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.

learn more… | top users | synonyms (1)

1
vote
5answers
34 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 ...
16
votes
3answers
180 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?
1
vote
1answer
29 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 ...
0
votes
2answers
67 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 ...
1
vote
0answers
29 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
votes
1answer
114 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 ...
1
vote
0answers
44 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 ? ...
1
vote
1answer
94 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 ...
5
votes
3answers
100 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 ...
1
vote
1answer
21 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. ...
12
votes
2answers
146 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, ...
12
votes
1answer
208 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 ...
6
votes
2answers
80 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 ...
1
vote
0answers
41 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 $\text{t}_{0} \in (−1, 1)$ ...
3
votes
3answers
123 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?
2
votes
1answer
34 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 ...
0
votes
2answers
49 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?
7
votes
4answers
59 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 ...
2
votes
0answers
51 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 + ...
0
votes
1answer
38 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.
1
vote
3answers
80 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?
1
vote
3answers
52 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
votes
1answer
49 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
votes
1answer
44 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
votes
2answers
38 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
votes
1answer
159 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.
0
votes
1answer
30 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 ...
2
votes
1answer
58 views

How to solve Kepler's equation $M=E-\varepsilon \sin E$ for $E$?

I'm trying to create a program to solve a set of Kepler's Equation and I cannot isolate the single variable to use the expression in my program. The Kepler Equation is $$M = E - \varepsilon ...
3
votes
5answers
114 views

Finding the roots of $x^n+\frac{1}{x^n}=k$

Find the roots of $$x^n+\frac{1}{x^n}=k$$ when $n$ is an integer number and the $k$ is positive integer number. So far I found one root which is $x=\frac{1+\sqrt{5}}{2}$ when $n$ is even.
4
votes
0answers
51 views

On the location of the roots of a polynomial

Consider the following two polynomials \begin{align} p(s)&:=s^n+\alpha_{n-1}s^{n-1}+\cdots+\alpha_1s+\alpha_0,\\ q(s)&:=s^{n-1}+\alpha_{n-1}s^{n-2}+\cdots+\alpha_2 s+\alpha_1, \end{align} ...
0
votes
0answers
45 views

Quintic Roots of 1

z^5=1 has five roots. How does z^5=32 relate to those roots? Its basically those roots, but multiplied by 32 right?
0
votes
0answers
41 views

roots of complex polynomials with real coefficients in conjugate pairs?

I used the Argument Principle and applied Rouche's Theorem to show that a polynomial with real coefficients had 4 zeroes inside the unit disk. I then argued that, since these roots must come in ...
0
votes
1answer
15 views

For each number $d$ dividing 12, list the a's with $1 \leq a < 13$ and $e_{13} (a) = d$

For each number $d$ dividing 12, list the a's with $1 \leq a < 13$ and $e_{13} (a) = d$ Can some explain the method of solving this number theory problem. Giving me a hard time, thanks.
0
votes
1answer
32 views

If $g$ is a primitive root modulo $37$, which of the numbers $g^2, g^3,.., g^8$ is a primitive root modulo 37?

If $g$ is a primitive root modulo $37$, which of the numbers $g^2, g^3,.., g^8$ is a primitive root modulo $37$? This problem is a problem bothering me. Any help would be much appreciated.
2
votes
1answer
44 views

For any positive number $k$, find the value of $1^k + 2^k + 3^k+…+(p-1)^k$(mod $p$)

For any positive number $k$, find the value of $1^k + 2^k + 3^k+...+(p-1)^k$(mod $p$) and prove that your answer is correct. A Little confused about this problem. Any help? Would love to see a ...
3
votes
1answer
51 views

Zeros of a function

Show that all zeros of $$f(z)=\sin z +z\cos z$$ are real. I tried to use zeros of $\sin z$ and $\cos z$ are real even though I couldn't get any ideal.
0
votes
1answer
25 views

Find the root of C [duplicate]

Can u help me to find a root for C (except c = 0) in below equation. $$ce^{-c}-{10\over5}(1-e^{-c})^2=0$$ by expanding this I got, $$ce^{-c}-2 + 4 e^{-c}-2e^{-2c}=0$$ now grouping, ...
1
vote
1answer
28 views

Not able to use fzero function in Matlab

I am new to Matlab. I am trying to solve a non-linear equation using this inbuilt Matlab function called fzero() but it's not giving me the results. The main file ...
1
vote
0answers
43 views

Why does this equation have four roots?

