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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6
votes
2answers
115 views

Find the limit analytically when the sine functions have square roots?

Find the limit analytically of the following: $\lim \limits_{x \to 0} \frac {\sin(\sqrt{2x})}{\sin(\sqrt{5x})} $ The closes thing we learned in class about this was that $\sin(x)$ over $x$ will ...
1
vote
0answers
21 views

Showing polynomials as products of roots

How do I show rigorously that any polynomial $a_nx^n+a_{n-1}x^{n-1}+...a_1x+a_0$ can be written as $a_n(x-b_1)(x-b_2)...(x-b_n)$ for real $a_i$ and real or complex $b_i$
2
votes
1answer
112 views

Polynomial tending to infinity

Take any polynomial $(x-a_1)(x-a_2)\ldots(x-a_n)$ with roots $a_1, a_2,\ldots,a_n$ where we order them so that $a_{i+1}>a_i$ is increasing so $a_n$ is the biggest root. It doesn't matter whether ...
1
vote
2answers
68 views

find the number of solutions to $p(z) = z^6 + 9z^4+z^3+2z+4$

Let $p(z) = z^6 + 9z^4+z^3+2z+4$ find then number of roots in each quadrant of the complex plane find in which quadrant exists a root which is inside the unit circle using the Argument priniciple ...
1
vote
0answers
51 views

What methods are known to visualize patterns in the set of real roots of quadratic equations?

I came across a previous awesome question about the visualization of the distribution of polynomial roots and tried to do a simpler version applied to the set of real roots of quadratic equations ...
0
votes
2answers
48 views

Find $\prod\limits=(\alpha_1+1)(\alpha_2+1)…(\alpha_n+1)$ where $\alpha_i$ are complex roots of a complex polynomial

The complex roots of a complex polynomial $P_n(z)=z^n+a_{n-1}z^{n-1}+\cdots+a_1z+a_0$ are $\alpha_i$, $i=1,2,...,n$. Calculate the product $(\alpha_1+1)(\alpha_2+1)\cdots(\alpha_n+1)$ By the ...
3
votes
2answers
48 views

Polynomial with real roots

Consider the polynomial: $$f=X^4+4X^3+6X^2+aX+b$$ We know that $f$ has four real roots. Let $x_1,x_2,x_3,x_4$ be the roots of this polynomial. How can one compute ...
4
votes
3answers
152 views

Is $\sqrt[3]{-1}=-1$?

I observe that if we claim that $\sqrt[3]{-1}=-1$, we reach a contradiction. Let's, indeed, suppose that $\sqrt[3]{-1}=-1$. Then, since the properties of powers are preserved, we have: ...
0
votes
2answers
40 views

How to find the value of $(a+b+c)(a+b+d)(a+c+d)(b+c+d)$ from the following equation?

I have a question about polynomial. Given a polynomial: $$x^4-7x^3+3x^2-21x+1=0$$ Given too that the roots of this polynomial are $a, b, c,$ and $d$. Find the value of ...
1
vote
3answers
86 views

How do I solve an equation like this?

How do I solve following equation for $X$: $$ AX^n + BX^{n-1} + CX^{n-2} + \dotsb + YX + Z = 0, $$ where $A,B,C,\dotsc,Z,n$ are known?
1
vote
1answer
54 views

is there a way to prevent false roots?

Is there a systematic way of preventing false roots when squaring a root equation? The testing of the roots is quite tedious in some problems. My first thought was absolute values in some form
1
vote
1answer
17 views

Durand-Kerner with derivative in denominator

The correction term for Durand-Kerner root finding method is $w_k = -\frac{f(z_k)}{\prod_{j\not=k}(z_k - z_j)}$ Wikipedia Talk page mentions that it is also possible to use derivative in ...
1
vote
2answers
38 views

multiple sets of complex roots of a number?

