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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37 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) = ...
2
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0answers
72 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
69 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
54 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
83 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
122 views

Rate of convergence of an iterative root finding method similar to Newton-Raphson

We are defining an algorithm as follows: Let $f(x)$ be a function with a root in $[a,b]$. We define a series $\{x_k\}_{k=1}^{\infty}$ as follows: $x_{k+1}=x_k-f(x_k)\frac{b-a}{f(b)-f(a)}$. ...
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1answer
70 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
93 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 ...
2
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3answers
177 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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1answer
92 views

Find the coefficients such that all four roots of $(x^2-px+q)(x^2-qx+p)$ are natural numbers

Find all ordered pairs $(p,q)$ of natural numbers such that all $4$ the roots of $$f(x)=(x^2-px+q)(x^2-qx+p)$$ are natural numbers. I got a solution of the problem (see below) but I want some ...
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3answers
171 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
60 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
440 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
527 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
81 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
46 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
47 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
141 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
66 views

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

Please explain how you solved it, thanks. $f(x)=\sqrt{\log_x2 - \log_2x}$
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1answer
49 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 ...
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2answers
89 views

How to find the zeros of $f(x) = 2x(5-x)$

How do I get the zeros, if $f(x) = 2x(5-x)$. I have told by my classmate that in order to get the zeros of $f(x) = 2x(5-x)$, I need to distribute $2x$ to $(5-x)$. So I distribute it to make it ...
2
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2answers
75 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
23 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
95 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
426 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 ...
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4answers
47 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 ...
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3answers
65 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
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2answers
461 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 ...
2
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1answer
63 views

Number of roots and Rouché's Theorem

Given a polynomial $p(z)=z^4 +6z+3$, I want to show that it has exactly one root $z_1$ with $|z_1|<1$. I am pretty sure it will be easy to show this using Rouché's Theorem. Using this I would have ...
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1answer
59 views

Weird doubt about complex roots of second grade equation

I'm at the beginning of complex numbers study. I have the following equation: $$ x^2-6i=0$$ It's a second grade eq. so I expect to get two solutions. But: $$x=\sqrt{6}\sqrt[4]{-1}$$So I get 4 ...
3
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0answers
244 views

Trigonometric functions of angle fractions

I've just encountered a problem that seems to me interesting enough so that some result exists on the subject. I was working on a problem in complex analysis, in which I needed the fifth root of a ...
2
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1answer
158 views

How many quadratic polynomials exist given the two zeroes? ($1$ or $\infty$)

I was reading some book which had this question: Q. The number of [quadratic] polynomials having zeros $-2$ and $5$ is: (A) 1 (B) 2 (C) 3 (D) More than three? Sol. (A) 1. But ...
2
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2answers
610 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 ...
2
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2answers
53 views

Determine a quartic equation

I am working on a puzzle from Popular Science from the 1980's. This was a puzzle that existed before pocket calculators or programmable computers. It results in one needing to solve a seeming quartic ...
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2answers
64 views

Finding the zeros of a polynomial equation.

Find the exact solutions of $x^3 + 5x^2 -2x -15 =0$. While making notes for my students (in high school), I came across this problem. Using the Rational Root Theorem there don't seem to be any ...
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2answers
42 views

ODE with complex char roots gives strange solutions

$y''-4y'+5y=0$ has char roots - $\{e^{(2+i)x},e^{(2-i)x}\}.$ So its solutions is $e^{2x}\cos(x), e^{2x}\sin(x).$ But when i plug, e.g., first of them into original eq. i get: $-4 e^{2x} cos(x) + 8 ...
0
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1answer
48 views

Only root of a sum?

I have the following equation: $\sum\limits_{i=1}^{k}\left(n_{i}-n\cdot p_{i}\right)\log p_{i}=0$ where $p_{1},p_{2},...,p_{k}$ are the unknown variables with the condition: ...
2
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1answer
42 views

Extend rolle's theorem to complex functions?

