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

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1answer
35 views

Show that the roots of P' lie in the same half plane as the roots of P

The problem statement is: Part(a) Assume P(z) is a non-constant polynomial with all of its roots in some half plane H. Show that the derivative P'(z) must also have all of its roots in H. Part (b) ...
2
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3answers
76 views

Applying trigonometry in solving quintic polynomials?

So I came across the unsolvable quintic polynomial noticing that solutions can be found by connections with ellipses and such here. But more importantly, I was considering methods we use (or at least ...
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3answers
68 views

Theorem 1.21 in Baby Rudin: How do we obtain $\sqrt[m]{\sqrt[n]{a}}=\sqrt[mn]{a}$ for any $a>0$ and $m, n \in \mathbb{N}$?

Here is Theorem 1.21 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition: For every real $x>0$ and every integer $n > 0$ there is one and only one positive real $y$ such ...
1
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1answer
30 views

Factorize polynomial in Matlab

I want to factor (break up) a polynomial $P(x)$ into first orders $(x + a_i)$ for real roots and second orders $(x^2 + b_ix + c_i)$ for complex roots. That is to say, $$P(x) = \prod (x + a_i) \prod ...
3
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1answer
34 views

Why are the factors of some solutions to a Pell equation also a solution?

I came across this observation while trying to answer this post using the Pell equation $x^2-2y^2=1$. Define, $$P(m) = \frac{ (3+2\sqrt{2})^m+(3-2\sqrt{2})^m}{2}$$ $$Q(m) = \frac{ ...
1
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1answer
12 views

Combining the rule of signs and index shifting.

When applying the rule of signs to a polynomial, one can determine possible positive and negative roots. But we can also apply a substitution or index shifting as follows:$$P(x)\to P(x\pm n)$$ Then ...
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9answers
296 views

Why is $x=2 \implies (x-2)(x-3)=0$ false?

Let $P(x)$ be the equation $x=2$ and $Q(x)$ be the equation $(x-2)(x-3)=0$ By definition of implication I see that $P(x)$ implies $Q(x)$... As I see it, any premise that is false can give any ...
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0answers
37 views

Zeros of a sum of real powers

Consider the complex function $$f(z)=\sum_{k=1}^n z^{\alpha_k}\,, \quad z\in\mathbb C,\;\Re z>0$$ where $\alpha_k$ are real numbers (assume positive without loss of generality). What can I say ...
1
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1answer
51 views

$x^n+a_{n-1}x^{n-1}+\cdots +a_1x+a_0=0$ has real coefficients which satisfy $0<a_0 \le a_1 \le \cdots \le a_{n-1} \le 1$ prove that $z$ is a root

Suppose that the coefficients of the equation $x^n+a_{n-1}x^{n-1}+\cdots +a_1x+a_0=0$ are real and satisfy $0<a_0 \le a_1 \le \cdots \le a_{n-1} \le 1$. Let $z$ be a complex root of the ...
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2answers
89 views

How can one find the zeroes of $f(x)=ae^{bx}+cx+d$?

A certain physics problem I have been working on has turned into a math problem. Particularly, I want to find the solutions of some equation of the form $$f(x)=ae^{bx}+cx+d = 0$$ where $a, b, c,$ ...
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4answers
49 views

The minimum number of non real roots of the equation $x^4-2x^3+2x^2-x=k$ is?

The minimum number of non real roots of the equation $x^4-2x^3+2x^2-x=k$ is? k is any real number. I plotted this on https://www.desmos.com/calculator/vpfpjwyxz8.It seems that the answer will ...
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3answers
164 views

Roots of the Chebyshev polynomials of the second kind.

