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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4
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1answer
127 views

How do you solve 5th degree polynomials?

I looked on Wikipedia for a formula for roots of a 5th degree polynomial, but it said that by Abel's theorem it isn't possible. The Abel's theorem states that you can't solve specific polynomials of ...
0
votes
2answers
57 views

Showing that an equation has no solution in $\Bbb Z$

Show that $x^3 + 10x^2 + 6x + 2 = 0$ has no solutions in $\Bbb Z$. This seems rather trivial to do but I don't know how to rigorously show this is true. Having graphed this and attempted to factor,I ...
1
vote
2answers
34 views

Constructible Solutions

We know that if a cubic equation with a rational coefficients has a constructible root, then the equation has a rational root. Now let; $$x^3-2x+2\sqrt{2}=0$$ Could $\sqrt{2}t$ be a viable ...
2
votes
1answer
91 views

What are the roots of $ x^{2} = 2^{x} $? [duplicate]

What are the roots of $ x^{2} = 2^{x} $? I drew the graphs and found $ x = 2 $ and $ x = 4 $, and there is one other root in $ [-1,0] $. Can anyone describe an algebraic method to obtain all roots?
0
votes
0answers
16 views

Hermite-Bielher Theorem for entire functions

In a question asked by Bobby Ocean, the following theorem is cited: Hermite-Kakeya Theorem(for polynomials) - Given two real-valued polynomials, $f$ and $g$, then $f(x)+g(x) r$ has only real zeros ...
0
votes
1answer
33 views

Nonzero root of equation

Is there a condition so that a polynomial has at least one nonzero root? Suppose we have the equation $$\sum_{i=0}^m \alpha_i x^i=0$$ Do the coefficients have to satisfy a specific condition so ...
9
votes
2answers
91 views

On discriminants and nature of an equation's roots?

Edited: All equations in the post are assumed to have all real coefficients and are minimal polynomials. While trying to ascertain if the Brioschi quintic $B(x)=x^5-10cx^3+45c^2x-c^2=0$ could ever ...
2
votes
2answers
148 views

Computationally finding roots of a recursive function

I'm having a pretty complex function $h(n,d) = f(n,d) -n$ where $n \in \mathbb{N}$ and $d \in [1,9] \subset \Bbb{R}$. $f(n,d)$ is recursively defined. $$f(n, d) = \begin{cases} n<0\quad f(|n|,d) ...
0
votes
1answer
37 views

If we randomly pick a n-degree polynomial, is the probability of getting complex roots higher than the probability of getting all real roots?

Intuitively, I feel the probability of getting all roots of a polynomial to be real to be less. But is there a proof for this statement?
2
votes
1answer
102 views

In the limit of $N \rightarrow \infty$, find solution $z$ to $\text{e}^{-(z+N)} \sum \limits_{k=0}^{N} \frac{(z+N)^k}{k!}=\frac{1}{2}$

Fix an integer $N$, and consider the unique positive solution $z$ to the following equation: $$\text{e}^{-(z+N)} \sum \limits_{k=0}^{N} \frac{(z+N)^k}{k!}=\frac{1}{2}$$ For $N = 0$, we find that $z ...
1
vote
1answer
33 views

A polynomial with only one real $x$-intercept without imaginary roots has a root equal to $-c_n/(nc_{n-1})$.

Given coefficients $c_n, c_{n-1}, c_{n-2}, \ldots$ of the polynomial $c_n x^n+c_{n-1}x^{n-1} + \cdots +c_{1}x+c_0,$ prove that for $c_nx^n+c_{n-1}x^{n-1} + \cdots +c_1 x+c_0 = 0,$ $x=-c_n/(nc_{n-1})$. ...
1
vote
4answers
156 views

$ax^{13}+bx^{12}+c=0$ by hand. Is there any chance?

