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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2
votes
2answers
358 views

Tangent at average of two roots of cubic with one real and two complex roots

I was able to easily prove that the tangent at the average of two roots of a real cubic polynomial passed through the third root of the function. But I have only done this for functions with three ...
0
votes
3answers
37 views

Square Root Confusion

well we know that $$\sqrt{x^2} = \pm x$$ Then if $$x^2=y^2$$ then $$\pm x= \pm y$$ Does this mean $x = y$ or $-x = -y$ or $x = -y$ or $-x = y$ or all is true? Which is true among these?
-3
votes
0answers
33 views

Polynomial and a field [on hold]

How to prove that if a polynomial $$f(x) = ax^3+bx^2 +cx +d,$$ where $a,b,c,d \in K$, where $K$ is a subfield of $\mathbb{C}$, has a root in $K(\alpha)$ then $f$ has a root in $K$. $\alpha \in ...
8
votes
3answers
655 views

closed-form expression for roots of a polynomial

It is often said colloquially that the roots of a general polynomial of degree $5$ or higher have "no closed-form formula," but the Abel-Ruffini theorem only proves nonexistence of algebraic ...
3
votes
3answers
121 views

explicit solution for transcendental equation

Does anyone knows whether there is an explicit, analytical solution for transcendental equations of the form $A x + B \tanh(C x) + \coth(x) = 0$, where $A, B$, and $C$ are positive real constants?
1
vote
1answer
30 views

Effect on existing roots of polynomial when adding small higher-order term

How do existing roots of a polynomial change when adding higher-order term with a small coefficient? Given a sufficiently small coefficient of the new higher-order term, the existing roots shouldn't ...
0
votes
0answers
30 views

Problem with $x^0$ where $x \in(-∞, ∞)$

I understand that when $x \in(-∞, ∞)$, is means that $x^0 = 1$. I also understand why this is. But isn't the following still true that when $x \in(-∞, ∞)$, it means that $x^y = \sqrt [1/y] {x}$? And ...
0
votes
0answers
16 views

“root” of a right-continuous function

Suppose $f:[0,1] \longrightarrow [-1,1]$ is a right-continuous function such that $f(0) < 0$, $f(1) > 0$, and $f$ only changes sign once in the interval $[0,1]$. Suppose we define the "root" of ...
5
votes
4answers
5k views

Is it true that a 3rd order polynomial must have at least one real root?

While solving a problem a friend said - this polynomial is $3^{rd}$ order ($ax^3+bx^2+cx+d$), so it must have a real root. I didn't want to sound stupid and I said sure. I can't figure out if he's ...
1
vote
0answers
16 views

Interval of Polynomial Root Finding

Let's say we have a polynomial of a given degree. You don't have any tools to figure out the amount of roots in this polynomial. All you know is the function and you cannot graph it. How would you ...
0
votes
1answer
26 views

Prove or disprove this relation between one root of the quadratic and the cubic equation of a certain form, and linear recurrences.

It is well known that the n-anacci (higher degree Fibonacci, that is Tribonacci and so on) numbers can be computed in closed form from roots of polynomials in the way Eric Weisstein at Mathworld ...
0
votes
2answers
32 views

Solving recurrences whose characteristic equations have complex roots

In my Discrete Mathematics lecture notes, there is a section regarding solutions for linear recurrences whose characteristic polynomials have complex roots. There is a particular statement which I am ...
0
votes
2answers
31 views

Roots of quadratic equation

If the roots of $ax^2+bx+c$ are $\alpha$ and $\beta$, express $\frac1\alpha-\frac1\beta$ in terms of $a$, $b$ and $c$. I know how to express $\alpha+\beta$ or $\alpha\beta$ which is usually enough, ...
2
votes
1answer
80 views

Irreducible polynomial in $\mathbb Q[x]$ of degree $n$ having exactly $n-2$ real roots.

I need to show that for any $n$ there is an irreducible polynomial in $\mathbb Q[x]$ of degree $n$ having exactly $n-2$ real roots. I know that from a previous exercise that if $f(x) \in ...
2
votes
1answer
31 views

Show that $z^n+nz-1$ has $n$ zeros in $D(0,R)$

Let $n\geq 3$. Show that the polynomial $z^n+nz-1$ has $n$ zeros in $D(0,R)$, where $$R=1+\left(\frac{2}{n-1}\right)^{1/2}.$$ I was hoping to use Induction and Rouche's Theorem. For the base case ...
1
vote
2answers
50 views

Polynomial $f(x)$ over $\mathbb{R}$ has $k$ distinct roots, then $f(x) + a$ too

I am trying to learn Galois theory by myself. When reading a section for applications to polynomials, I got stuck in the following exercise: If $f(x) \in \mathbb{R}[x]$ is any polynomial having ...
0
votes
1answer
59 views

Proving that if the sequence $X_n$ converges to $x$, then ${X_n}^a$, where $a$ is a positive rational, converges to $x^a$.

