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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1answer
46 views

reconstructing a quadratic equation from roots

I have this quadratic equation $$2x^2+x-3=0$$ that I wish to reconstruct from its roots. $$D=b^2-4ac=25$$ $$x_1=\frac {-b\pm \sqrt D} {2a} = 1 \text{ and } \frac {-2} 3$$ Now, I've always learned that ...
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1answer
47 views

How to get the polynomial which roots are almost “equal and opposite of sign ?”

Respected All I got stuck in it and need your help. We know that if $\alpha_1, \cdots, \alpha_5$ be the roots of $p(x):=x^5+ax^4+bx^3+cx^2+dx+e=0$ then the equation which roots are opposite in sign ...
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1answer
68 views

find a quadratic polynomial p ( x ) and a number n such that p ( x ) and a number $n \pmod n $ has at least 2015 roots?

I understand what the question is asking for, but I don't know how to prove my answer. Let's say I took an equation of the form: $x^2+ 6x+ 8 \equiv0 \pmod {15}$. The first four roots are ...
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1answer
75 views

Secant method and false position method exercise

We have $f(x)=x^2-6$. I have to find $p_3$ if $p_0 = 3$ and $p_1 = 2$ by using a) Secant method b) False position method So for the first one I have $p_2=p_0- \dfrac ...
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2answers
52 views

Nature of The Roots of The Quadratic Equation $(a-1)x^2+(4a-2)x+4a+1=0$ [closed]

For which values of the real parameter $a$ are the roots of the quadratic equation: $$ (a-1)x^2+(4a-2)x+4a+1=0 $$ a) Real b) Positive
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1answer
25 views

Slight problem with solving a trigonometric equation.

I had to prove the identity $4\cos^3x-3\cos x=\cos 3x$ and then use it to solve the equation $(4\cos^2x-3)(4\cos^23x-3)(4\cos^29x-3)=1$. After proving the identity I proceeded to simplify the ...
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2answers
62 views

Find root of equation using bisection method?

Question : Find an approximate value of $\sqrt[3]{25}$ using Bisection Method. Since it doesnt state the accuracy in the question,how many iterations am I going to do to get that approximate value? ...
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1answer
52 views

What is meant by positive root of $x^3-x^3-1$?

I am a bit confused. I think there must be a mistake. In a text I read: The entropy is $2\ln p$, where ...
1
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4answers
75 views

how to factor this cubic polynomial

Let $f(t)=36t^3-19t+5$ be a cubic polynomial. How we can factor $f$ to its roots? Mathematica says that $f(t)=(-1+2 t) (-1+3 t) (5+6 t)$. How?
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1answer
58 views

Multivariate polynomials at bounded evens

Univariate polynomials Given $n$, is there a degree $cn^{c'}$ polynomial $p(x)\in\Bbb R[x]$ and a degree $dn^{d'}$ polynomial $q(x)\in\Bbb R[x]$ with fixed $c,c',d,d'>0$ such that $$m\in\Bbb ...
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0answers
185 views

Prove equation has only one root in a specific interval

Prove that the following equation has only one solution in the interval $[-\text{min}(a_i), +\infty]$: $f(x) = \left(\sum_{i=1}^n \frac{1}{a_i + x}\right)\times \left(\sum_{i=1}^n \frac{a_i b_i}{(a_i ...
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2answers
103 views

Degree Polynomials and Zeroes

"Find a degree $3$ polynomial that has zeros $-3, 4$ and $8$ and in which the coefficient of $x^2$ is $-18$." I've been trying to solve this problem, but I keep getting it wrong. I've worked with ...
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0answers
50 views

Does this simple problem using Vieta's formulas have deeper connections to elliptic curves?

A friend posed the following question to me: Suppose $p(x)=x^3+ax+b$ has one real root, $x_1$, and two non-real roots, $x_2$ and $x_3$. Compute $x_1$ in terms of $x_2$. By Vieta's formulas, ...
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3answers
117 views

Polynomials in one variable with infinitely many roots.