I'd like to know if is there any way to get an approximation for the roots of the equation below by hand. $$ax^{13}+bx^{12}+c=0.$$ You are allowed to use calculator to calculate powers, logarithms, ...
0
votes
1answer
68 views

Use Fermat's little theorem to find the roots of $2x^{219}+3x^{74}+2x^{57}+3x^{44}$ in $\mathbb{Z}_5$

The solution says $2x^{219}+3x^{74}+2x^{57}+3x^{44}=_52x^3+3x^2+2x^1+3x^0$ but I don't see how they arrived at that, even with Fermat's theorem
10
votes
0answers
69 views

Property of a polynomial with no positive real roots

The following is an exercise (Exercise #3 (a), Chapter 3, page 28) from Richard Stanley's Algebraic Combinatorics. Let $P(x)$ be a nonzero polynomial with real coefficients. Show that the ...
0
votes
0answers
24 views

How many digits of an $n$ digit positive integer do I need to know to correctly guess the first $m$ digits of the $k$ th root of $n$.

Please help me solve this puzzle, which in my opinion only sounds tough: How many digits (counting from the left) of an $n$ digit positive integer do I need to know to correctly guess the first $m$ ...
4
votes
2answers
56 views

Galois group of a polynomial of degree seven

Let $K$ be the splitting field over $\mathbb{Q}$ w.r.t. the polynomial $x^7 - 10x ^5+15x+5$. I think its Galois group is the symmetric group $S_7$. I tried to prove it using a theorem which says: "If ...
1
vote
1answer
31 views

how to show whether there are non-real roots in the unit disc of this equation?

For the equation $e^z=e^2z$, $|z|\leq 1$. I have shown there are no roots on the imaginary axis and boundary of the unit disc by simple computing. And let $z=x+iy$ ($x^2+y^2\leq 1$). $e^z=e^2z$ ...
1
vote
1answer
48 views

Find roots of a polynomial with coefficients all positive integers

Given the polynomial: $P(x) = x^7 + 9x^6 + 31x^5 + 55x^4 + 63x^3 + 55x^2 + 33x + 9 $ How do find it's roots ? After a some time with try- error, I could verify: $ P(x) = (x + 1)^3 (x + 3)^2(x + i)(x ...
1
vote
1answer
31 views

Roots of quadratic equation by completing the square or other method?

I'm trying to find solution(s) to the following equation: $x^2 - 5x + 3 = 0$ It seems like it can't be factored normally so I tried solving by completing the square: $x^2-5x=-3$ $x^2-5x+6.25=-0.5$ ...
1
vote
1answer
43 views

Newton's sums on a polynomial

Let $S$ denote the sum of the $2011th$ powers of the roots of the polynomial $(x − 2^0)(x − 2^1)\cdots(x − 2^{2010}) − 1$. How many $1$’s are in the binary expansion of $S$? Progress: I used Newton's ...
1
vote
0answers
17 views

Roots of an equation with normally distributed variable

Consider the following equation: $p\left(1-\int _{\mu}^{x} f(y)dy\right) \left[p\left(1-\int _{\mu}^{x} f(y)dy\right)+(1-p)q \right]-xf(x)p(1-p)q=0$, where $p,q \in [0,1]$, $f(\cdot)$ is the ...
1
vote
4answers
40 views

Show that $f(x) = x^3 -3x^2 -1 = 0$ (unique root) on the open interval $]3.1, 3.2[$

Given the function: $$ f(x)=x^3 -3x^2 -1 $$ *Show that $f(x)=0$ admits a unique root on the interval $]3.1,3.2[$I first thought of using this rule $f(a).f(b) < 0$ since the function is monotonic on ...
-1
votes
3answers
51 views

form a quadratic equation whose roots are $2α+β$ and $α+2β$

$g(x)=x^2+kx+2k-3$, where $k$ is a constant. Given that $g(x)=0$ has roots $α$ and $β$, form a quadratic equation whose roots are $2α+β$ and $α+2β$ Answer- $x^2+3kx+(2k^2+2k-3)=0$
4
votes
0answers
151 views

Find the complex (or real) roots of $e^{\frac{3 x}{2}}+2 \cos \left(\frac{\sqrt{3} x}{2}\right)$

Define for natural $n\geq 2$ $$G(x,n)= \sum _{k=0}^\infty \frac{x^{k n}}{(k n)!}= \frac{\sum _{k=0}^{n-1} e^{x e^{\frac{2 i \pi k}{n}}}}{n}= G(x e^{\frac{2 i \pi}{n}},n)= \prod_{m=1}^\infty ...
0
votes
1answer
71 views

Factoring the expression $(\sqrt{x^2} -a)^2 + M = 0$

Where, M stands for all other terms in the equation. This is a typical format that you'll see when taking affine sections of an n-torus. I think I figured out how to do it correctly, without violating ...
0
votes
5answers
55 views

Given that the roots of the equation $9x^2+bx+4=0$ are $4a$ and $a$ and that $b>0$, find the value of $b$.

