left(1-\lvert a_n\rvert\right) \lt \infty$
I'm going over old exam problems and I got stuck on this one.
Suppose that $\mathbb{f}\colon \mathbb{D} \to \mathbb{C}$ is analytic and bounded. Let $\{a_n\}_{n=1}^\infty$ be
the non-zero zeros of ...
6
votes
3answers
106 views
How can I find all the solutions of $\sin^5x+\cos^3x=1$
Find all the solutions of $$\sin^5x+\cos^3x=1$$
Trial:$x=0$ is a solution of this equation. How can I find other solutions (if any). Please help.
6
votes
1answer
71 views
Bounding the roots of the sum of two monic polynomials with real coefficients.
Let $P_1(z)$ and $P_2(z)$ be monic polynomials with real coefficients and roots $\{z_1^{(1)},z_1^{(2)},...\}$ and $\{z_2^{(1)},z_2^{(2)},...\}$, respectively. Are there any results relating the ...
2
votes
2answers
73 views
Condition For No Existence Of Real Root
$2x^{4}+5ax^{3}-2bx^{2}+1=0$ has no real root in $(5,2014)$
Find the conditions for $a$ and $b$
I am suspicious of even the existence of its solution and at a loss.
3
votes
3answers
46 views
Roots of a cubic equation
I have the following equation:
$s^3+as+b=0$
Now I want the values for a and b for which the given equation has the following complex roots:
$c \pm di$
I don't really care about the remaining root.
...
0
votes
1answer
21 views
Relationship between 2 Dimensional Quadratic systems and roots
Given four points
$(x_1, y_1)
(x_2, y_2)
(x_3, y_3)
(x_4, y_4)$
How does one construct a system of two equations:
$a_1x + a_2x^2 + a_3y + a_4y^2 + a_5xy = c_1$
$b_1x + b_2x^2 + b_3y + b_4y^2 + ...
1
vote
1answer
29 views
Rouche's theorem for two functions that have the same number of roots
I hope this is not too long. Thanks in advance!
Edit: I edited it for a great deal, most of the information was unnecessary.
Let us define a function $h(z) = f(z) + g(z)$. We know that $f(z)$ has ...
-5
votes
1answer
120 views
Root of a quadratic equation that has modulus $1$
Let us suppose $\alpha \in \mathbb C$ and $|\alpha|=1$ and $\alpha$ satisfies a monic quadratic equation. Then prove that $\alpha^{12} =1$.
Show me the right way to solve this. Thanks in advance.
3
votes
3answers
96 views
Solve $\sin(z) = z$ in complex numbers
Show that $\sin(z) = z$ has infinitely many solutions in complex numbers.
Little Picard theorem should help, but using big Picard theorem is undesirable.
Thanks a lot!
0
votes
1answer
29 views
Complex numbers and absolute values
If i have equation:
\begin{align}
P = \left|\psi\right|^2
\end{align}
where $P$ is a probability and we know there is no negative probability. This means $P$ must belong to $\mathbb{R}$. If i want ...
1
vote
1answer
32 views
Conditions that Roots of a Polynomial be Less than Unity
Is is the case that Samuelson's result is a more specific result of Rouche's Theorem, or the Routh–Hurwitz stability criterion? Is it not the goal for a polynomial to be stable that all of its roots ...
1
vote
0answers
29 views
Fixed Point Iteration Scheme
I have been asked to "Find a fixed point iteration scheme for minimising $f(x) = e^{cos (x)}$".
Does anybody know what a fixed point iteration scheme actually is? I know it's not Fixed Point ...
13
votes
5answers
698 views
Show that $\sqrt[3]{2+\frac {10} 9\sqrt 3}+\sqrt[3]{2-\frac {10} 9\sqrt 3}=2$
Find $\displaystyle\sqrt[3]{2+\frac {10} 9\sqrt 3}+\sqrt[3]{2-\frac {10} 9\sqrt 3}$.
I found that, by calculator, it is actually $2$.
Methods to denest something like $\sqrt{a+b\sqrt c}$ seems ...
1
vote
1answer
39 views
Solution of a polynomial in interval $(0,1)$
Let $\displaystyle a_0 + \frac{a_1}{2} + \frac{a_2}{3} + ... + \frac{a_n}{n+1} = 0$, where $a_i$'s are some real constants.
How can we prove that the equation $a_0 + a_1x + a_2x^2 + ... +a_nx^n = 0$ ...
5
votes
3answers
64 views
Minimum degree of a polynomial passing through points
If $P(x)$ is a polynomial such that $P(a_{1})=b_{1}, P(a_{2})=b_{2}, \ldots , P(a_{k})=b_{k}$, how can I find the polynomial which has minimum degree and for whom the relations above are true?
0
votes
0answers
28 views
How to effciently solve a radical equation of the form $0=\sum_{j=1}^n a_j\sqrt{|b_j-x|}$?
Given a radical equation of the form
$$0=\sum_{j=1}^n a_j\sqrt{|b_j-x|}$$
where $b_j>0$ and the sign of $a_j\in\mathbb R$ matches that of $b_j-x$, is there any more efficient (analytical?) solution ...
4
votes
3answers
59 views
roots of the polynomial equations and relation among the coefficients
If the equation $x^4 + ax^3 + bx^2 + cx + 1 = 0$ ($a,b,c$ are real numbers) has no real roots and if at least one root is of modulus one, then what is the relation between $a,b$ and $c$?
2
votes
2answers
37 views
Roots of cubic polynomial lying inside the circle
Show that all roots of $a+bz+cz^2+z^3=0$ lie inside the circle
$|z|=max{\{1,|a|+|b|+|c| \}}$
Now this problem is given in Beardon's Algebra and Geometry third chapter on complex numbers.
What might ...
3
votes
5answers
110 views
How to find the number of real roots of the given equation?
The number of real roots of the equation $$2 \cos \left( \frac{x^2+x}{6} \right)=2^x+2^{-x}$$ is
(A) $0$, (B) $1$, (C) $2$, (D) in finitely many.
Trial: $$\begin{align} 2 \cos \left( ...
2
votes
4answers
135 views
Interception with $x$-axis - not so trivial?
I want to find the interception with the x-axis of the following function:
$f(x) = \frac{1}{4}x^4-x^3+2x$.
So putting $0 = \frac{1}{4}x^4-x^3+2x$ I would get $0 = x(\frac{1}{4}x^3-x^2+2)$ but how to ...
2
votes
1answer
52 views
Rouché's Theorem on $z^{10} + 10z + 9$
Please note: this question was asked before, but the solution provided does not work as far as I know; see How to find the number of roots using Rouche theorem?
We have $f(z) = z^{10} + 10z + 9$ and ...
1
vote
0answers
31 views
How to solve an equation in three variables fixing two of the variables?
Also, I have the following equation, I want to solve it for $b$ keeping $a$ and $c$ fixed.
$5b^5+(60-5a)b^4+(125+50c-80a)b^3+(594c-445a-775)b^2+(2324c-1005a-3270)b+3000c-750a-3000=0.$
Also how to ...
2
votes
3answers
20 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
72 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
101 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
54 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
93 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, ...
2
votes
2answers
42 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
69 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
149 views
Roots and coefficients of a 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
113 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
78 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
79 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
93 views
Location of zeros of an analytic function
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
69 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 $n \to \infty$.
1
vote
1answer
48 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
44 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
63 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
39 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 ...
4
votes
4answers
192 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
32 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
73 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
83 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
58 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
101 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?
1
vote
3answers
118 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!
3
votes
2answers
92 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??
