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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0answers
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How to find the root of a polynomial

I don't know how to solve the following equation: $x^5-h_1x^4-h_2x^3-h_5=0$, where $h_1=\beta_1+\beta_2$, $ h_2=\beta_1+\beta_2-\beta_1\beta_2-\frac{\beta_1\beta_2}{\beta_1+\beta_2}$, ...
2
votes
1answer
154 views

The roots of $x^3+4x-1=0$ are $a$, $b$, $c$. Find $(a+1)^{-3}+(b+1)^{-3}+(c+1)^{-3}$

This is a question in A level Further Pure mathematics pastpaper Nov 2010. The roots of the equation $x^3+4x-1=0$ are $a$, $b$ and $c$. i) Use the substitution $y=1/(1+x)$ to show that the equation ...
2
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6answers
6k views

How to find the roots of $x^4 +1$

I'm trying to find the roots of $x^4+1$. I've already found in this site solutions for polynomials like this $x^n+a$, where $a$ is a negative term. I don't remember how to solve an equation when $a$ ...
1
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2answers
7k views

How to find the inverse of $y=x^3-5x^2+3x+c$

I know how to find the inverse of $y = x^3$, but once you add/subtract another term, that is where I become lost. I have used wolframalpha to get an answer. I am lost upon how this was obtained. ...
0
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4answers
113 views

For $\sqrt[3]{-1+i}$, is $r$ (when put in polar form) $\sqrt[6]{2}$?

And when you put that into the nth root form... It becomes $2^{1/18}\cos\theta + 2^{1/18}\sin\theta$? $n$th root form given is: $\sqrt[n]r\cdot\cos(\theta+2\pi k)n$
0
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2answers
96 views

Multiple root in a polynomial

I'm doing some old multiple tests. It seems I'm pretty stuck around the topic off complex numbers, could someone elaborate how to: Show that 1 is a multiple root of 2nd degree in p$p(x)=x^3-x^2-x+1$
1
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1answer
102 views

Real Positive Zeros of Equation

During my research on physical problem, I faced the following simple equation: $r^{2k+1}+ab\,r-a=0$ With: $-1\leq k\leq1\:,\:0<r\:,\: a,b\in\mathbb{R}$ I need to put bounders on $a,b,k$ such ...
1
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1answer
52 views

Try to solve root in inequality got wrong result

I am very confused. So I have to solve this inequality. The result is $13/24$. But if I try to solve it myself, I get $17/24$. Because: $$\sqrt{\left(\frac{-5}{24}\right)^2 + \frac{1}{4}} = ...
3
votes
3answers
197 views

How to find the roots of $x³-2$?

I'm trying to find the roots of $x^3 -2$, I know that one of the roots are $\sqrt[3] 2$ and $\sqrt[3] {2}e^{\frac{2\pi}{3}i}$ but I don't why. The first one is easy to find, but the another two roots? ...
1
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1answer
87 views

Prove $\sum_{i=0}^{n}\left(x_{i}^{n}\prod_{0\leq k\leq n}^{k\neq i}\frac{x-x_k}{x_i-x_k}\right)=x^n$

Suppose $x_0$ , $x_1$ , $x_2$ , ... , $x_n$ are distinct real numbers , prove that : $$\large{\displaystyle{\sum_{i=0}^{n}\left(x_{i}^{n}\prod_{0\leq k\leq n}^{k\neq ...
2
votes
4answers
88 views

Solve for $x$ in this equation

How do I solve for $x$ algebraically? $$\dfrac{x^2(x^2-1)}{x+3} = 12$$
1
vote
2answers
457 views

find number of solution for given equation

suppose we we have following equations and conditions Let $k$ be the number of real solutions of the equation $e^x+x-2=0$ in the interval $[0, 1]$ and and let $n$ be the number of real solutions ...
3
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0answers
31 views

Study a particular polynomial sequence

Let us define the following sequence of polynomials for every two non-negative integers $i,d$: $$s_i^d(w)=\sum_{j=0}^{d+1} (-1)^j {d+1\choose j} (j+1)^i w^{d+1-j}.$$ Conjecture: The sequence ...
2
votes
1answer
166 views

Fixed-point method in many-dimensions

A well known method of easily solving multi-dimensional non-linear root finding problems, is to bring the equations into the form: $$\bf x = g(x)$$ And then iterating. The problem is, one has to find ...
1
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1answer
387 views

how to prove cubic root of 25 is irrational

how to prove cubic root of 25 is irrational using mathematical induction? I've been trying to do it for hours but can't get it, help plz guys :S
0
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2answers
112 views

Quartic (degree 4) polynomial complex number problem

Can you find a quartic (degree 4) polynomial $p(x) = ax^4+bx^3+cx^2+dx+e$ with real coefficients $a$, $b$, $c$, $d$, $e$ whose roots are precisely $x=5$, $x=-2$, $x=3$ and $x=1+i$ ? Guys please help ...
4
votes
2answers
125 views

