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
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2answers
88 views

Locating the roots of a cubic polynomial.

Given a cubic polynomial $f(x) = ax^{3} + bx^{2} + cx +d$ with arbitrary real coefficients and $a\neq 0$. Is there an easy test to determine when all the real roots of $f$ are negative? The ...
1
vote
2answers
66 views

What is the relationship between the concept of a square root and a number's prime factorization?

Essentially what I am asking is if there is some kind of correlation between a number such as √385 and it's factorization (which is 5,7,11). Is it possible to use a number's (especially very large ...
0
votes
0answers
183 views

Matlab Coding finding zeros without using fzero or roots function

So i am a completely new at Matlab. I'm basically suppose to develop a function in Matlab that finds the zeros of a cubic polynomial. real and complex. I'm pasting below what I have so far. I started ...
1
vote
1answer
27 views

Find all the values of $k$, if any, such that $f=t^4+2t^3-3t^2+2kt+k^2$ is divisible by $g=t+2$ in $\mathbb{Z}_{7}[t]$

Find all the values of $k$, if any, such that $f=t^4+2t^3-3t^2+2kt+k^2$ is divisible by $g=t+2$ in $\mathbb{Z}_{7}[t]$. I solve it in the normal way but I do not sure that my way is correct or ...
0
votes
1answer
292 views

How to find the roots of a 2 variable polynomial of 2nd degree?

The following polynomial is just an example: $$(3-3y)(x^2-y)$$ and is what does it mean to find the critical points of this polynomial? These are the maxima minima. Are they always concerned with ...
0
votes
0answers
57 views

Find Zeros / Factors of a polynomial

I have been told that to find factors of a polynomial (nth degree) we have to find the factors of constant term and that of coefficient of leading term of the polynomial in concern. The possible ...
3
votes
2answers
114 views

Solve $x+y+z=1; x^2+y^2+z^2=35; x^3+y^3+z^3=97$

It may be surprising that I can't get any analytical way of verifying that one of the solutions of $$x+y+z=1$$ $$x^2+y^2+z^2=35$$ $$x^3+y^3+z^3=97$$ is $x=-1, y=-3$ and $z=5$. Although it may be ...
1
vote
2answers
106 views

How to count the real roots of a quartic equation?

Suppose I have a quartic equation with real coefficients, such as: $$a x^4 +b x^3+c x^2+d x +e=0$$ I want to know the number of its real roots. Search engines lead me to symbolic expressions for all ...
2
votes
2answers
115 views

Solving for $x$ : $a^x+b^x=c$

Well the question is to solve for $x$ in $$a^x+b^x=c \tag{a,b,c are constants}$$ Well as of me, I tried to put $\ln{}$ on both sides which does not seem to help. Apart from this I don't seem to ...
1
vote
3answers
72 views

Find coefficients of polynomials $f(x)=x^2+ax+b$ and $g(x)=x^2+cx+d$ $(a,b,c,d \in \mathbb{R})$

Roots of polynomial $f(x)=x^2+ax+b$ are cubes of the roots of polynomial $g(x)=x^2+cx+d$. Sum and product of roots of polynomial $g(x)$ are equal. Find coefficients $a,b,c,d$ so that polynomial $f(x)$ ...
3
votes
3answers
115 views

Negative roots of a cubic equation

Under what conditions will the cubic equation $ax^3 + bx^2 + cx + d$ where $a,b,c,d \in \mathbb R$ yield roots which have negative real parts? (All roots must have negative real parts) Motivation: I ...
0
votes
2answers
53 views

Showing that the roots of the quadratic are real

If $x^2+bx+c=0$ has real roots, show that the roots of the equation $x^2+bx+c(x+a)(2x+b)=0$ are real for all real values of $a$. I could do it by standard way by proving determinant is postive. ...
3
votes
1answer
91 views

Finding the root for a multivariate function.