It is known that the roots of the Chebyshev polynomials of the second kind, denote it by $U_n(x)$, are in the interval $(-1,1)$ and they are simple (of multiplicity one). I have noticed that the roots ...
1
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1answer
31 views

residue equation for the denominator in a Padé approximant for $e^{-x}$

I had success in computing the roots numerically for the Bessel polynomial $\theta_n(x) = x^ny_n(1/x)=\sum\limits_{k=0}^n\frac{(n+k)!}{(n-k)!k!}\frac{x^{n-k}}{2^k}$ by using this residue equation I ...
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3answers
35 views

Using substitution in complex equation

$x^{10}-2x^5+\frac{4}{3}=0$ I substituted $t=x^5$ Then I got with the quadratic formula the result $t_1=1+\frac{\sqrt\frac{4}{3}}{2}i$ $t_2=1-\frac{\sqrt\frac{4}{3}}{2}i$ How do I calculate the ...
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1answer
26 views

solving a series of nonlinear equations for the zeros of Bessel polynomials

I am trying to find the zeros of the Bessel polynomials numerically. I found an article by Shafique Ahmed on the zeros of Bessel polynomials --- where Bessel polynomials are related but not the same ...
2
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1answer
49 views

Factoring the quartic $\left(x^{2}+x-1\right)\left(x^{2}+2x-1\right)-2sx\left(2x-1\right)^{2}$

Define $Q{\left(s;x\right)}$ to be the quartic function of $x$ with real parameter $s$ such that $0\le s\le1$ given as ...
2
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1answer
82 views

How to solve in radicals this family of equations for any degree $k$?

Part I. Given any constant $a,b$, the equation in $x$, $$\left(\frac{x+\sqrt{x^2+4a}}{2}\right)^{k}+\left(\frac{x-\sqrt{x^2+4a}}{2}\right)^{k}=b\tag1$$ is solvable in radicals for any degree $k$. ...
3
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0answers
59 views

$\tan(\alpha)=\alpha$ has not complex roots? [duplicate]

By the Strum theory it is easy to see that the equation $\tan(\alpha)=\alpha$ has not complex roots and it's real roots occur at zero and near $(2n+1)\pi/2,\ n\in Z^{+}$. I am interesting to know ...
0
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0answers
31 views

roots of Padé approximating polynomials to the exponential function

I need to obtain (numerically) the roots of the denominator in the Padé approximation to the exponential function $e^{-x}$, in Python. I can calculate its coefficients in closed form (see below). But ...
0
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2answers
46 views

Roots (Algebra)

This question consists of multiple questions but I am stuck on the very last one but without showing the first two the last one will be hard to understand so I'll show all my work: 13a) $w$ is one ...
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3answers
52 views

Proving Sum of Roots is Zero

The question is if $w$ is any real number , the equation $z^k = w$ has $k$ roots, $k ∈ n$. Show that the sum of the k roots is zero. Usually with these root questions I convert the right hand ...
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0answers
38 views

Rouché's theorem for the real part of a complex valued function

Rouché's theorem states that for two holomorphic functions $f,g$ in some closed $K$, if $|f-g|<|g|$ in $\partial K$ then $f$ and $g$ have the same number of roots inside $K$. Is this also true for ...
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2answers
40 views

If $ 3x^2 + 2\alpha xy + 2y^2 + 2ax - 4y + 1 $ can be resolved into two linear factors, then prove the following.

Prove that : $ \alpha $ is a root of the equation $ x^2 + 4ax + 2a^2 + 6 = 0 $. What does it mean by "can be resolved into two linear factors"? If it means $( ax + b ) ( cx + d )$ , is it necessary ...
9
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2answers
117 views

$f(g(h(x)))=0$ has $8$ real roots

Find all quadratic polynomials $f(x),g(x)$ and $h(x)$ such that the polynomial $f(g(h(x)))=0$ has roots $1,2,3,4,5,6,7$ and $8$. I don't know what to do. Making a $8$ degree equation is quite ...
1
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1answer
32 views

Pth Root of Polynomial Over Finite Fields for Yun's Algorithm

While I was implementing Yun's algorithm in java, I could not figure out an algorithm to find the $p$th root of a polynomial in $\mathbf{F}_p$ where the polynomial is a perfect power of $p$. How would ...
2
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3answers
62 views

Finding roots of Equation involving trig. functions.

