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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0
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2answers
33 views

Discriminant of $f(x)=x^3+ax+b$

Suppose we have the polynomial $f(x)=x^3+ax+b$, with roots $\alpha, \beta, \gamma$ in $\mathbb{C}$, and let $\Delta = (\alpha - \beta)(\beta - \gamma)(\gamma - \alpha)$. Is there any quick way of ...
3
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4answers
104 views

Find roots of $3^x+x^3=17$

Find $x$ in the following equation: $$3^x+x^3=17$$
2
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3answers
45 views

$f(x^2 + 1) = f(x)g(x)$ $\forall$ x $\in\mathbb{R}$ $\Rightarrow$ no. of roots of $f(x)=$?

If two real polynomials $f(x)$ and $g(x)$ of degrees $m$ $(\ge2)$ and $n$ $(\ge1)$ respectively, satisfy $f(x^2 + 1) = f(x)g(x)$; for every $x \in \mathbb{R}$ , then what can be said about the ...
2
votes
2answers
91 views

Solve exponential-polynomial equation

Solve the equation in $\mathbb{R}$ $$10^{-3}x^{\log_{10}x} + x(\log_{10}^2x - 2\log_{10} x) = x^2 + 3x$$ To be fair I wasn't able to make any progress. I tried using substitution for the ...
9
votes
2answers
241 views

Are rational numbers + numbers constructed with a root = algebraic numbers?

I'm a math newbie, so an intuitive explanation is the most helpful for me, but don't pull your punches with the formulas, if you feel like it. We can construct the rational numbers using the division ...
0
votes
2answers
71 views

Roots of $x^4-3x^2+x-\sin x$

Is there a logical/qualitative argument to find out the number of roots of $x^4-3x^2+x-\sin x$ ? I tried plotting graphs for $x^4-3x^2+x$ and $\sin x$ to check the points of intersection but it only ...
1
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1answer
36 views

Basic question from LA: Why do we find Roots of a polynomial?

This may sound like a basic question, but I am sorry to say that I did not find it's answer which completely satisfy my query. Here is the question: "What is the need to find roots of a polynomial ?" ...
0
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0answers
26 views

Prove that $x^3+px+q$ has multiple root $\Longleftrightarrow$ the discriminant is $0$. Is my solution correct?

$f'=x^2+p$ $f'=0,p=-3x^2$ Using $p=-3x^2,f=x^3-3x^3+q$ $f=0,q=2x^3$ Then the discriminant is: $$D=-108\cdot\big((2x^3/2)^2+(-3x^2/3)^3\big)=-108\cdot(x^6-x^6)=0$$ Even if it's correct, can you ...
1
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0answers
19 views

Restrictions on the coefficients that allow a polynomial in a field of characteristic 0 to be solvable by radicals and the associated formula.

We know that a general polynomial $p(x) \in \mathcal{F} \left[ x \right]$, $\deg{ p } = n$, (char(${\mathcal{F}}) = 0$) is not solvable by radicals if $n \geq 5$. However, I was wondering what ...
13
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2answers
658 views

Can we introduce new operations that make quintics solvable?

I have heard from various sources that the typical arithmetic operations (addition, subtraction, multiplication, division, rational exponentiation) are not sufficient to express in general the roots ...
7
votes
2answers
453 views

Approximating roots of the truncated Taylor series of $\exp$ by values of the Lambert W function

To everyone: don't bother writing up another answer, i'm giving this bounty Antonio's answer. It just doesn't let me yet (24 hours delay). If you map the nth roots of unity $z$ with the function ...
5
votes
1answer
626 views

How to get the roots of a quartic function when given a quadratic factor

We have the function $$x^4 + 4x^3 - 17x^2 -24x + 36 = 0.$$ $x^2 -x - 6$ is a factor of this function. Find all the roots of the polynomial. So we have $(x-3)(x+2)$, and since it is a quartic we ...
0
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0answers
27 views

Finding roots of a fractional exponential equation.

If we consider a polynomial equation its easy to find the number of roots associated with the expression by applying Descartes Rule. This method, however, doesn't work with non integer exponents. ...
0
votes
2answers
41 views

n'th and (n-1)'th derivative of $\frac{1}{x^n - a}$

I have a function of the form $f(x) = \frac{1}{x^n - a}$, and I need to programmatically find the n'th and (n-1)'th derivative of the function. Since the function has this specific form and that the ...
1
vote
1answer
24 views

Effect on roots of function on taking the derivative of the function

Suppose there is a function $$f(x)=(x-1)^{15}(x-2)^{20}(x-3)^{25}(x-4)^{30}$$ As we take the derivatives of the function, what will happen to the number of real roots and the number of distinct real ...
3
votes
2answers
53 views

Solutions for quartic

Suppose I have an equation in the form $(x-a)^4 + (x-b)^4 = c$. What is a clever way to find all four solutions? I have tried expanding and then used long division. However, I believe a better way is ...
0
votes
0answers
15 views

Roots and Weights

I use a Mathematica package to compute roots and weights (and other things) but the package gives me only the expression of the roots in $\omega$-basis (basis of fundamental weights) and in the ...
0
votes
4answers
80 views

Why all such polynomials have $-1$ as a root?