Background In a practical problem I need to find the solution to: $$f(\bar{x}) - \bar{p} = \bar{0}$$ where $ f : \mathbb{R}^2 \rightarrow \mathbb{R}^2 $. I don't know the exact expression for $f$ ...
0
votes
1answer
84 views

Relation between coefficients of a quadratic if one root is the square of the other.

If one root of the equation $ax^2+bx+c=0$ is the square of the other prove that $b^3+ac(c+a)=3abc$ I couldn't understand how to start the problem I considered the two roots as $p$ and ...
4
votes
4answers
113 views

What is the solution to the equation $9^x - 6^x - 2\cdot 4^x = 0 $?

I want to solve: $$9^x - 6^x - 2\cdot 4^x = 0 $$ I was able to get to the equation below by substituting $a$ for $3^x$ and $b$ for $2^x$: $$ a^2 - ab - 2b^2 = 0 $$ And then I tried \begin{align}x ...
2
votes
2answers
140 views

Prove that $\sin x - x\cos x = 0$ has only one solution in $ [-\frac{\pi}{2}, \frac{\pi}{2}]$

I have to prove that $\sin x - x\cos x = 0$ has only one solution in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$. While it seems obvious that one solution might be $x=0$, I don't know how to do a ...
1
vote
1answer
71 views

Fourth root of unknown positive constant-4th order DE

I am attempting to solve a fourth order homogeneous linear differential equation: $${d^4y\over dx^4}-ay=0$$ The auxiliary equation is $$m^4-a=>m^4=a$$ But I don't how to find the roots of ...
5
votes
1answer
303 views

Nature of roots of a biquadratic equation

(Biquadratic $\rightarrow$ Quartic (degree 4)) The Question: (from a book i am practicing from) Find the nature of the roots of the equation $$f(x) = 45 x^4-144 x^3+146 x^2-56 x+12=0$$ (By nature i ...
1
vote
5answers
131 views

Can the roots of $f(x)=x^4-x^3+2x^2-x-1$ be found algebraically?

Can the roots of $f(x)=x^4-x^3+2x^2-x-1$ be found algebraically? Are there multiple methods for doing so?
1
vote
3answers
66 views

$k2^x+2^x=8$, find the possible values of $k$ [closed]

Find all the possible values of $k$ such that equation $$k2^x+2^x=8$$ has a single root. Find the root in the case. Can anyone give some hints for me? I have no idea how to solve it.
4
votes
2answers
88 views

Is a polynomial $f$ zero at $(a_1,\ldots,a_n)$ iff $f$ lies in the ideal $(X-a_1,\ldots,X-a_n)$?

This is probably a very silly question: If $R$ is an arbitrary commutative ring with unit and $f\in R[X]$ a polynomial, then for any element $a\in R$ we have $$f(a)=0 \Longleftrightarrow X-a ~\mbox{ ...
0
votes
5answers
75 views

Find the cubic equation of roots $α, β, γ$.

Taken from Fitzpatrick $4$ unit course textbook. The question says: If the cubic equation $\ ax^3+bx^2+cx+d$ has roots $α, β, γ$. Find the cubic equation who's roots are $α^2, β^2, γ^2$ I keep ...
7
votes
4answers
3k views

Guessing one root of a cubic equation for Hit and Trial

Suppose I have a cubic equation as $$15x^3-4x^2-25x+14=0$$ By Hit and Trial method I know that one of the roots is $x=1$. And hence I can solve the cubic equation wit ease as it will take the form ...
2
votes
3answers
380 views

Finding the roots of a different Quadratic equation from the roots of a Given Quadratic equation

The Question: If $\alpha$ and $\beta$ are the roots of the equation $ax^2+bx+c=0$... Then find the roots of the equation $ax^2-bx(x-1)+c(x-1)^2=0$ My Attempt: The new equation can be ...
11
votes
1answer
493 views

Sum of roots of cubic = -coefficient of quadratic term?