In a problem of classical mechanics, I encounter the following equation: $$\mu \sin^4 \theta + \cos \theta = 0 \qquad \mu > 0 \qquad \frac{\pi}{2} < \theta < \pi,$$ where $\mu$ is some ...
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1answer
22 views

Is it possible to represent a root as a simple rational function with an exponent?

Using the following function:$$y=\frac{mx^p+b}{d}$$... where $m$, $p$, and $b$ may be any integer ... where $d$ may be any integer $\gt0$ ... and where $x$ may be any rational number $\ge0$ Is it ...
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2answers
50 views

When does $nx^4+4x+3=0$ have real roots?

Find all positive integers $n$ such that the equation $nx^4+4x+3=0$ has real roots. I think the answer must also include the cases with $2$ real roots. But my main question is, how do I start? ...
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0answers
32 views

Factorization of polynomial $f(x,y)$

The motivation is solving the following equations: $$ f(x,y)=0, x=L-kL, y=ks $$ $k$ is the variable, $L$ and $s$ are constants. The plan is: First, to factorize the polynomial as $(a_1x+b_1y+c_1) ...
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1answer
33 views

Algebra Roots (Cubic/Complex)

Show that the equation $3z^3+(2-3ai)z^2+(6+2bi)z+4=0$ (where both $a$ and $b$ are real numbers) has exactly one real root, and find this root. I've dealt with quadratics in this form but never with ...
3
votes
2answers
57 views

There are exactly two values for which $x^2=x\sin x+\cos x$ holds

I'm asked to show there is exactly two values for which $x^2=x\sin x+\cos x$ I have no idea where to start but I was thinking about taking the difference $h(x)=x^2-x\sin x-\cos x$ to show that $h'(x) ...
1
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2answers
89 views

Setting up a polynomial that has what I am referring to as a “wiggle root” at $x=0$

EDIT: I am loathe to delete/restructure this post, so I will include the following: A function has a "wiggle root" at $c$ if $f(c)=0$ and $f(x)$ crosses the $x$-axis. I am trying to set up a ...
2
votes
1answer
62 views

About equality of nested radicals.

Allow me, please, reformulate this problem. The equal numbers $$a=\sqrt{13}+\sqrt{10+2\sqrt{13}}$$ $$b=\sqrt{5+2\sqrt3}+\sqrt{18-2\sqrt3+2\sqrt{65-26\sqrt3}}$$ have the same minimal polynomial (over ...
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0answers
29 views

An exercise about repeated root

I want to prove the following: Let $K$ be a field, let $g(X) \in K[X]$ be an irreducible polynomial, and let $L$ be a splitting field for $g(X)$ over $K$. Prove that $g(X)$ has at least one ...
2
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1answer
51 views

Does this stand also for $f$?

We consider the differential equation $Ly=f$ in the ring of exponential sums $\mathbb{C}[e^{\lambda x} \mid \lambda \in \mathbb{C}]$ so we have that $f=\sum_{i=0}^n C_i e^{\lambda_i x}$. If we apply ...
0
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1answer
35 views

Two polynomials having the same roots are identical

The polynomials $f$ and $g$ have same roots(no repeated roots) and $\{x : f(x) = 2015\} = \{x : g(x) = 2015\}$ = S. Here S is non-empty. Show that $f = g$. Here $x$ is a complex number. A little help ...
3
votes
1answer
79 views

What about $\lim_{n\to\infty}\frac{\sum_{k=1}^n s_k\mu(k)}{n}$, for the zeros of Dirichlet eta function $s_k=1+\frac{2\pi k}{\log 2}i$ with $k\geq 1$?

Let for integers $k\geq 1$ the corresponding zeros of Dirichlet eta function of the form $$s_k=1+\frac{2\pi k}{\log 2}i,$$ then we can consider the following puzzle, when we multiply previous ...
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1answer
45 views

If the real polynomial $x^4+ax^3+bx^2+cx+d$ satisfies $d < 0$ and $3 a^2 < 8 b$, it has precisely $2$ real roots

Let $a,b,c,d$ be real numbers. If $d < 0$ and $3a^2 < 8b$, show that $$x^4+ax^3+bx^2+cx+d = 0$$ has exactly two roots. I know that you have to use IVT to prove that there are at least $2$ ...
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1answer
22 views

Prove that between every two roots of f there is a root of g and vice versa

Suppose $f$ and $g$ are two continuous and differentiable functions such that $f' = g$ and $g' = -f$. Prove that between every two consecutive roots of $f$, there is a root of $g$ and between every ...
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3answers
33 views

Solve $ \frac{1-\sqrt{1-x^2}}{1+\sqrt{1-x^2}} = 27\frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}-\sqrt{1-x}}$. Is my solution correct?