$(7x+1)^{1 \over 3}+(8+x-x^2)^{1 \over 3}+(x^2-8x-1)^{1 \over 3}=2$ I figured the roots are 0, 1, -1 and 9 but why?
3
votes
2answers
32 views

Show that $x^a+x-b=0$ must have only one positive real root and not exceed the $\sqrt[a]{b-1}$

If we take the equation $$x^3+x-3=0$$ and solve it to find the real roots, we will get only one positive real roots which is $(x=1.213411662)$. If we comparison this with $\sqrt[3]{3-1}=1.259921$, we ...
0
votes
2answers
78 views

Show that the equation $ \cos(x) - kx = 0$ has a unique solution in $[0, \pi/2]$ for all $k>0$

Show that the equation $$f(x;k) \equiv \cos(x) - kx = 0$$ has a unique solution in $[0, \pi/2]$ for all $k>0$.
3
votes
2answers
50 views

Roots of $x^4 -6x^3 +x^2+10x +1=0$

How can one prove that the following function has 4 real roots? $$x^4 -6x^3 +x^2 +10x+1=0$$ The problem is that roots don't seem to be possible to compute by hand.
-1
votes
1answer
10 views

Prove the iterative scheme converges to the root in [0.4,0.6]

Prove that the iterative scheme $$X_{r+1} = g(X_r) = e^{X_r^{2}-2X_r}$$ with a suitable starting point, converges to the root in $[0.4,0.6]$, by showing that $g$ is a contraction mapping on this ...
2
votes
1answer
37 views

Most efficient way to find polynomial roots

Given a polynomial: $$z^7+10z^6+42z^5+96z^4+129z^3+102z^2+44z+8$$ find it's roots. I started off by using Horner's method (I believe one of the roots has to be $1$, so that's my starting point) but ...
1
vote
3answers
32 views

How to find roots of $\sin (x) - a$?

How to find roots of $\sin(x) - a$, where $a \in [0, 1)$ and $x \in [0, 2\pi]$?
-1
votes
1answer
41 views

How to solve the non-linear equation $-(a+c\,e)\left(\exp(-b/(a+c\,e))-1\right)-c\,d=f$ for $c$?

I have this non linear equation: $$-(a+c\,e)\left(e^{-\frac{b}{a+c\,e}}-1\right)-c\,d=f$$ The only unknown is $c$. All the coefficients ($a$, $b$, $c$, $d$, $f$) are real non-null costants. How can I ...
1
vote
2answers
365 views

Will a 2nd degree function always have maximum 2 roots?

Will a function of 2nd degree always have maximum 2 roots? For example: $$f(x) = x^2 - ln(x^2 +1) -1 $$ EDIT: More specific; if you have a function with $$k * x^2$$ where k is a real number, and ...
0
votes
1answer
33 views

Roots of an equation using Maple

I am using Maple to find the roots of a non-linear equation in one variable. When I solve the equation, I get only 2 negative roots whereas if I plot the graph of the function, it also shows that the ...
0
votes
0answers
24 views

How to find the root of this non-linear equation?

I am trying to solve this non-near equation using Matlab but it doesn't give me the correct answer (as shown in the document that I am doing it from). The Matlab code gives me imaginary root. Could ...
0
votes
1answer
28 views

Find monic quartic polynomial f(x) with rational coefficients whose roots include…(Algebra)

Find a monic quartic polynomial f(x) with rational coefficients whose roots include $x=2-3\sqrt{2}$ and $x=1-\sqrt{3}$. How could you find the other roots?