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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4
votes
3answers
152 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?
1
vote
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 ...
0
votes
2answers
67 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
votes
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 ...
2
votes
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 + ...
0
votes
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.
1
vote
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?
1
vote
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
votes
1answer
69 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
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
votes
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
votes
1answer
186 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
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 ...
3
votes
2answers
115 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 ...
1
vote
5answers
121 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
55 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
50 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
56 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
25 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
35 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
52 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
57 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
27 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
85 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
52 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?
1
vote
2answers
36 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
85 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
55 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
50 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
39 views

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

How to find roots of $\sin(x) - a$, where $a \in [0, 1)$ and $x \in [0, 2\pi]$?
0
votes
1answer
53 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
386 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
49 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
1answer
63 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?
0
votes
1answer
40 views

Let $r_0,r_1,…,r_m$ be the real roots of $a_nx^n+a_{n-1}x^{n-1}+…+a_0$.Is there a closed-form expression for $\sum_{i=1}^mr_i -\sum_{i=1}^m1/r_i$?

Let $r_0, r_1, ... ,r_m$ be the real roots of $a_nx^n+a_{n-1}x^{n-1}+...+a_0$, with $a_0\ne0.$ Is there a closed-form expression for $$ \ \ \ \ \ \sum_{i=1}^mr_i - \sum_{i=1}^m \frac{1}{r_i} \ \ \ ...
1
vote
0answers
37 views

Find the positive root of the equation $ce^{-c}-2(1-e^{-c})^2=0$

Can you help me find a root for $c$ in the equation below? $$ce^{-c}-{10\over5}(1-e^{-c})^2=0$$ By expanding this I got, $$ce^{-c}-2 + 4 e^{-c}-2e^{-2c}=0$$ now grouping, ...
2
votes
2answers
95 views

How to show the equation $x^4 + 2cx^3 + 6x^2 + 60x =-11$ has exactly two real solutions?

How can we show that $x^4 + 2cx^3 + 6x^2 + 60x =-11$ has exactly two real roots? $c$ is any element in the interval $(-2,2)$.
0
votes
2answers
48 views

When looking for zeros of a rational function, why is the numerator equated to zero and not the denominator?

If you have a function $F(x)=\dfrac{a(x)}{b(x)}$ and you are asked to find the zero(s) of the function, why do you set the numerator equal to zero, and not the denominator?
0
votes
1answer
34 views

Solving for the roots of a polynomial

Suppose we have a polynomial of the form: $$-x^3+3x^2+9x-27=0$$ Is there an easy way to find the solutions of $x$? I know that they will be factors of $27$, so I begin by factoring $27$ into ...
0
votes
3answers
56 views

Finding the roots of a polynomial on a complex plane [duplicate]

I use an online calculator in order to calculate $x^5-1=0$ I get the results x1=1 x2=0.30902+0.95106∗i x3=0.30902−0.95106∗i x4=−0.80902+0.58779∗i x5=−0.80902−0.58779∗i I know that this is the ...
0
votes
1answer
31 views

Show uniqueness of a point

I use the banana function $F(x_1,x_2)=(1-x_1)^2+100(x_2-x_1^2)^2$ and I found the minimum point X to be (1,1). I need to show the uniqueness of that point. Could you please help me on how to show ...
0
votes
1answer
33 views

Counting function for the number of zeros of a continuous positive function?

Let $f(x)$ within $x\in[a,b]$ an absolute continuous function with $f(x)\geq0$ $f(x_m)=0$ for all absolute minima $x_m$ no other zeros than at $x_m$ I am trying to define a counting function for ...
1
vote
2answers
49 views

Finding zeroes of $x^3-5x^2+11x+17$

I'm trying to find all the zeros of $x^3-5x^2+11x+17$. I figured the possible zeros as being +/- 1, +/- 17$. The book says that -1 is supposed to be a factor, but I tried dividing the polynomial by ...
0
votes
0answers
30 views

Polynomial Root Multiplicity Testing.

I would appreciate some help here. Either a reference or a proof or just a statement that helps me to conduct research of my own. Long ago when I was studying polynomials intently I read about a ...
1
vote
0answers
29 views

Show that $\sin z$ has only one series expansion

The question goes: An extension of the real function $\sin x$ into a complex analytic function is by defining $\sin z = z- z^3/3! + z^5/5!- \cdots$. Show that this is the only way6 to extend $\sin x$ ...
1
vote
1answer
64 views

Estimating the modulus of the roots of $\sinθ_1z^3+\sinθ_2z^2+\sinθ_3z+\sinθ_4=3$

If $θ_1,θ_2,θ_3,θ_4$ are four real numbers, then any root of the equation $$\sinθ_1z^3+\sinθ_2z^2+\sinθ_3z+\sinθ_4=3$$ lying inside the unit circle $\vert z\vert$=1, satisfies which inequality? ...
3
votes
0answers
131 views

On polynomials over finite fields?

Pick prime $q\in\Bbb Z$ such that $q>B^{3t}>(mB)^{2t}$. Suppose I have a multilinear polynomial $g(x)\in \Bbb Z[x_1,\dots,x_n]$ of degree $t$ with $m^t$ non-zero coefficients that bound by ...
0
votes
2answers
50 views

Number of roots of $x^3+2x-1$ in $\mathbb Q$

How many roots does $x^3+2x-1$ have in $\mathbb Q$? I know that it has one real and two complex conjugate roots because the determinant is $-59$.
1
vote
1answer
50 views

$P_n(x):=1+ \sum_{m=1}^n\dfrac{x^m}{m!}$ has no real root for even $n$ and exactly one real root for odd $n$

Is it true that $P_n(x):=1+ \sum_{m=1}^n\dfrac{x^m}{m!}$ has no real root for even $n$ and exactly one real root for odd $n$ ? I can only prove that the polynomial cannot have any multiple roots . Am ...