Why all polynomials of this form have $-1$ as a root? $ x^5+x^4+x^3+x^2+x+1 $ and similar polynomials like $ x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1$
0
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0answers
38 views

Number of roots of a polynomial (Proof)

What might be a simple proof to show that the maximum number of roots of a polynomial is equal to the degree of the polynomial? For example a quadratic polynomial can have a maximum of 2 roots. Can ...
2
votes
1answer
43 views

Solving a problem using Householder's method

For the following points on a plane: $(-1,1),(0,0),(1,1),(1,-1)$, we look for a polynomial $p(x)=a+bx$ such that: $$ \sum_{i=1}^4{(p(x_i)-y_i)^2} = min $$ How do I formulate this as problem as a ...
1
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1answer
45 views

Number of integer roots possible of the following polynomial

Let $p(x)$ be polynomial with integer coefficients, such that $p(0)$ and $p(1)$ are both odd. What is the maximum possible number of integer roots this polynomial can have?
-4
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0answers
27 views

differentiate the given function. Simplify your answers [closed]

In Exercise 1 through 28, differentiate the given function. Simplify your answers y=√2X
0
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0answers
18 views

Approximately minimising a transcendental function.

I currently have a closed form solution for the error probability of a certain type of wireless channel. By letting all $S_i$ terms denote constants, using $U(\cdot,\cdot,\cdot)$ to denote the ...
2
votes
0answers
41 views

How find the range value $a^2+b^2$ if $\cos{(a\sin{x})}=\sin{(b\cos{x})}$ have no solution

if the equation $$\cos{(a\sin{x})}=\sin{(b\cos{x})}$$ have no zero solution,then $a^2+b^2$ range of value $A:[0,\dfrac{\pi}{4})$,$B: [0,\dfrac{\pi^2}{2})$,$C: ...
1
vote
1answer
28 views

How do I find zeros in D(0,2)

$p(z) = z^8 - 20z^4 + 7z^3 + 1$. I know there is 4 real roots, but how do i figure out how many zeroes are there in $D(0,2)$?
3
votes
2answers
60 views

Determine the number of zeros in the first quadrant

This is a homework question: $$f(z) = z^2 - z + 1$$ sorry for the poor code!
1
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0answers
33 views

Find the number of zeros of the polynomial in the first quadrant [closed]

$$p(z) = z^6+9z^4+z^3+2z+4$$ Please help! I have an exam coming up and I don't completely understand!!!
2
votes
3answers
64 views

Determine the number of zeros in the first quadrant $f(z) = z^4- 3z^2 + 3$ [closed]

Determine the number of zeroes of the following function which are in the first quadrant: $$f(z) = z^4- 3z^2 + 3$$ Help please!!! I'm not that good at complex variables!
1
vote
2answers
39 views

How is the nature of the roots of a third degree polynomial determined?

Given a polynomial $p(x) = x^3-bx^2+cx-d = 0 $ such that all three roots are real positive integers. How does one figure out if the three roots are distinct? The coefficient of $x^3$ is 1. In the case ...
1
vote
3answers
100 views

Prove that $f(x)=(e^x-1)(x^2+3x-2)+x$ has exactly one positive root, exactly one negative root and one root at $x=0$.

Prove that the function $f(x)=(e^x-1)(x^2+3x-2)+x$ has exactly one positive root, exactly one negative root and one root at $x=0$. My work so far: $f(0)=0$ Thus, $x=0$ is a root. For the ...
5
votes
1answer
93 views

Prove that the equation $1+x+\dfrac{x^2}{2!}+\cdots+\dfrac{x^n}{n!}$ cannot have a multiple root.