Sum of Square Roots Problem

Consider these two lists A = {1,25,31,84,87,134,158,182,198} B = {2,18,42,66,113,116,169,175,199} Now for both lists, add 1,000,000 to each of the numbers then take the sum of their square roots. ...
2
votes
2answers
192 views

looking for a technique to solve an indefinite integral of one over the square root of a cubic polynomial

I am looking for a technique to solve an indefinite integral of $$ \int \frac{dx}{\sqrt{ax^3+bx^2+cx+f}} $$ I honestly have no idea where to start with this and I cannot find anything like this in ...
0
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3answers
211 views

number of roots of an equation

Plotting the equation $x^3-x^2 \sin(x)+\cos(x)$ I see that $x^3-x^2 \sin(x)+\cos(x)=0$ has only one real solution, is there a simpler way to see that it cannot have 3 real solutions?
3
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5answers
1k views

How to solve the cubic equation $x^3-12x+16=0$ [closed]

Please help me for solving this equation $x^3-12x+16=0$
0
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1answer
168 views

Complex n-th root question

Let $m$ and $n\neq0$ be any two integers.Show that $z^{m/n}=\left(z^{1/n}\right)^m$ has $n/(n,m)$ distinct values, where $(n,m)$ is the greatest common divisor of $n$ and $m$. Prove that the sets of ...
1
vote
1answer
133 views

Why does calculator say that $\sqrt{4} = 2$? [duplicate]

Possible Duplicate: Square roots — positive and negative I know that the square roots of four are 2 because $2^2=4$ and $-2$ because $(-2)^2=4$ Are there any conventions in ...
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2answers
319 views

How to do this Intermediate value theorem proof?

Use the Intermediate Value Theorem to show that the equation $x^3+x+1=0$ has a solution. How to do this? :S Thank you very much!
0
votes
1answer
324 views

Creating a Bisection/Secant Hybrid…when to switch between algorithms?

As an optional assignment in a Numerical Analysis class I have the task of creating a hybrid root finding algorithm that uses both the Secant and Bisection method. I have only started learning about ...
0
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2answers
51 views

roots written as exponents

It appears that i'm not quite sure anymore how to write roots as exponents, and how to work with them. I know $\sqrt[3]{a}$ is written $a^{\frac{1}{3}}$, but I don't know how to handle them in things ...
5
votes
4answers
2k views

How do I cube/square a logarithm?

Btw, please don't give me the answer. I just wanna know how to raise a logarithm to its cube cause I'm stuck in this part, but don't solve it for me. $$\log \sqrt[3]x = \sqrt[3]{\log x}$$ I tried ...
2
votes
3answers
546 views

Relations between the roots of a cubic polynomial

How do I solve the last two of these problems? The roots of the equation $x^3+4x-1=0$ are $\alpha$, $\beta$, and $\gamma$. Use the substitution $y=\dfrac{1}{1+x}$ to show that the equation ...
0
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1answer
40 views

How would you find the roots to this question?

I have a homework problem that I arrived. With Mathematica, the limit is 0. So by using $\epsilon= 10^{-6}$ (it is -6, not -0, sorry for the cutoff). $\sin(n^2)/\sqrt{n} <\epsilon =10^{-6}$ ...
0
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0answers
281 views

Integral of absolute value of polynomial?

Let $a(x)$ and $b(x)$ be integer irreducible polynomials where $b$ is U-shaped in the interval mentioned below and has 2 distinct real zeros. The zeros of $b$ cannot be expressed by radicals. Also $b$ ...
6
votes
2answers
2k views

roots of a cubic polynomial

Consider a cubic polynomial of the form $$f(x)=a_3x^3+a_2x^2+a_1x+a_0$$ where the coefficients are non-zero reals. Conditions for which this equation has three real simple roots are well-known. What ...
0
votes
1answer
90 views

Discriminant and roots of $ x^{n^2} \pm (x-1)^{n^2}$?

When considering the polynomials $x^{n^2} \pm (x-1)^{n^2}$ ( $n$ integer > 1 ) i noticed some things that appeared weird to me. Discriminant($x^{n^2} + (x-1)^{n^2}) = (n^2)^{n^2}$. ...
2
votes
2answers
108 views

bound the distance of two roots of multivariate polynomial systems

Consider a system of multivariate polynomial equations $\vec{x}= f(\vec{x})$ with integer coefficients, $f$ is at most of degree 2. Suppose $\vec{x}_1$ and $\vec{x}_2$ are two real roots of $f$, is ...
4
votes
3answers
527 views

solution to equation $a \cdot \cos(\theta) - b \cdot \sin(\theta) = c$

Does the equation $$ a \cdot \cos(\theta) - b \cdot \sin(\theta) = c$$ have a closed-form solution for $\theta$? What about the case where $a^2 + b^2 = 1$?
1
vote
0answers
321 views

Transforming root-equations into polynomials

Let's define special polynomials as polynomials in $\mathbb{Q}[X]$, where we allow to make roots, too. Examples: $\sqrt{X^4+1}$, $\sqrt[3]{X}+\sqrt{X+1}$, $\sqrt{X+\sqrt{X+1}}$ How can I transform a ...
1
vote
2answers
88 views

Find $y=\sqrt{x}$ where $x$ and $y$ positive integers in polynomial time?