Given that the roots of the equation $9x^2+bx+4=0$ are $4a$ and $a$ and that $b>0$, find the value of $b$.
1
vote
1answer
18 views

$f$ has zero of multiplicity $m$ at $\alpha$ and $k$ at $\beta$, where $m+k-1=n$. Prove that $f^{(n)}$ has at least one zero in $(\alpha, \beta)$.

Let $f\in C^n[a,b]$. Suppose that $f$ has a zero of multiplicity $m$ at $\alpha$ and a root of multiplicity $k$ at $\beta$, where $m\geq1$, $k\geq1$, and $m+k-1=n$. Prove that $f^{(n)}$ has at least ...
3
votes
0answers
43 views

What is the number of zeros of antiderivatives of $(x-1)(x-2)^2(x-3)^3(x-4)^4$?

For each $x \in \mathbb{R}$, let $f(x) = (x-1)(x-2)^2(x-3)^3(x-4)^4$. This defines a function $f : \mathbb{R} \to \mathbb{R}$. There is a unique natural number $k$ such that every antiderivative of ...
0
votes
0answers
16 views

Understanding Ferrari's Solution

I'm trying to understand how Ferrari's Solution works. Thanks to this post I understand that we are solving this for $y$ to find the perfect square: $$y^3 + \frac{5\alpha y^2}{2} + (2\alpha^2 - ...
0
votes
2answers
62 views

How to solve the equation $(25{ x }^{ 2 }-1)(10x+1)(2x+1)=11$? [closed]

How to solve this equation? $$(25{ x }^{ 2 }-1)(10x+1)(2x+1)=11$$
3
votes
2answers
131 views

The trigonometric solution to the solvable DeMoivre quintic?

Using the relations for the Rogers-Ramanujan cfrac described in this post, $$\frac{1}{r}-r = x$$ $$\frac{1}{r^5}-r^5 = y$$ and eliminating $r$ yields, $$x^5+5x^3+5x = y$$ This is the case $a=1$ ...
0
votes
0answers
16 views

Algorithm for complex roots of high degrees.

Is there an algorithm to find complex roots of equations of high degrees? Let's suppose I'm given an even function of degree greater than 6 that does not have real roots, how am I supposed to find its ...
0
votes
1answer
17 views

Exponential trigonometric function zeroes

How would I find the zero of this exponential function: $f(x)= sin e^x$ After i set the equation to zero, what do i do? Does this involve finding the derivatives?
3
votes
3answers
310 views

Solving Cubic when There are Known to be 3 Real Roots

When solving for roots to a cubic equation, the sign of the $\Delta$ tells us when there will be 3 distinct real roots (as long as the first terms coefficient, $a$, is non-zero.) Namely when $\Delta$ ...
0
votes
1answer
65 views

Finding All 3 Roots of a Cubic

I'm trying to find all real roots of a cubic. I wanted to use Cardando's Method but I'm not sure I'm correctly understanding how to obtain all 3 roots given the depressed cubic: $$t^3 + pt + q = 0$$ ...
1
vote
0answers
28 views

Let $p=a_0+a_1x+…+a_nx^n$ how can I add $a_{n+1}x^{n+1},a_{n+2}x^{n+2},…$ while maintaining some properties?

Let $p_n=a_0+a_1x+...+a_nx^n$ be a polynomial with only real roots. I need a way to algorithmically find real coefficients $a_{n+1},a_{n+2},...\neq 0$ such that the polynomial ...
0
votes
1answer
26 views

When computing contour integration with sines and cosines in the integrand, must we always first look at Euler's formula?