I am not sure if this question was asked before but I couldn't find the right keywords to choose for searching. So today I discovered a weird problem: If we take this equation: $$x^2=1=e^{(0i)}$$ ...
1
vote
1answer
31 views

Tricks for a Specific System of Polynomial Equations

I'm looking for all the complex solutions to the following 3 equations (and for this consider $a$ to be some given constant, so that there are really just 3 unknowns in solving): ...
5
votes
4answers
174 views

Express roots in polynomials of equation $x^3+x^2-2x-1=0$

If $\alpha$ is a root of equation $x^3+x^2-2x-1=0$, then find the other two roots in polynomials of $\alpha$, with rational coefficients. I've seen some other examples [1] that other roots were ...
1
vote
1answer
22 views

Zero of holomorphic function

Let $\Omega \subset \mathbb{C}$ be an open set that contains the unit ball $D$ and let $f \in \mathcal{O}(\Omega)$ a non constant map s.t. $|f(z)| = 1$ for all $z \in \partial D$. Show that $f$ has a ...
-2
votes
1answer
30 views

Determine values of $a$ given $c$ in quadratic equation and a point $(1,2)$ [closed]

The point $(1,2)$ is on the graph of the quadratic function $f(x) = ax^2 + bx + 1$. Determine the values of $a$, such that the graph of $f(x)$ intersects the $x$-axis at two distinct points. This ...
1
vote
0answers
21 views

Please check this perturbation solution of polynomial root and truncation order.

I have a quintic polynomial where the coefficients depends on a parameter $c$, i.e. $$ a_0(c)+a_1(c)x+a_2(c)x^2+a_3(c)x^3+a_4(c)x^4+x^5 $$ I know that the roots of the polynomial are real and ...
2
votes
0answers
23 views

Find cubic equation roots: what choice to make for cube root to avoid circularly permuting the roots?

My work involves solving a cubic equation similar to this $a x^3+b x^2+c x +d$ which has three roots, $x_1, x_2, x_3$ (as given in https://en.wikipedia.org/wiki/Cubic_function.) ...
4
votes
0answers
60 views

How to prove the Riemann hypothesis holds for the first non-trivial zero? [duplicate]

The Riemann hypothesis states that all non-trivial zeros of the Riemann zeta function $\zeta(z)$ lie on the critical line $\Re(z)=1/2$. The MathWorld page on this topic mentions that the hypothesis ...
3
votes
2answers
92 views

Is my proof for the existence of roots of an odd-degree polynomial correct?

$\color{crimson}{\text{Problem}}$ Show that if $f:\mathbb{R}\to\mathbb{R}$ be a polynomial of odd degree with real coefficients then it has at least one real root. $\color{green}{\text{Proof}}$ Let ...
1
vote
2answers
78 views

Find the roots of a function with logarithms (possibly using lambert W function)

I am wondering if anyone can help me find an analytical solution to the roots of the following function: $$f(b) = c\log \left( \frac{b}{a} \right) + (n-c)\log \left( \frac{1-b}{1-a} \right),$$ $a,b ...
2
votes
0answers
41 views

How to find all roots of the quintic using the Bring radical?

Finding one root $x_1$ of the quintic equation $x^5 + x = -a$ by using the Bring radical is described on Wikipedia. The root is $x_1 = -a +a^5 -5a^9+35a^{13}+...$, and is found by reversion of the ...
0
votes
0answers
27 views

Square root in a general field

In $\mathbb{Q}$, $\mathbb{R}$ and $\mathbb{C}$ there are obvious ways to calculate the square root of a quadratic residue. For finite fields of order $p$ we can use the Tonelli–Shanks algorithm. How ...
1
vote
2answers
42 views

How to solve the equation $x^3+y^3=0$ for real numbers $x$ and $y$?

I'm finding stationary points of the function $f(x,y)=2(x-y)^2-x^4-y^4$, but stuck in the equation $x^3+y^3=0$ while solving the equations $f_x=0$ and $f_y=0$. Please help me. Thanks in advance.
1
vote
1answer
36 views

If all derivatives are zero at a point, what does this imply?

