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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35 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
80 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
55 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
39 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
36 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
28 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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0answers
15 views

solve equations with exponentials of the unknown

folks, I'm trying to solve $$A = B Y + C Z$$ where $A$, $B$, $C$ are known functions defined on $\mathbb R^3$, the unknowns are basically $z$ defined on $\mathbb R^3$ and $$ Y = ...
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1answer
46 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
54 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
112 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
92 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 ...
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2answers
108 views

are all polynomial equations solvable

Has anyone read the Book named " Monad science" published by Lambert Academic Publishing on 28 Febuary,2014 ...
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2answers
49 views

Square root of negative integer

Can I write: $-\sqrt{(2)}$ = $\sqrt{(-2)}$ and vice versa? Or, say, we have, $(-\sqrt{(x - 4)}$ Can this be changed into $(\sqrt{(4 - x)}$ by taking the minus sign inside the square root? How?
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3answers
43 views

Elementary Symmetric Polynomials, Roots of cubic polynomials

I'm given $a_1, a_2, a_3$ as roots of the equation $x^3 + 7x^2 - 8x + 3$ and need to find the cubic polynomials with roots $a_1^2, a_2^2, a_3^3$ and $\frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}$. ...
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3answers
308 views

Polynomials with Integer Coefficients and irrational roots

Is there a polynomial with integer coefficients which has √2 +√7  as a root?
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3answers
70 views

One of the roots of $3x^2 + p = 5x$ , is $2$ . Determine p and the other root. [closed]

Grade 11 Maths Questions .One of the roots of $3x^2+p=5x$, is $2$. Determine the value of p and the other root.
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0answers
23 views

Finding a base for a series to sum to a constant

I'd like to find the value of $r$ that solves the following equation: $$\sum_{n=1}^N r^{\frac{-1}{n}} = C \,,$$ where $N$ and $C$ are positive constants. An approximate method would also work fine ...
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1answer
30 views

Approximations to the Roots of a Function

I want to find approximations to the root of a function in two variables using the Newton-Raphson method. I can use the method on a function in a single variable but I'm lost as to how you can use it ...
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0answers
15 views

Find roots of characteristic equation $p(a,k)=0$

I need help understanding a derivation I've seen in a paper. I have a characteristic equation expressed as a polynomial $p(a,k)$, which means I can represent the characteristic equation as $p_a(a,k) ...
1
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1answer
23 views

Analytic expression for zeroes of sum of two sinusoids

I'm after a closed-form expression for the zeroes of the following function $$ p(z) = d_1 d_2 + d_1\cos(k_1 z) + d_2\cos(k_2 z) $$ $d_1$, $d_2$, $k_1$ and $k_2$ are all real constants. I'm after the ...
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0answers
21 views

Show that the full series have no zeros in $A$

Let us consider the convergent series $h:Ω→ℂ$ given by $$h(α,β)=∑_{n=2}^{∞}(-1)ⁿ⁻¹a_{n}(α)n^{iβ}$$ where $α,β$ are reals in the domain of convergence and $a_{n}(α)$ is a real increasing sequence with ...
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1answer
44 views

Estimating the multiplicity of a root (numerically)

I'm working on a modified root finding script that uses the Newton method, but with a modification such that I estimate the order of the root to get faster convergence. The basis of my motivation is ...
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2answers
35 views

Separability of $f(x) = (x-1)^2(x-3)$ over $\mathbb{Q}$

This is an example in Ash, Basic Abstract Algebra, ch.3.4 page 73 at the bottom (or here on page 11). It states that $f(x) = (x-1)^2(x-3)$ over $\mathbb{Q}$ is separable. But, $f'(x) = ...
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0answers
51 views

Cube root equations

I am interested in finding a general method of solving equations involving cube roots such as $$x^{1/3} + (x-16)^{1/3} = (x-8)^{1/3}.$$ I have a solution for this particular one: $$\{8 - (12 \cdot ...
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0answers
35 views

Finding roots and studying the sign if a polynomial?

We have two polynomials $g(x):= 1+x+\cdots+x^{2m+1}$ and $f(x):= 1+x+\frac{x²}{2}+\cdots+\frac{x^n}{n}$. For the first one, we wish to find the real roots and study the sign as $x$ varies. I ...
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0answers
23 views

What is a contraction mapping and the use in iterated function?

Like the title said, I really appreciate if anybody can explain the contraction mapping in simple terms with examples for iterated function in numerical analysis. I have looked at the Wikipedia page ...
2
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0answers
36 views

Showing that the n first derivatives of (x²-1)^n have at least r roots (for the r-th derivative)?

I have f(x) = (x²-1)^n. I want to show that, for r = 0,1,2,...,n, the r-th derivative is a polynomial (that's easy to show) that has no fewer than r distinct roots in (-1,1). I guess I need to use ...
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0answers
16 views

Solve two equations to have the same roots (perhaps with computer symbolically)

I have a function: $$ P(z) \equiv (\cos(k_1 z) - d_1)(\cos(k_2 z) - d_2) $$ where $d_1$ and $d_2$ are both functions in $z$ and $k_1$ and $k_2$ are constants. I believe that there exists a function, ...
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0answers
52 views

What is the order of convergence and multiplicity at each root?