For example, in computing $$\int_{Cr}\frac {\cos(z)}{(z^2+a^2)^2}dz$$ over a semi-circular contour, must I first look at $$\int_{Cr}\frac {e^{iz}}{(z^2+a^2)^2}dz$$ compute this integral first, ...
1
vote
1answer
25 views

“Extending” the calculation of the golden ratio using square roots (not silver-ratio)

I'm looking at the following formula: $x =\frac{-n+\sqrt{n^{2}+4n}}{2}$ For $n=1$ this this gives $0.618...$ and then $\frac n x$ gives $1.618...$ which is $\phi$, the golden ratio. What ...
4
votes
1answer
54 views

Find number of roots of the equation $e^x(x^4 + 4x^3 + 12x^2 + 24x + 24) + 1 = 0$

Find number of roots of the equation $e^x(x^4 + 4x^3 + 12x^2 + 24x + 24) + 1 = 0$ Using Descartes rule, number of positive roots is zero and there can be a maximum of 4 negative roots. Also, for ...
0
votes
0answers
33 views

Solving degree 3 polynomial to get real root

Get the real number $z$ such that $z^3 - 63z - 162 = 0$ using Cardano's method. I haven't learned about Cardano's method how can I get the real number $z$ that is a root of the following polynomial?
1
vote
2answers
116 views

Can the roots of the derivative of the polynomial in complex variable be as close as we want them to be from the roots of the polynomial itself?

The (probably) famous Gauss-Lucas theorem states that the roots of the derivative $P'(z)$ are contained in the convex hull of the roots of $P(z)$, where $P(z)$ is complex variable polynomial. I am ...
0
votes
1answer
43 views

Complex root of a real polynomial

Let $n\in\mathbb{N}$. We consider, in $\mathbb R[X]$, the polynomial $$C_{n}=X(X+1)^{2n}-2^nX.$$ Determine the values of $n$ for which the complex number $i$ is a root of $C_n$. My thoughts: ...
3
votes
2answers
88 views

$x^3-3x+1=0$ has three real roots $x_{1}\;x_{2}\;,x_{3}$. Then what is the value of $\{x_{1}\}+\{x_{2}\}+\{x_{3}\}$?

If the equation $x^3-3x+1=0$ has three real roots $x_{1}\;x_{2}\;,x_{3}\;,$ Where $x_{1}<x_{2}<x_{3}$. Then the value of $\{x_{1}\}+\{x_{2}\}+\{x_{3}\} = \;,$ Where $\{x\}$ Represent ...
1
vote
2answers
83 views

My proof that this entire analytic function's range is within epsilon of any complex number,

If $f(z)$ is an entire analytic function assuming the values $0$ and $1$, show that for any complex number $a$ and any real number $ϵ>0$ there is a point $z_0$ such that $|f(z_0)−a|<ϵ$ My work: ...
1
vote
1answer
48 views

Trying to find splitting fields over Q of $x^{19} -1$

I'm trying to find subfields L of C which are splitting fields over Q For $x^{19}-1$ I've found the roots, but since you can't express them in exact form I don't see what to do next.
-1
votes
1answer
56 views

Find the units and the zero divisors in $F_{3}[x]$ mod $m(x) = x^{2} + x$ [closed]

Find the units and the zero divisors in $F_{3}[x]$ mod $m(x) = x^{2} + x$ I am completely confused about how to proceed with this question. In the case of integers, for example mod 4, I could ...
1
vote
1answer
34 views

Minimization optimization - where have I gone wrong?

Following @littleO's advice, I've set about to minimize $\sum_n ((x-x_n)^2+(y-y_n)^2+(z-z_n)^2-d^2)^2$. Going using an exact Hessian (because the function is smooth definite) as follows: $\textbf{H} ...
0
votes
1answer
41 views

Explanation of Chandrupatla's algorithm for root finding?

Is there any writeup of Chandrupatla's algorithm for root finding, besides his original article? A new hybrid quadratic/bisection algorithm for finding the zero of a nonlinear function without using ...
1
vote
5answers
66 views

What are roots of $x^{3}+3x^{2}+4x+1$?

There are no divisor of 1 in this polynomial for which would be satisfied $x^{3}+3x^{2}+4x+1=0$. How to find roots here?
1
vote
2answers
31 views

How do I find the poles of this difference equation?

I have an equation: $$y(n) = 0.634x(n) - 0.634x(n-2) + 0.268y(n-2)$$ I completed a $z$ transform and got: $$ H(z) = \frac{1-0.268z^{-2}}{0.634 - 0.634z^{-2}}$$ What is the next step to find the ...