Let's say I have a function $f$ which for all positive $n$ and some complex point $z_0$ satisfies $f^{(n)}(z_0) = 0$. What does this say about the function's analyticity or holomorphicity? Obviously, ...
0
votes
2answers
33 views

Algebra calculating to zero point without using rational root theorem

I have to find the points where x equals zero in the following equation, without using rational root theorem. The equation is: (3-x)(1-x)²+(1-x) = 0 I know the answer is x=2 and x=1. I get the x=1, ...
4
votes
4answers
145 views

Finding roots of the polynomial $x^4+x^3+x^2+x+1$

In general, how could one find the roots of a polynomial like $x^4+x^3+x^2+x^1+1$? I need to find the complex roots of this polynomial and show that $\mathbb{Q (\omega)}$ is its splitting field, but I ...
2
votes
0answers
51 views

Solving for $x$ in $M ^ {M ^ M} = x ^ {1 / (x-1)}$ where $M = 5 ^ {\sqrt{5} / 10}$

Methods used by analogy, for example $x ^ x = 3 ^ 3 \implies x = 3$, Determine the value of $x$ in $$M ^ {M ^ M} = x ^ {1 / (x-1)}$$ if $M = 5 ^ {\sqrt{5} / 10}$.
2
votes
2answers
39 views

Is “the nth root of x” well-defined without further qualification?

(No, I'm not asking if $\sqrt{-1} = +i$ or if $\sqrt{-1} = -i$. Yes, I know $+i$ is the principal square root.) Consider the cube root of -8. If asked to evaluate it, I would say -2, and I think we ...
0
votes
0answers
30 views

Integral roots of a polynomial

I have one doubt. Suppose, $f_{n}(x)=a_0x^n+a_1x^{n-1}+a_2x^{n-2}+,...,+a_{n-1}x+a_n=0$ be a polynomial with an integral coefficients. If for some $n$ ( say $n=2 \ or \ 3$) , $f_{n}(t)=0,$ where, $t ...
1
vote
2answers
32 views

Zeros of Trigonometric Equation

I'm studying the function $$ f(x) = \log(x + 1) + \cos(x)/2 $$ The first derivative is: $$ f'(x) = 1/(x + 1) − \sin(x)/2. $$ To find the first two positive critical points (without Wolfram and the ...
0
votes
0answers
31 views

Is this true :${(a+ib)}^{(k+ij)}=0$ iff $0<a=k<1$ and $b<j$?

let $z=a+ib ,s=k+ij$ are two complex numbers and let $f(z,s)$ be a complex function defined as follow :$$f(z,s)=z^s={(a+ib)}^{(k+ij)}$$ and $a,b,j, k$ are non -nul real numbers . .After some ...
0
votes
0answers
21 views

Explanation of symmetric sum in a solution

Can someone explain me why $x+y=5$ in $\text{E8}$ clearly.
1
vote
1answer
121 views

drawing $\sqrt[3]{2}$

I know that drawing cubic root of "2" ($\sqrt[3]{2}$) is not possible with just a ruler and a compass. But is there a way or a tool to draw this? I mean, a segment line with a length of $\sqrt[3]{2}$. ...
3
votes
0answers
60 views

Finding exact roots

I know of the rational root theorem to find all rational zeros and Newtons method of approximating zeros, but what if all the solutions are irrational/imaginary and you need exact answers for the ...
1
vote
4answers
148 views

Sum of non-real roots of equation?