Working through Ian Stewart's "Galois Theory, Third Edition," he states at the end of the second paragraph on page 13: "Because we know that $\alpha_1+\alpha_2+\alpha_3$ is minus the coefficient of ...
0
votes
0answers
73 views

How to do polynomial composition/substitution? (Vincent-Alesina-Galuzzi)

For the polynomial $$ p(x) = \sum_{i=0}^n c_i x^i, $$ of real coefficients and real variable, obtain the coefficient of $$ q(x) = \left(1 + x\right)^n p\left( \frac{a + b x}{1 + x} \right), $$ as ...
0
votes
0answers
69 views

Nonlinear System of Equations: Criteria for Existence of Solution

Let $\Omega \subset \mathbb{R}^n$ and $F: \Omega \rightarrow \mathbb{R}^n$ is at least once continuously differentiable (but not necessarily a polynomial). we want to find a point $x^* \in \Omega$ ...
0
votes
3answers
50 views

Calculate roots from $\frac{x \cosh(x) - \sinh(x)}{x^2}$

I want to solve the following equation $$f(x) = \frac{x \cosh(x) - \sinh(x)}{x^2} = 0$$ Because the term above is undefined for $x = 0$ I calcuted $$\lim_{x \rightarrow 0}\frac{x \cosh(x) - ...
1
vote
1answer
32 views

Checking whether points form a polygon in complex plane

If z^8=(z-1)^8 then the roots are 1) concyclic 2) form a polygonal 3)none I found the roots to be 1+cot(k.pi/8) for k is a natural number and less than 8. Then couldn't figure it out.
5
votes
1answer
114 views

Proof of why $\sqrt[x]{x}$ is greatest when $x=e$

Stated above question. If the mathjax I used was wrong, it should be: Why does the xth root of x reach the greatest y at x=e
0
votes
1answer
98 views

Newton-Raphson For Integer Factorization

Per my earlier question on Naive Grouping for factorization here, below is the modified Newton-Raphson method (integers only) for the polynomial $N -x^2 - yx - x = 0$. ...
0
votes
2answers
89 views

Find conditions for $a$ and $b$ such that $P(x)=x^4-(a+b)x^3+(ab+2)x^2-(a+b)x+1$ has only real roots. [closed]

I need to find conditions for a and b such that $$P(x)=x^4-(a+b)x^3+(ab+2)x^2-(a+b)x+1$$ has only real roots. Any hints on how I should do that?
3
votes
1answer
98 views

How to calculate the $n$-th member of sequence $a_{n+1}=\sqrt{y+a_{n}}$

I was searching for a smooth continuous concave function $$f:R^{+}\times R^{+}\to R^{+}$$ so that $$f(x+1,y)=\sqrt{y+f(x,y)}\quad\text{and}\quad f(0,y)=0.$$ But I couldn't find a general function, ...
2
votes
1answer
56 views

Roots of polynomial outside a vertical strip of $\mathbb C$

Let $P(z)$ be an arbitrary polynomial with real coefficients. I'd like to guarantee that all roots of $P$ have real parts outside the interval $(0, 1)$. Is there some simple condition on P that will ...
4
votes
4answers
56 views

Unknown both as a exponent and as a term in an equation

Let's say I have an equation $e^{x-1}(x+1)=2$. According to Solving an equation when the unknown is both a term and exponent it's impossible to solve this using elemetary functions. If so, then how do ...
3
votes
0answers
74 views

How do i find formula for the recurrence relation :$x_{n+1}= x^2_{n}-x^2_{n-1}$ with :$x_{-1}=0,x_{0}=\frac{3}{4}$? [closed]

How do i find formula for the recurrence relation according to the below initial conditions: $x_{n+1}= x^2_{n}-x^2_{n-1}$ with :=$x_{-1}=0,x_{0}=\frac{3}{4}$ ? Note:$n=0,1,2\cdots$ Thank you for ...
2
votes
0answers
23 views

Question on real polynomial in projective space

Hi all I was given this question and desperately in need of help. I am given a homogeneous polynomial of degree 4 of two variables x and y, with real coefficients with 4 real distinct projective roots ...
1
vote
1answer
49 views

Constructing matrices with eigenvalues equal to roots of a given polynomial

Suppose we are given a polynomial e.g. $$x^4+Ax^3+Bx^2+Cx+D,\tag1$$ and we need to construct a matrix, whose eigenvalues would be equal to roots of this polynomial. One way, rather inelegant, is to ...
1
vote
5answers
71 views

Is there any notation for general $n$-th root $r$ such that $r^n=x$?