I've been stuck on this problem for a while. I splitted a into $p/q$, so it would be $({X_n}^p)^{1/q}$, and I got the convergence of ${X_n}^p$ to be $x^p$ since it is just induction using the product ...
1
vote
1answer
28 views

Find the fixed points of a function $f(x) := exp(x - 2)$ using a recursive algorithm

I need to find the fixed points (i.e. when $f(x) = x$) of the following function $f(x) := exp(x - 2)$. I understood that the fixed points should be the intersecation points between $f(x)$ and a ...
1
vote
0answers
62 views

Solving quartic equation? (Cardano/Ferrari)

today I've written a little Code-Snippet that is based upon the steps that are mentionned in this wikipedia-Article to solve a general quartic polynom. Here's my matlab-implementation: ...
1
vote
1answer
32 views

How to upper-bound the smallest positive root of a polynomial?

Is there any algorithm for (upper-)bounding the smallest positive root of a polynomial of an arbitrary degree if it exists, or detecting that it does not exist otherwise? Edit: I'm looking for a ...
1
vote
1answer
122 views

Find the smallest root of the function $e^{-x} = \sin (x)$

I have the following problem: Find the smallest root of the function $e^{-x} = \sin (x)$ and focus the root with Newton's method to $8$ decimal accuracy. Any suggestions?
1
vote
1answer
27 views

Finding a root by bisection method in Excel

Working on a maths assignment and we're trying to use Excel for a bisection method. $$\frac12 e^{x/2}+\frac{1}{2x}-\frac32=0$$ Here is a pic, I can't get the formula to work with the exponent. This ...
0
votes
3answers
52 views

Find the real root $\alpha$ of the cubic equation $z^3-2z^2-3z+10=0$

Find the real root $\alpha$ of the cubic equation, $$z^3-2z^2-3z+10=0$$ The exam paper is giving just 2 marks for this and the mark scheme isn't very helpful. My idea is that you can use some of this ...
0
votes
1answer
45 views

The roots of the cubic equation $z^3-2z^2+pz+10=0$ are $\alpha$, $\beta$ and $\gamma$. Show that $\alpha^2+\beta^2+\gamma^2=p+13$

$$z^3-2z^2+pz+10=0$$ $$ax^3+bx^2+cx+d=0$$ $$\Rightarrow\,\,\,\,\,\,\,\,\,a=1,\,\,\,\,\,\,\,\, b=-2,\,\,\,\,\,\,\,\, c=p,\,\,\,\,\,\,\,\, d=10$$ ...
2
votes
3answers
42 views

Prove $x^7+3x^5+1$ has exactly one real root using Bolzano's theorem and the MVT.

Prove $f(x)=x^7+3x^5+1$ has exactly one real root using Bolzano's theorem and the MVT. What I did: $f(-1)=-3$ $f(0)=1$ As $f$ is continuous, there exists a $c \in (-1,0) /f(c)=0$ Then computed ...
0
votes
1answer
37 views

general theorem on roots of a polynomial needed to show it's identically zero.

Polynomial degree k, one variable, if it's zero at k+1 values, then it's identically zero. Can someone point me to a proof of this? I know derivatives at points can count as these roots (if k-degree ...
1
vote
3answers
120 views

Solve $ \left(\sqrt[3]{4-\sqrt{15}}\right)^x+\left(\sqrt[3]{4+\sqrt{15}}\right)^x=8 $ [closed]

I don't know what can I substitute for $x$ so that equation becomes satisfied. Any assistance will be greatly valued. Thanks!
0
votes
0answers
10 views

Root finding of a Hermite interpolating polynomial

Consider a Hermite interpolation problem. I have an approach for obtaining the roots of interpolating polynomial. I would like to present an example for this approach. Can you suggest me an applicable ...
1
vote
2answers
36 views

Entire functions of order 0

Sorry, this may be a stupid question, but I am just beginning to learn about this and cannot find the answer anywhere I have looked so far. Clearly if we have any polynomial $P(z)$, then it is easy to ...
2
votes
2answers
764 views

Roots of biquadratic equation

This question also was a part of my today's maths olympiad paper: If squares of the roots of $x^4 + bx^2 + cx + d = 0$ are $\alpha, \beta, \gamma, \delta$ then prove that: $64\alpha\beta\gamma\delta ...
1
vote
1answer
41 views

What are 'Regular Products'?

When looking at the functional equation for the Riemann zeta function, I came across the statement: For $s$ an even positive integer, the product $\sin{(\frac{\pi s}{2})}\Gamma({1-s})$ is regular. ...
0
votes
1answer
40 views

I have to show $p=p(x-\lambda)$ if and only if they have the same zeros in $F$

Suppose $F$ is a field, $|F|\geq n \geq 2$. Given $\lambda \in F$ I know $p,p(x-\lambda)\in F[x]$ are irreducible monic polynomials. I have to show $p=p(x-\lambda)$ if and only if they have the same ...
1
vote
3answers
61 views

exact roots of $e^{ax}-x=0$

How can I find the general solution to (not a numerical approximation) $e^{ax}-x=0$ as a function of $a$. I thought maybe something like $\frac{ln(x)}{a}$.
0
votes
1answer
56 views

Help required! Polynomials

Let $D(p) = p^{20} - p^{18} - p^{16} - \dots - p^2 - 2$ Prove that the sum of fourth powers of all the real roots of $D(p) = 8.$ Please help.
3
votes
1answer
54 views

How to remove duplicate roots from a polynomial?