Let $x$ be a positive integer and let $y$ be a real number such that $$y=\sqrt{x}$$ Objectives: If $y$ is an integer, find it in polynomial time. If $y$ is not an integer, prove that there is no ...
1
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1answer
154 views

two functions $ f(x) $ and $ g(x) $

let be two functions $ f(x) $ and $ g(x) $ with an infinite set of roots $ a_{n} $ and $b_{n} $ so $ f(a_{n}) =0= g(b_{n}) $ also they satisfy the same functional equation $ f(1-s)=f(x) $ and $ ...
15
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1answer
1k views

Showing that the roots of a polynomial with descending positive coefficients lie in the unit disc.

Let $P(z)=a_nz^n+\cdots+a_0$ be a polynomial whose coefficients satisfy $$0<a_0<a_1<\cdots<a_n.$$ I want to show that the roots of $P$ live in unit disc. The obvious idea is to use ...
2
votes
2answers
137 views

Solving polynomials in $\mathbb{Q}[X]$ exactly

I wanted to write an equation solver for rational polynomials in one variable $X$. However, such solutions do not need to be in $\mathbb{Q}$. What I wanted was to display solutions "lossless", i.e. ...
1
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3answers
159 views

Equations - Solving for x

I have this problem: $$9x^3 - 18x^2 - 4x + 8 = 0$$ However, I'm not sure how to find the values of $x$. I moved the 8 over and factor out an $x$, but the trinomial it created can't be factored. ...
5
votes
1answer
491 views

Finding all roots of polynomial system (numerically)

I want to numerically find all the roots of a system of polynomials (n equations in n variables). Since I can compute the Jacobian for the system (analytically or otherwise), I can use the Newton ...
3
votes
4answers
113 views

Easy way to find roots of the form $qi$ of a polynomial

Let $p$ be a polynomial over $\mathbb{Z}$, we know that there is an easy way to check if $p$ have rational roots (using the rational root theorem). Is there an easy way to check if $p$ have any roots ...
1
vote
2answers
187 views

Irreducible polynomials with integer coefficients over Q

Suppose p(x) is an irreducible polynomial over Q of degree n, with integer coefficients. If p(x) has two roots r1 and r2 satisfying r1r2 = 5, prove that n is even. Attempt at solution: Because the ...
5
votes
1answer
1k views

Find the number of real roots of the polynomial

Find the number of real roots of the polynomial $$f(x)=x^5+x^3-2x+1$$ If I use Descarte's Rule then I get $$f(x)=x^5+x^3-2x+1$$ there can't be more than two positive real roots. Again ...
4
votes
2answers
211 views

Why isn't this square root $+$ or $-$?

I was tasked with proving the identity $\tan(\frac x 2) = \dfrac {\sin(x)}{1+\cos(x)}$ I used the quotient identity for tangent and the half angle identities for sine and cosine to get $ \pm \dfrac ...
4
votes
2answers
1k views

root-finding methods to invert numerically a function

let be the equation $ y-f(x)=0 $ the idea is to get $ s=g(y) $ that is x as a function of 'y' can this be made by a root finding algorithm ?? i mean you treat $ y $ as a numerical free parameter and ...
4
votes
1answer
1k views

What are the best methods for solving cubic and quartic equations by computer programs?

We know that there are closed form formulas for real roots of degree 4 and 3 polynomials, but people sometimes advise to use numerical (e.g. Newton) methods anyway. They claim that closed formulas ...
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1answer
68 views

Understanding a theorem of Marden's on the moduli of zeros of polynomials

My question is concerning Theorem 3.2 in this paper of Marden's. The gist of the theorem is stated below. Theorem 3.2. Every polynomial of the form $$ f(z) = \sum_{j=0}^{n} (b_j - ...
2
votes
4answers
480 views

Approximate solution for the root of a non-linear function

I have been working with a system which involves computing the roots of functions that look like \begin{equation} e^t (g\cos(\omega t) + b) = c \end{equation} where $t$ is the independent variable ...
3
votes
1answer
461 views

Multiple choice question - number of real roots of $x^6 − 5x^4 + 16x^2 − 72x + 9$

The equation $x^6 − 5x^4 + 16x^2 − 72x + 9 = 0$ has (A) exactly two distinct real roots (B) exactly three distinct real roots (C) exactly four distinct real roots (D) six distinct real roots
-1
votes
1answer
138 views

Find a degree 5 polynomial $f \in \mathbb Z_5[x]$ with exactly 4 distinct roots

Find a degree 5 polynomial $f \in \mathbb Z_5[x]$ so that it has exactly 4 distinct roots and factorize it as product of irreducible factors. I'm really struggling in finding such polynomial, so ...