Prove that the equation $$1+x+\dfrac{x^2}{2!}+\cdots+\dfrac{x^n}{n!}$$cannot have a multiple root. Using induction and the result that $f(x)=0$ have a root $\alpha$ of multiplicity $r\implies ...
1
vote
0answers
25 views

Find all integers $m$ and positive integers $n > 1$ so that $m + \sum_{k=1}^n x^k/k!$ has a rational root

If $m = 1$, then $m + \sum_{k=1}^n x^k/k!$ has no rational root for $n > 1$. And clearly the polynomial has a rational foot for all integers $m$ if $n = 1$. So, besides those cases, for what ...
1
vote
3answers
53 views

A polynomial's roots

Let $Q_n(x) = (x^2-1)^n$ and $P_n(x) = Q_n^{(n)}(x)$. Using Rolle's theorem, prove that $P_n$ has exactly $n$ roots.
0
votes
1answer
20 views

Show that a Polynomial has certain factorization

$P(x)$ is a polynomial in $x$ of degree $\leq n-1$. Show that $P(x)$ has $n-1$ distinct roots and thus has the factorization $$k\Pi_{i=2}^n(x-a_i)$$, where the constant $k$ is the coefficient of ...
0
votes
1answer
38 views

Inverse Quadratic Interpolation and the secant method

I am currently completing a maths project that aims to approximate the roots of functions using MATLAB. The two root finding methods that I have used are inverse quadratic interpolation and the ...
4
votes
2answers
80 views

If $a,b,c(a,b,c\in\mathbb{R} )$ satisfy $b^2-4ac<0$ then equation $f(x)=0$ has complex root

I would appreciate if somebody could help me with the following problem: Q: show that ($n>2, n\in\mathbb{N}$) Let $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+ax^2+bx+c, ...
1
vote
1answer
263 views

Solution of Bessel equation

Prove that for a Bessel function in its normal form that is: $$u'' + \left(1 + \frac{1-(4*p^2)}{4x^2}\right)u=0$$ if $p > \frac12$ then every interval of length $\pi$ contains at most one zero of ...
2
votes
3answers
88 views

Numerical Solution of $\frac{x}{1-e^{-x}} -5 = 0$

I am working on a problem at the moment which cuts down to the following question: How do I get a numerical solution for: $$\frac{x}{1-e^{-x}} -5 = 0?$$ I've been thinking about using Newton's ...
2
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0answers
26 views

Finding product of roots of equation of unknown degree when a root is given

If $7^{\frac13} + 7^{\frac23}$ is a root of equation of minimum possible degree with rational coefficients, then what is the product of roots of this equation? How do I solve it?
3
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3answers
2k views

Prove using Rolle's Theorem that an equation has exactly one real solution.

So the question is; Prove that the equation $x^7+x^5+x^3+1=0$ has exactly one real solution. You should use Rolle’s Theorem at some point in the proof. And I have, Since $f(x) = x^7+x^5+x^3+1$ ...
1
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1answer
24 views

Breakaway Point in Root-Locus

Can anyone explain me why the breakaway points in Root-Locus are only on the real axis?
1
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1answer
30 views

Why does the Uniqueness Principle imply real identities are true in the complex analogue?

Uniqueness principle theorem :If $f$ and $g$ are analytic functions on a domain $D$, and if $f(z)=g(z)$ for $z$ belonging to a set that has a non isolated point, then $f(z)=g(z)$ for all $z\in D$. ...
1
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0answers
45 views

Analytic solutions of an equation

I am trying to find analytic solutions of this equation for $x$ with parameter $a$ ($x>0, a>0$): ...
2
votes
1answer
23 views

Open mapping principle complex?

Shows that if $f(z)$ is a non-constant analytic function on a domain D, then the image under f(z) of any open set is an open set. What I have so far: Since $f(z)$ is non-constant and and analytic, it ...
2
votes
1answer
150 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 ...
1
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1answer
28 views

Is $n=2$ the only root of $M(n!)$…?

Wolfram can help till $n=9$, but are there other value larger than $2$ for which $$ M(n!)=0, $$ where $M(n)$ is Merten's function.
2
votes
2answers
96 views

Real roots plot of the modified bessel function

Could anyone point me a program so i can calculate the roots of $$ K_{ia}(2 \pi)=0 $$ here $ K_{ia}(x) $ is the modified Bessel function of second kind with (pure complex)index 'k' :D My conjecture ...
6
votes
2answers
674 views

How was Euler able to create an infinite product for sinc by using its roots?

In the Wikipedia page for the Basel problem, it says that Euler, in his proof, found that $$\begin{align*} \frac{\sin(x)}{x} &= \left(1 - \frac{x}{\pi}\right)\left(1 + \frac{x}{\pi}\right)\left(1 ...
1
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0answers
35 views

Why doesn't Logz/z have zeros?

Our book claims that $\frac {Logz}{z}$ has no zeros, where Logz is the principle branch of the complex natural logarithm. However, $Logz=log|z|+iArg(z)$, correct? So $Log1=log|1|+iArg(1)=0+i0=0.$ ...
2
votes
5answers
129 views

How to solve $x^4-8x^3+24x^2-32x+16=0$

How can we solve this equation? $x^4-8x^3+24x^2-32x+16=0.$