If $f(z)$ is a polynomial of degree n with n distinct real roots $r _1$<,..., <$r_n$, then there exists exactly one root of $f '(z)$ in between any consecutive root of $f(z)$. The context of ...
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6answers
117 views

How to find all solutions of $4^x-3^x=1$?

I have problem with equation: $4^x-3^x=1$. So at once we can notice that $x=1$ is a solution to our equation. But is it the only solution to this problem? How to show that there aren't any other ...
0
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3answers
141 views

$P(x)=x^5+a_4x^4+\cdots+a_0$ has roots $1,2,3,4$ and $k$. Find $P(5) -P(0)$.

A polynomial $P(x)$ with leading coefficient $1$ is of degree $5$, and its distinct roots are $1, 2, 3, 4$ and $k$. Find the value of $P(5) -P(0)$. I have no clue on what my initial steps should be.
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4answers
97 views

Why is the other root negative even though the coefficients are rational?

About quadratic equations, I have the follow question, What is the equation, with rational coefficients, knowing that one root is X1 = 1 + sqrt(3) ? So, to solve the problem, I must know that ...
0
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2answers
90 views

Relationship between f and f' in terms of number of real/non-real roots

$f$ is a polynomial of degree $n\ge1$ and $\forall x,x\in \Bbb R \rightarrow f(x)\in\Bbb R$. Prove that: (a)$f$ has at most one more real root than $f'$ (b)$f'$ has no more non-real roots than $f$ ...
1
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1answer
50 views

Four real roots of $x^4+2x^3+mx^2+2x+1=0$ iff $m$ is…

The Equation $x^4+2x^3+mx^2+2x+1=0$ has $4$ different real roots iff: a) $m<3$; b) $m<2$; c) $m<-6$; d) $1<m<3$; e) $-6<m<2$
1
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1answer
79 views

Roots of unity over $\mathbb{Q}$

I want to show the following proposition from Algebra, Hungerford V.8.9. If $n > 2$ and $\xi$ is a primitive $n$th root of unity over $\mathbb{Q}$, then $[\mathbb{Q}(\xi + \xi^{-1}) : ...
3
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2answers
70 views

If $x\in\mathbb R$, solve $x^{\lfloor x\rfloor}=\frac{9}{2}$, where $\lfloor x\rfloor$ is the integer part of $x$.

If $x\in\mathbb R$, solve $x^{\lfloor x\rfloor}=\frac{9}{2}$, where $\lfloor x\rfloor$ is the integer part of $x$. Of course, $x=\lfloor x\rfloor+\{x\}$, where $\{x\}$ is the fractional part of ...
18
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2answers
724 views

Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least n+1 zeros on $(-1,1)$

Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$ such that $f(x)=0$ on $\mathbb{R}$\ $(-1,1)$. Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least n+1 zeros on $(-1,1)$ My attempt : ...
5
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1answer
104 views

Prove that a mapping $f:[-1,1]^2\to\mathbb R^2$ with certain properties has the value $(0,0)$.

The mapping $f:[-1,1]^2\to\mathbb R^2$ is known to be continuous. Also the image of the upper edge of the rectangle is contained in the upper half-plane, the left edge's image is contained in the left ...
2
votes
2answers
88 views

Prove that 1 has n distinct roots of order n

I am trying to show that 1 has n distinct roots of degree n, or in other word that the equations $$z^n=1$$ has n different roots over the complex field. I know that the fundamental theorem of Algebra ...
1
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2answers
119 views

Solve the equation $x^{2n} + 1 = 0.$ Use these solutions to find a factorization of $x^{2n} + 1$ with real coefficients.

I am asked to solve the equation $x^{2n} + 1 = 0,$ and to use these solutions to find a factorization of $x^{2n} + 1$ with real coefficients. I am given the hint that pairing factors arising from ...
2
votes
5answers
389 views

proving zeros of a polynomial are not real

I'm working on a optimization problem and need to show that \begin{equation} \frac{1}{2}x^4 - x^3 -x + 100 = 0 \end{equation} has no real solution in order to prove certain properties about the ...