Can a non-zero polynomial in one variable have infinitely many roots ? Can a non-zero polynomial in one variable have uncountably many roots ? Motivation : over $\mathbb Z/12\mathbb Z$, ...
4
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2answers
107 views

Find the number of polynomial zeros of $z^4-7z^3-2z^2+z-3=0$.

Find the number of solutions of $$z^4-7z^3-2z^2+z-3=0$$ inside the unit disc. The Rouche theorem fails obviously. Is there any other method that can help? I have known the answer by Matlab, but ...
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0answers
38 views

root of equation using bisection method

We have the equation $f(x)=x^3-7x^2 + 14x -6 $ .I have to find the root of the equation using bisection method in the interval of $]1.3 , 2 [$ First I find $f(1.3)=2.567 >0 $ and $f(2)=2>0$ I ...
3
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1answer
71 views

What is the value of this integral (using the Argument Principle),

F(z) is given by $$F(z) = e^zz^{-2}(z-1)(z^2-4)(z+8)^7$$ What is the value of the integral $$\int_0^{2\pi} \frac{F'(3e^{i\theta})}{F(3e^{i\theta})}d\theta \space \space ?$$ I think the relevant ...
5
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6answers
171 views

How $\frac{1}{\sqrt{2}}$ can be equal to $\frac{\sqrt{2}}{2}$?

How $\frac{1}{\sqrt{2}}$ can be equal to $\frac{\sqrt{2}}{2}$? I got answer $\frac{1}{\sqrt{2}}$, but the real answer is $\frac{\sqrt{2}}{2}$. Anyway, calculator for both answers return same numbers. ...
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2answers
87 views

How to simplify this expression with radicals? $3\sqrt2 - \sqrt{32} + \sqrt{\frac{80}{16}}$

I don't understand how I could calculate this: $3\sqrt2 - \sqrt{32} + \sqrt{\dfrac{80}{16}}$ My answer is $-\sqrt2 + \sqrt5$, but the real answer should be $\dfrac{9-4\sqrt2}{4}$.
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2answers
245 views

Find coefficient in quartic given product of roots

The product of two of the roots of $$x^4 -11x^3 + kx^2 + 269x - 2001=0$$ is $-69$. Find k. This is a question I have recently received, and I am required to take a test on related questions ...
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1answer
57 views

Sum of roots: Vietas formulas

The equation $x^4-x^3-1=0$ has roots $\alpha, \beta, \gamma, \delta$. Find the equations with roots $\alpha^6, \beta^6, \gamma^6, \delta^6$. I was able to do this using the substitution $y=x^3$. I ...
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1answer
49 views

roots of functions involving several sines

Is it possible to find exact solutions (in $\mathbb{R}$) of equations of the type $$\alpha_1\sin(\beta_1 t)+\alpha_2\sin(\beta_2 t)+1=0$$ for $\alpha_i,\beta_i\in\mathbb{R}$? In a comment to this ...
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1answer
99 views

Replacing numbers by roots of quadratic

We have $10$ numbers in the interval $(0,1)$, not necessarily distinct. At any moment, we can choose two of them, $a$ and $b$. If the quadratic $x^2-ax+b$ has two (possibly identical) real roots, we ...
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5answers
357 views

Sum of roots: Vieta's Formula

The roots of the equation $x^4-5x^2+2x-1=0$ are $\alpha, \beta, \gamma, \delta$. Let $S_n=\alpha^n +\beta^n+\gamma^n+\delta^n$ Show that $S_{n+4}-5S_{n+2}+2S_{n+1}-S_{n}=0$ I have no idea how to ...
15
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3answers
286 views

How can I prove $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}…}}}}=2$ [duplicate]

How can I prove $$\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}...}}}}=2$$ I don't know which method can be used for this?
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1answer
42 views

Prove/disprove number of zeros inequality

Having a continuous differentiable function $f(x)$, and denote $Z(\cdot)$ number of zeros (assume real line), and $(\cdot)^\prime$ first derivative, I would like to know if following inequality ...
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2answers
82 views

How to find all solutions of the equation $\sin x+\cos x=0$ which belong to $(-\pi, \pi)$?