Let $f(x) = x^3 + 3x^2 − 4$. Find two of its roots using Newton’s method. Start with $x_0=2$ and $x_0=−1$ in each case and calculate up to 3 iterations. What is the order of convergence at each root? ...
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1answer
29 views

Number of needed iterations in finding p'th root of a number with newton method

I need to write a parallel code for finding p'th root of n with newton method. I know how the serial code must be. The only method I found to get rid of the do-while loop in the code is finding a ...
2
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0answers
57 views

Descartes Rule of Sign for exponential sums

I have the following exponential sums ($x\in\mathbb{R}$) $$f(x)=\sum_{i=1}^Na_iP_i(x)b_i^x$$ where $P(x)$ is some monomial, e.g., $x^2, x^3,\dots$, so $f(x)$ looks like ...
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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
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3answers
67 views

Solving $\arcsin(1-x)-2\arcsin(x)=\pi/2$

\begin{eqnarray*} \arcsin(1-x)-2\arcsin(x) & = & \frac{\pi}{2}\\ 1-x & = & \sin\left(\frac{\pi}{2}+2\arcsin(x)\right)\\ & = & \cos\left(2\arcsin(x)\right)\\ & = & ...
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3answers
62 views

How to determine root multiplicity from ONLY the graph?

If you were given the graph of a function, without the function's equation, is there a way to determine exact multiplicity (not just parity) of the roots of the function?
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2answers
54 views

Calculate the integral $\int_{\varGamma}\frac{3e^{z}}{1-e^{z}}dz$

I am looking to solve $$\int_{\varGamma}\frac{3e^{z}}{1-e^{z}}dz,$$ where $\varGamma$ is the contour $|z|=4\pi/3$. We have been asked first to consider $e^{z}=1$ and $e^{z}=-1$ which I get to be ...
5
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3answers
311 views

Roots of functions / polynomials

Please excuse the naivity of this question, but it is a concept that I just have not been able to grasp entirely. My question is, why are the roots of a function, or a system of polynomials so ...
2
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4answers
83 views

Rolle's theorem prove polynomial has only 1 root

Prove that $x^3-x-4=0$ has exactly one real root: This is my working so far: suppose $f(x) = x^3-x-4$ has $2$ roots : $a,b$ $f(a) = f(b) = 0$ $f'(x)=3x^2-1$ $f'(x)$ exists on $(a,b)$ so $f$ is ...
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3answers
75 views

roots of $x^2 - (6 k + 3 )x + 8 k^2 = 0$ are $a$ and $2 a$ . Find the value of $k$ and of $a$. [closed]

The roots of the quadratic equation $x^2 - (6 k + 3 )x + 8 k^2 = 0$ are $a$ and $2 a$ . Find the value of $k$ and of $a$.
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2answers
39 views

Need help with a proof concerning zero-free holomorphic functions.

Suppose $f(z)$ is holomorphic and zero-free in a simply connected domain, and that $\exists g(z)$ for which $f(z) =$ exp$(g(z))$. The question I am answering is the following: Let $t\neq 0$ be a ...
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2answers
42 views

How many solutions to $f'(x)=0$

How many solutions to $f'(x)=0$, when $f(x)=(x-1)(x-2)...(x-n)$ I know that $f$ is a polynomial of degree $n$, so $f'$ has at most $n-1$ roots It depends on whether $n$ is odd or even ? Thanks
2
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2answers
137 views

Solve for p in $\frac1{20} = (1 - p)^{19}p$

I need help to solve for $p$, where $p$ is a probability, i.e. is between $(0,1)$. $\frac1{20} = (1 - p)^{19}p$ How would one solve for $p$? Thnx
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2answers
61 views

What is the domain of this function? (Don't know how to solve it, logarithms…)

Please explain how you solved it, thanks. $f(x)=\sqrt{\log_x2 - \log_2x}$
1
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1answer
38 views

Find coefficients so that polynomial has at least one rational root

I have the following problem: Given $P(X) = X^5 + 15aX^4 + 12bX^3 -18X^2 -1$ Find $a,b \in \Bbb Z$ so that $p$ has at least one rational root. Prove that for any $a, b$ the ...
2
votes
2answers
52 views

Numerically finding roots of function - converges?

Well this question was in my homework, I have difficulty to "proof" it (or more correctly: seeing how I would solve it). Consider a floating point system ($s \cdot b^e$ where $1\leq s \leq 10 - 1 ...
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0answers
20 views

What is the possible structures (closed, discrete, etc…) of the set $A$

Let $f$ be a non identically zero holomorphic function on the set $B=(a,b)×ℝ$. Let $g$ be a non identically zero harmonic (not holomorphic) function on the set $B=(a,b)×ℝ$. Assume that there is a set ...
0
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2answers
82 views

Intriguing Equation

How many ordered tuples of 7 integers ${\{x_{i}\}}_{i=1}^{7}$ are there, such that $$\sum _{i=1}^{7}{x_{i}}-\prod_{i=1}^{7}{x_{i}} =6$$ where $1\le x_i\le 8$. I tried taking ${ \{ x_{ i }\} }_{ ...
1
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1answer
29 views

Finding a polynomial with product and sum of its zeroes

A was reading a book with this question in it: Q. Find a quadratic polynomial, the sum of whose zeroes is 7 and their product is 12. Hence find the zeores of the polynomial. Sol. Let the ...
0
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4answers
38 views

Finding the Zeros of A Function

In my Algebra II class we are learning how to find the zeros of a function, but I find the process very confusing despite the many efforts of my algebra teacher to explain them to me. I understand ...
1
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3answers
44 views

Relation betwen coefficients and roots of a polynomial, K.A.Stroud

I am stuck on example 3, page 4 of Advanced Engineering Mathematics. The equation to be solved is $x^3+3x^2-6x-8=0$, The solution gives the roots as $-4, 2,-1$. Is it possible for someone to show me ...
5
votes
2answers
347 views

Polynomial $p(a) = 1$, why does it have at most 2 integer roots?

The question that I am trying to answer is : Suppose is $p(x)$ is a polynomial with integer coefficients. Show that if $p(a) = 1$ for some integer a then $p(x)$ has at most two integer roots. I have ...