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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122 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
103 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
245 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
135 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
34 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
182 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
272 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
104 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
345 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
56 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
81 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 $a$ ($x>0, a>0$): ...
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1answer
68 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
50 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 ...
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4answers
596 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
706 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 $i$ ...
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1answer
112 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 ...
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0answers
112 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
71 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) & ...
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6answers
227 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 ...
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2answers
108 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
261 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
66 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
81 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
96 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
192 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
103 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 ...
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1answer
201 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
711 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 ...
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1answer
204 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} ...
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2answers
43 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 ...
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2answers
179 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
1k 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 ...
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4answers
61 views

Let $f(x)=x^2+17x+a$, $g(x)=x^2-17x-a$, $r$ a root of $f$ and $-r$ a root of $g$. Determine the roots of $f$.

Let $f(x)=x^2+17x+a$ and $g(x)=x^2-17x-a$. Suppose $r$ is a root of $f$ and $-r$ is a root of $g$. Determine all roots of $f$. From the descriptions, I can conclude that $f(x)-g(x)=2a$. But that ...
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1answer
48 views

How do you call the following iterative solving method

I have the following implicit equation $$ x= f(x) $$ which I solve by starting with some value for $x$, then setting $x$ to the new value $f(x)$ and so forth until convergence. How is that method ...
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2answers
44 views

Finding root for the segment - found the formula but it doesn't work for some values - wrong formula?

I have the segment, defined as $(x_1, y_1)$, $(x_2, y_2)$. I know that $y_1\ge 0$ and $y_2 < 0$. I want to compute the root point for that segment. I decided to do it that way: ...
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0answers
107 views

Approximating the smallest positive root of a function

Suppose we have a smooth function $f:\mathbb{R}\rightarrow\mathbb{R}$. Let $S$ denote the set of all positive roots of $f$ and let $x^*$ denote the minimum of $S$ (assuming such a thing exists). What ...
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1answer
197 views

Solving multivariate polynomial to find closest point to a $3$ (or more) circles

My requirement is to find the point closest to three circles. So lets say the three circles are $C_1$, $C_2$, $C_3$. I want to find the point in the space such that the SUM of its distance from $C_1$, ...
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1answer
260 views

An Application of Rouche's Theorem to Two Cases

Here is my question - it is an example sheet question, completely non-examinable: [I have managed this first part, but am including it to help give a sense of where the question is going.] $(i)$ ...
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1answer
396 views

Show that $ z \sin(z) = 1 $ has only real solutions.

Here is my question - it is an example sheet question, completely non-examinable: Show that the equation $ z \sin(z) = 1 $ has only real solutions. [Hint: Find the number of real roots in the ...
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1answer
273 views

Solution of cubic modulo some prime

Let $f(x)=x^3+3x+12$. Now if we have the relation $$f(x)\equiv0\pmod p$$ for some prime $p$, then what are the values of $p$ for which this equation is solvable for $x$? I know that the cubic ...