Given a polynomial equation (with real coefficients of any degree with any number of repeating roots), let say $x^5 + 6x^4 - 18x^3 - 10x^2 + 45x - 24 = 0$, ... (A) it can be written as $(x-1)^2 ...
0
votes
0answers
17 views

Root locus vs Matlab's root locus function

Today during a lecture out teacher was demonstrating Matlab's rlocus function, but before that he decided to do it on chalkboard. Below we have a simple transfer ...
1
vote
1answer
25 views

Algebraic vs. analytic definition of the multiplicity of a polynomial's root

Let $f(x) = a(x - c_1)^{d_1}(x - c_2)^{d_2} \dots (x - c_n)^{d_n}$ be a polynomial over the complex numbers ($n, d_i \in \{1, 2, \dots\}$, $a \in \mathbb{C}\setminus \{0\}$), where the roots $c_1, ...
1
vote
1answer
29 views

N-th roots equation

I am facing the following equation and I do not have any idea about how to solve it. $\dfrac{(n^c-1)^a}{n^{ac}}$ = $\dfrac{1}{2}$. I am free to choose $c$ (any constant). $a$ on the other hand can be ...
0
votes
1answer
19 views

Finding roots of complex polynomial with conjugates

I am having problem with the following question... I know that I should use De Moivre's formula somewhere... but can't quite get to it $$ (-15w + 34\bar{w})^4 = -1 $$ will be happy to get some help, ...
2
votes
2answers
51 views

Roots of a Polynomial Minus It's Constant Term

Suppose we have a sequence of integers $a_1,\dots,a_n$. Is there any way to determine the roots of the polynomial $$P(x) = (x+a_1)\dots(x+a_n) - a_1\dots a_n$$ Clearly $P(0) = 0$, but can anything ...
1
vote
4answers
160 views

Find the square root of $404.11$.

Find the square root of $404.11$ without using calculator accurate upto $2$ decimal places . It is clear that $20<\sqrt{404.11}<21$ so it will be $20.ab$ without trial and error what ...
1
vote
0answers
76 views

nth-root of continued fraction with Raney transducers

There are some algorithms for doing basic arithmetic by using regular continued fraction expansions. These algorithms are mainly due to Gosper (1972) and Raney (1973). These two approaches use ...
1
vote
1answer
40 views

The number of zeros of a polynomial that almost changes signs

Let $p$ be a polynomial, and let $x_0, x_1, \dots, x_n$ be distinct numbers in the interval $[-1, 1]$, listed in increasing order, for which the following holds: $$ (-1)^ip(x_i) \geq 0,\hspace{1cm}i ...
1
vote
1answer
23 views

Is Gershgorin bound of roots sharp?

Gershgorin circle theorem tells that the eigenvalues of a matrix $A$ lie in the union of the associated Gershgorin circles. $A=\begin{pmatrix} 0 & 0 & \dots & 0 & -a_0 \\ 1 & 0 ...
0
votes
1answer
31 views

Is there closed form solution for this infinite polynomial or high-order polymonial?

The equation is as follows \begin{align} \sum_{N=1}^{\infty}P(N)x^N=Z, \end{align} where $P(N)$'s are real number satisfying $0\leq P(N)\leq 1$. Another equation is \begin{align} \sum_{N=1}^{\bar ...
4
votes
2answers
86 views

Solution of $Ax^5+Bx^3=C$

I have to find the positive solution of the type $Ax^5+Bx^3=C (A,B,C>0)$. It is well known that a polynomial of degree greater than $4$ does not admit an expression for the roots but I hope :D In ...
2
votes
1answer
323 views

Working with casus irreducibilis

I read about casus irreducibilis here. As an example of casus irreducibilis, it says we can factor $x^3 - 15x - 4$ to find $4$ as a root and it also has two other real roots. Using Cardano's method we ...
0
votes
1answer
26 views

Roots of trigonometric equation

In the following trigonometric equation $$1 + \alpha^2 \cos^2 (n \theta) = 0$$ The complex solutions are $$\cos (n \theta) = \pm i/\alpha$$ So I thought that the correspondant angles were $$n ...
1
vote
1answer
39 views

Numerically find the minimum distance between given point and the curve of given function.

Let $f\colon\Bbb{R}^n\to\Bbb{R}$ and $\mathbf{x}_0\in\Bbb{R}^n$. How could I (numerically) find the minimum Euclidean distance between the curve $f(\mathbf{x})=0$ and $\mathbf{x}_0$, granted that $f$ ...
3
votes
2answers
30 views

Elementary bound theorem of a monic real polynomial

An elementary bound theorem on the roots of a real monic polynomial states that $$M := \operatorname{max} (1, |a_0| + \cdots + |a_{n-1}|) := \operatorname{max} (1, B)$$ is an upper and lower ($-M$) ...