Could you please help me understand and answer this question? Find all  the  solutions of this equation $$ \sin x+\cos x=0 $$ which belong  to  the interval $(-π; π)$ Progress Divided by ...
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0answers
30 views

finding root of 3rd degree math equation

I need to solve the following equation and give a simple formula for $y$ such that with the known value of $x$ we can easily compute value of $y$. $$x = \frac{(c+ky)y^{2}}{2}$$ $c$ and $k$ are ...
9
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1answer
125 views

Prove that $ ax^2+bx+c=0 $ has at least one root in $(0,1)$ if $10a+12b+15c=0$

If $10a+12b+15c=0$, Prove that $$ ax^2+bx+c=0 $$ has at least one root in $(0,1)$. Progress I tried to solve this by Rolle`s theorem ($f'$ has a root between any two roots of $f$), but could not ...
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0answers
60 views

Real roots of an nth order polynomial

Given an nth order polynomial, is there any algorithm that can calculate all the roots ? Is there any algorithm that can calculate ALL the roots of the equation ? ...
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1answer
136 views

Determining existence of roots of a polynomial in the unit disk (possibly with Rouché's theorem?)

I'm studying for my PhD prelim exam in complex analysis, and I ran into this example problem. Show that the polynomial $$p(z)=z^{47} − z^{23} + 2z^{11} − z^5 + 4z^2 + 1$$ has at least one root ...
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3answers
112 views

Rules for whether an $n$ degree polynomial is an $n$ degree power

Given an $n$ degree equation in 2 variables ($n$ is a natural number) $$a_0x^n+a_1x^{n-1}+a_2x^{n-2}+\cdots+a_{n-1}x+a_n=y^n$$ If all values of $a$ are given rational numbers, are there any known ...
2
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1answer
31 views

Finding the value of $y=b^2(3a^2+4ab+2b^2)$ if $a^2(2a^2+4ab+3b^2)=3$ and $a$ and $b$ are distinct zeros of $x^3-2x+c$

If $a$ and $b$ are distinct zeroes of the polynomial $x^3-2x+c$ and $$a^2(2a^2+4ab+3b^2)=3$$ $$b^2(3a^2+4ab+2b^2)=y$$ Evaluate $y$ I tried for many hours but couldn't solve this question. ...
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2answers
176 views

Solve $x^7-5x^4-x^3+4x+1=0$ for $x$

Solve for $x$ $$x^7-5x^4-x^3+4x+1=0$$ This equation has been bugging me since the past few days. I have found, using the Rational Root Theorem that $x=1$ is a root of this equation. However, ...
16
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1answer
369 views

Something strange about $\sqrt{ 4+ \sqrt{ 4 + \sqrt{ 4-x}}}=x$ and its friends

We have the nice radical identity involving $d = 163$, $$-\sqrt{ 44- \sqrt{ 44 - \sqrt{ 44-x}}}=x,\quad\quad x = 2-2\sum_{n=1}^{27}\cos\left(\frac{2\pi\, t_1(n)}{163}\right)=-6.15824\dots$$ where ...
7
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2answers
96 views

Evaluate this Trigonometric Expression

Evaluate $$ \sqrt[3]{\cos \frac{2\pi}{7}} + \sqrt[3]{\cos \frac{4\pi}{7}} + \sqrt[3]{\cos \frac{6\pi}{7}}$$ I found the following $\large{\cos \frac{2\pi}{7}+\cos \frac{4\pi}{7} + \cos ...
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0answers
44 views

A problem related to complex polynomial

Let $$P_{t}(z) =a_{0}(t) + a_{1}(t)z + ...+a_{n}(t)z^n$$ be a polynomial where the coefficients depend continuously on a parameter $t \in (−1, 1)$. Assume that there exists $\text{t}_{0} \in (−1, 1)$ ...
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3answers
149 views

$x^4 + x^2 + 1 = 0$ has no solution in $\mathbb{R}$.

