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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6
votes
3answers
180 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
164 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
118 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
98 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
27 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
109 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 ...
-4
votes
1answer
184 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.
4
votes
3answers
535 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
100 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 ...
2
votes
2answers
125 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
68 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 ...
1
vote
1answer
89 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 $a_0 + a_1x + a_2x^2 + ... +a_nx^n = 0$ ...
6
votes
3answers
355 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
39 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 ...
5
votes
3answers
125 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
137 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
415 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
158 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 ...
3
votes
1answer
120 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
85 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
74 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
130 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
195 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
120 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
272 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
1k 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
236 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
320 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
269 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
287 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
220 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
92 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
266 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
114 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
361 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
41 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 ...
0
votes
1answer
38 views

The dependence of the number of solutions of the equation $x^3-3x=a$ on the parameter $a$

Find the dependence n (a) the number of solutions the equation given a parameter 1) $x^3-3x=a$ 2) $e^2x=ax$ 3) $x^ax=e (x>0)$ For example what I did for (1) is $x(x^2-3)=a$ $x=a , x^2-3=a$ ...
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
121 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
120 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
92 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
485 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
103 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, ...