What is the sum of all non-real, complex roots of this equation - $$x^5 = 1024$$ Also, please provide explanation about how to find sum all of non real, complex roots of any $n$ degree polynomial. ...
6
votes
3answers
81 views

Prove : The polynomial has no integral roots. [duplicate]

Q. Prove that a polynomial $f(x)$,with integer coefficients has no integral roots if $f(0)$ and $f(1)$ are both odd integers. My attempt: Let $$f(x)=a_0+a_1x+a_2x^2+\dots+a_nx^n$$ now $f(0)=a_0$ ...
0
votes
1answer
58 views

Number of real roots of polynomial derivative

Let $W(x)$ be a polynomial with n real roots and $P(x) = \alpha W(x) + W'(x)$. Prove that for any $\alpha \in \mathbb{R}$: $P(x)$ have at least $n-1$ real roots. I know that the degree of the ...
0
votes
0answers
9 views

Reduce multivariate polynomials by known roots?

Consider three multivariate polynomials $p_1(x,y,z)$, $p_2(x,y,z)$ and $p_3(x,y,z)$ with $x,y,z\in\mathbb{C}$. Imagine that the set of polynomials above is constructed such that they have exactly $6$ ...
3
votes
4answers
80 views

Condition for common roots of two Quadratic equations: $px^2+qx+r=0$ and $qx^2+rx+p=0$

The question is: Show that the equation $px^2+qx+r=0$ and $qx^2+rx+p=0$ will have a common root if $p+q+r=0$ or $p=q=r$. How should I approach the problem? Should I assume three roots $\alpha$, ...
1
vote
3answers
74 views

Finding all the values of $\sqrt[3]{7-4i}$

I'm reading about De Moivre's Formula and the Roots of Unity, and one of the exercises is to find all the different values of $$ \sqrt[3]{7-4i} $$ I know that you can find the $n$th root of 1 with ...
2
votes
0answers
41 views

Zero locus of 2-variate real polynomial are smooth curves

This seems like it should be an easy question, and probably already has already had answer in advanced mathematics, but I only know some basic calculus, so I would like to know how do I go about doing ...
-4
votes
2answers
52 views

Mathematics Radical Numbers Problem [closed]

If, $$\frac{\sqrt 5+1}{\sqrt 2-1} = x $$ then, $$\frac{\sqrt 5-1}{\sqrt 2+1} = ? $$
1
vote
1answer
22 views

Simplifying transfer functions in Z domain

I have difficulties to check whether the below transfer function is recursive or non-recursive: $$H(z)=\frac{1-z^{-1}+z^{-2}-3z^{-3}}{z^{-2}(1-z^{-1})}$$ I know that I have to multiply the num and ...
1
vote
1answer
102 views

Solve $x^3+x+3=0$ by Galois's theory

Solve with radicals the following equation $x^3+x+3=0$, using Galois Theory. I'm just starting learning this and I do not have many ideas.
0
votes
1answer
62 views

Lowest root of a quintic equation with 5 positive roots

I have a quintic equation $$ x^5-a_4 x^4+a_3 x^3-a_2 x^2+a_1 x - a_0=0 $$ with $a_n>0$ real coefficients, and I know that all 5 roots are real and positive (it is a characteristic polynomial). ...
0
votes
2answers
103 views

Finding roots of $2x^3-5x^2+18x+45$

solve $2x^3-5x^2+18x+45$ not exactly sure where to start on finding the zeros complex or real. There is one real zero and two complex I know that from graphing just cannot do it on paper to understand ...
0
votes
1answer
91 views

Find $r$ in the next formula

Lets suppose I have the next values $$0, 7, 8, 5, 6$$ And the next formula $$4250 = \frac{0}{(1+r)} + \frac{7}{(1+r)^2} + \frac{8}{(1+r)^3} + \frac{5}{(1+r)^4} + \frac{6}{(1+r)^5}.$$ What is the ...
2
votes
1answer
36 views

Weird square root disappearing and flipping fraction upside down?

So here I was, making 2 math problems, I was able to solve them, but 2 operations seem a bit intractable to me. Maybe you can help me understand why this is true: The first problem: $$x = \frac{1}{5} ...