I need to prove that the equation $x^4 + x^2 + 1 = 0$ has no solution in $\mathbb{R}$. How would I go about proving this?
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1answer
37 views

Find the equation which has key root $x=\sqrt{a}+\sqrt{b}+\sqrt{c}$

In my last question which was Proving $x=\sqrt{a}+\sqrt{b}$ is the key root to solve $x^4-2(a+b)x^2+(a-b)^2=0$ ,I could find the coefficients(were very easy) of fourth-degree equation, so I went to ...
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2answers
65 views

how many distinct real zeros a function has

$f(x)= x^4+2x^3-2x^2+1$ How many distinct real zeroes does $f$ have? Is it two because it crosses the $x$-axis twice or am I completely wrong?
5
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4answers
65 views

Proving $x=\sqrt{a}+\sqrt{b}$ is the key root to solve $x^4-2(a+b)x^2+(a-b)^2=0$

Proving the roots of $$x^4-2(a+b)x^2+(a-b)^2=0$$ are...... $$x=\sqrt{a}+\sqrt{b}$$ $$x=\sqrt{a}-\sqrt{b}$$ $$x=-\sqrt{a}+\sqrt{b}$$ $$x=-\sqrt{a}-\sqrt{b}$$ When $a$ and $b$ are ...
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0answers
53 views

Roots of a polynomial equation where coefficients follow a geometric progression

Given a positive constant $a\in\mathbb{R}$, , and a positive integer $n$, I am interested in the roots of $x^n + \sum_{i=0}^{n-1} a^i x^{n-i-1} = x^n + x^{n-1} + a x^{n-2} + a^2 x^{n-3} +\cdots + ...
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1answer
41 views

Find a solution for an equation

Is there any way to find the solution for $x$ in this equation: $$ x^2 = e^{2\mu} \left(e^{2x^2} - e^{x^2} \right) $$ Where $\mu$ has a constant value. I appreciate in advance.
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3answers
90 views

$f'(a)=0$ implies $x=a$ is not a simple zero of $f$

Let $a$ be the root of a polynomial $f(x)$ and let $f'(a)=0$. Then $x=a$ is not a simple zero of $f(x)$. What is the name of this theorem and does someone know a simple (high school level) proof?
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3answers
58 views

Solving $3t^2-\frac{12}{3}t+\frac{4}{3}=0$

I need to to solve: $$3t^2-\frac{12}{3}t+\frac{4}{3}=0$$ The solution manual factorizes this to $\dfrac{1}{3}(3t-2)^2$. How can you do this easily?
5
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1answer
67 views

Sum of square of absolute values of roots of a polynomial

If $\alpha_1,\dots,\alpha_n$ are roots of a polynomial $$P(z)=z^n+a_1z^{n-1}+\dots+a_{n-1}z+1,$$then how can one express the sum $$|\alpha_1|^2+\dots+|\alpha_n|^2$$in terms of $a_i$'s? Thanks.
2
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1answer
48 views

Applying Newton-Raphson method to $a\cdot b^{-2}=c\cdot d^4+e\cdot f(d)$

I am familiar with the method and it's application in classic problems, but I have troubles tackling the function I need to solve with it. So, variables in problem: Real numbers, all are known ...
0
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2answers
40 views

Solving for $x$ using $\ln$ or any possible way.

$$ 12.46x=1-(1+x)^{-20} $$ I tried solving for $x$ using $\ln$ and other methods but the only answer i got was 0.8. The correct answer is approximately to $0.05$.
8
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1answer
185 views

Show all roots of $\sum_{k=0}^n 2^{k(n-k)} x^k$ are real (December 6, 2014 Putnam problem)

Show that for each positive integer n, all roots of the polynomial $\sum_{k=0}^n 2^{k(n-k)} x^k$ are real numbers. I have no idea where to start. From this year's Putnam, problem B4.
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1answer
38 views

Conditioning of the calculation of roots for cubic polynomial

Let $P(x)=x^3+qx+r$. I have to show that the calculation of the three roots $\lambda_i(q,r),i=1,2,3$ can be extremely ill conditioned. For this I looked at the implicit derivative of ...