Find the roots of the following equation, if any: $$ \frac{1-\sqrt{1-x^2}}{1+\sqrt{1-x^2}} = 27\frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}-\sqrt{1-x}}. $$ My approach: The following constraints should ...
1
vote
1answer
31 views

Using calculus to determine how many roots are real

For the first one the calculus makes sense $f= x^3-3x+1$. So consider $f' = 3x^2 -3$ which has zeros at $\pm 1$. Then $f(1) = -3$ is negative and $f(-1) = 1$ is positive. S we know it actually crosses ...
2
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1answer
33 views

Help on solving the equation $\frac{\sqrt{a+x}}{\sqrt{a}+\sqrt{a+x}}=\frac{\sqrt{a-x}}{\sqrt{a}-\sqrt{a-x}}$

Could you give me some help on finding the roots (if any) of the following equation: $$ \frac{\sqrt{a+x}}{\sqrt{a}+\sqrt{a+x}}=\frac{\sqrt{a-x}}{\sqrt{a}-\sqrt{a-x}} $$ I tried to apply some classic ...
0
votes
1answer
25 views

Coefficient of the variable raised to the highest power in a polynomial with integer coefficients

Let $P(x) = a_nx^n+a_{n-1}x^{n-1} + ... + a_1x + a_0 $. Then, if there exists an integer $c$, which is a root of the polynomial, does it mean that the $a_n$ coefficient is equal to 1?
0
votes
1answer
24 views

Finding out if given polynominal is divisible by certain number knowing roots of the polynomial

The numbers $x_1 = 2, x_2 = 3, x_3 = 4$ belong to the roots of particular polynominal W(x). In addition, all of its coeffitients are integers. Is this polynominal divisible by 24 for any x? If the ...
2
votes
3answers
44 views

Can we predict if we are going to get extraneous root(s)?

We have extraneous roots when we perform operations on the given equation which are not invertible for all or some values of the variable in that equation. For example, if we have $x+2=0$, then, ...
3
votes
2answers
54 views

Factoring $x^3-3x-1$ in terms of $\alpha$ unknown

I never got a satisfactory answer here: Factoring $x^3-3x-1\in \Bbb Q[x]$ in terms of a unknown root But all context is contained below I want to factor $x^3-3x-1\in \Bbb Q[x]$ in terms of ...
2
votes
1answer
36 views

Can we apply Fundamental theorem of Algebra on entire, nonconstant functions?

I have the following question: Can we apply the Fundamental theorem of Algebra on entire, nonconstant functions $f:\mathbb{C}\to\mathbb{C}$? We can write such $f$ as ...
2
votes
2answers
31 views

$f \in \mathbb Z[x]$ be irreducible and suppose $f(x)$ has two roots in $\mathbb C$ with product $1$ , then degree of $f$ is even ?

Let $f \in \mathbb Z[x]$ be irreducible and suppose $f(x)$ has two roots in $\mathbb C$ with product $1$ . Then is it true that degree of $f$ is even ?
2
votes
2answers
73 views

Finding all roots of $x^4-10x^2+1$ without computational approximation

How can I find, by hand, all four roots of $f(x)=x^4-10x^2+1$? I can see that it has no linear rational roots, by the root test, and I can't see anyway to factorize it into two quadratic expressions. ...
6
votes
0answers
57 views

Can we express the roots of all polynomials in terms of roots of some special polynomials?

We can describe the roots of quadratic equations in terms of addition, subtraction, multiplication, division, and the square-root function $\sqrt a$ which computes a root of the special polynomial ...