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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36 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. ...
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2answers
105 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 ...
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1answer
31 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 ...
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0answers
48 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 ...
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2answers
63 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 ...
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4answers
90 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$
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0answers
56 views

Number of integer roots possible of the following polynomial [duplicate]

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?
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0answers
43 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: ...
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1answer
108 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 ...
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1answer
102 views

A number related to the roots of a quartic polynomial is a root of a cubic polynomial

So here is the problem, $a$ and $b$ are two distinct real roots of $f(x)=0$ where $f(x)=x^4-6x+3$, show that $(a+b)^2$ is a root of $g(x)=x^3-12x-36$. I have tried many methods, such as substitution, ...
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3answers
230 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!
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2answers
126 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!
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1answer
33 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 $D(0,2)$?
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2answers
163 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 ...
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1answer
132 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 ...
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0answers
30 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 ...
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3answers
258 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 ...
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3answers
102 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.
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1answer
27 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 ...
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1answer
325 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 ...
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2answers
91 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, ...
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3answers
96 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 ...
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0answers
52 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?
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1answer
78 views

Breakaway Point in Root-Locus

Can anyone explain me why the breakaway points in Root-Locus are only on the real axis?
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1answer
47 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 $z\in D$. ...
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0answers
51 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
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1answer
63 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 ...
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1answer
30 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.
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0answers
48 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 $Log1=log|1|+iArg(1)=0+i0=0.$ ...
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1answer
47 views

How do I find the roots of a quartic, without guessing?

I'm given a quartic function to sketch, and one of the things to find is the zeros/x-roots/x-intercepts. After a lot of guessing (and no success) I pulled it up on my trusty TI, to find the roots are ...
2
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4answers
558 views

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

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

Find all roots of $x^{6} + 1$

I'm studying for my linear algebra exam and I came across this exercise that I can't solve. Find all roots of polynomial $x^{6} + 1$. Hint: use De Moivre's formula. I guessed that two roots are $i$ ...
5
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1answer
110 views

Analyzing a fourth degree polynomial

Let $a,b$ and $c$ be real numbers. Then prove that the fourth degree polynomial in $x$ $acx^4+b(a+c)x^3+(a^2+b^2+c^2)x^2+b(a+c)x+ac$ has either 4 real roots or 4 complex roots. I have never solved a ...
4
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0answers
109 views

All roots of a polynomial lie on a circle.

I'm stuck in the following problem and I need your help to solve it. Given a number $\alpha$, $0 < \alpha < 1$. $A_j(x)$ is a sequence of polynomials of $x^{-1}$ such that: $A_0(x) = 1; \\ ...
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0answers
69 views

Number of zeros of Wronskian

Is there some relation between the number of zeros of a Wronskian and properties of given functions? Having Wronskian (e.g. $2$ x $2$) $$W(x)=\left|\begin{array}{c}f_1(x) & f_2(x)\\f'_1(x) & ...
4
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6answers
221 views

$f(x)=x^3+ax^2+bx+c$ where $1\ge a\ge b\ge c\ge 0$. If $\lambda$ is any root of the polynomial, show that $|\lambda|\le 1$

$f(x)=x^3+ax^2+bx+c$ where $1\ge a\ge b\ge c\ge 0$. If $\lambda$ is any root of the polynomial, show that $|\lambda|\le 1$. My attempt: As the polynomial is a cubic, it must have atleast one real ...
2
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2answers
104 views

Find the order of magnitude of the equation solution

Find the order of magnitude of the following equation solution: $$ x(\ln x)^{2001}=n $$
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1answer
237 views

Stuck on perturbation theory for finding a root of polynomial, with rescaling

I have been given the polynomial $$\epsilon x^3+x-2=0,$$ where epsilon is very small and I need to find the roots using perturbation theory. So far I have found the first root, 2, using the direct ...
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2answers
64 views

Sum of fifth power of roots

What is the sum of fifth power of roots of the equation $$x^3+qx+r$$.I tried expanding $$(a+b+c)^5$$ but it didn't work instead it is becoming more and more complex.
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1answer
80 views

Find the solutions of the equation…

How can I solve this equation? $$ \begin{equation*} \sqrt[3]{x-2}+\sqrt{x-1}=5 \end{equation*} $$ Frankly, I just have no idea at all!!! Thank you in advance!
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1answer
92 views

Working out the discriminant to a polynomial and using for working out “a”

For an equation: $$ x-b^2/x^3+a=0 \\$$ i.e. $$ x^4-b^2+ax^3=0 \\$$ If the discriminant is positive (i.e. $> or =0$) for real roots, what is the discriminant for these equations? Can you use the ...
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2answers
180 views

Can you find a Polynomial of Degree 7 that has 2 complex roots and 5 real?

Can you find a Polynomial of Degree 7 that has 2 complex roots and 5 real? The polynomial, call it $f(x)$ must be irreducible over $\mathbb{Q}$ (or over $\mathbb{Z}$ as Gauss' lemma can be used.) ...
3
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1answer
98 views

Finding an asymptotic expansion for a transcedental equation

I am new around here and was hoping you will be able to help me with the following. I have the equation: $x^3 - 3x^2 +(3-\epsilon ) x + \epsilon = sin(\frac{\pi}{2} x +\frac{\pi \epsilon}{2} ) $ and ...
0
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1answer
186 views

Improvement to regula falsi method?

The regula falsi algorithm is based on a linear interpolation between the points $a$ and $b$, which bracket a root we want to find. Would it be any improvement to use a parabolic interpolation ...
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0answers
620 views

Solving an 8th degree polynomial

I know that through the Abel Ruffini Theorem the general solution to a polynomial of degree five or more cannot be found explicitly. But are there are any other ways to find the roots of such a ...
5
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1answer
203 views

Limits of the solutions to $x\sin x = 1$

Let $x_n$ be the sequence of increasing solutions to $x\sin{x} = 1$. Define $$a = \lim_{n \to \infty} n(x_{2n+1} - 2\pi n) $$ and $$b = \lim_{n \to \infty} n^3 \left( x_{2n+1} - 2\pi n - \frac{a}{n} ...
2
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2answers
42 views

Roots of product of two functions

I wonder if the answer to this question is true: Having two functions $f(x)$, $g(x)$ where $f(x)$ has $N$ real roots, and $g(x)$ is positive for all $x$ (no real roots), does the product of ...
3
votes
2answers
173 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 ...
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2answers
928 views

How many iterations of the bisection method are needed to achieve full machine precision

Suppose that an equation is known to have a root on the interval $(0,1)$. How many iterations of the bisection method are needed to achieve full machine precision in the approximation to the location ...
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2answers
2k views

How to solve for a non-factorable cubic equation?

I want to know how one would go about solving an unfactorable cubic. I know how to factor cubics to solve them, but I do not know what to do if I cannot factor it. For example, if I have to solve for ...