As we know that the notation for the $n$-th principal root is $\sqrt[n]{x}$ or $x^{1/n}$. But the principal root is not always the only possible root, e.g. for even $n$ and positive $x$ the principal ...
0
votes
4answers
86 views

Prove that $f(x)=m$ has three distinct real roots for $m\in(0,8)$

We have $f:\mathbb{R}\rightarrow\mathbb{R},f(x)=x^5-5x+4$ and we need to show that $\forall m\in(0,8)$, $f(x)=m$ has three distinct real roots. How can I prove it?
-4
votes
4answers
82 views

Suppose that $\alpha$ root of the equation [closed]

Suppose that $\alpha$ root of this equation: $$x^4+x^2-1=0$$ Find the value of $$\alpha ^{6}+2\alpha ^{4}$$ "I want the way, not the roots of the equation." I tried, but I couldn't find any thing.
0
votes
1answer
49 views

When solving trigonometric irrational equations does the condition of existence of the radicand under an even root matter?

Hi everyone I would like to ask a thing about the following equation: $$\cos(x) + \sqrt[4]{1 - \frac{4}{3}\cos(2x) - \sin^4(x)} = 0$$ It is trigonometric and irrational, the root's index is 4 (even ...
2
votes
1answer
55 views

Does $\zeta(s)^2 \pm \zeta(1-s)^2$ have roots at the $\rho$s?

Maybe a strange (or stupid) question, but does $$\zeta(s)^2 \pm \zeta(1-s)^2$$ also have roots equal to the non-trivial zeros ($\rho$) ? At first sight you would expect so, however when I tried to ...
0
votes
1answer
39 views

Expansion and factorization to determine roots of equations

Let $(a,c)$ be the roots of the equation $x ^ 2 + ax - b = 0$. Let $(b,d)$ be the roots of the equation $x ^ 2 + cx + d = 0$. Find all the possible real values for $a, b, c, d$. NOTE: I have made ...
0
votes
3answers
73 views

Find the roots of equation involving $\arctan x$

I try to find the roots of the equation: $$y=x-2\arctan\left(x\right)$$ I know that one of them is $(0,0)$ but there are two others that should solve $$\dfrac{x}{2}=\arctan\left(x\right).$$ Is ...
1
vote
2answers
57 views

What type of equation is this? How to solve it?

$$m^4+a^4=0$$ , the answer I obtained is $$0+i1,0-i1$$ but the answer is given as a/sqrt(2)-a/sqrt(2),a/sqrt(2)+a/sqrt(2)
0
votes
1answer
57 views

suppose n is a natural number , prove equation $x^n+nx-1=0$ exist an unique real positive root $x_n$

suppose n is a natural number prove : equation $x^n+nx-1=0$ exist an unique real positive root $x_n$ ; and when $a>1$,$\sum_{n=1}^{\inf}x^a_n$ converges.
1
vote
1answer
52 views

How do I calculate the values of $\zeta(0.5+ie^x)$ for large $x$ ?

In wolfram alpha the values of $$\zeta(0.5+ie^x)$$ closed to zero then How do I know the real values of $\zeta(0.5+ie^x)$ for large real number $x$ ? Thank you for any help
1
vote
2answers
67 views

Why $ax^2+bx+c = a(x-r)(x-s)$, where $r$, $s$ are the roots?

When I was reading about math, I came across the following - Suppose the roots of the quadratic $ax^2+bx+c$ are $r$ and $s$. Then $ax^2+bx+c = a(x-r)(x-s)$ for all values of $x$. Is there ...
1
vote
0answers
31 views

Zeros of derivative of composition of polynomials

Let $f(x),g(x)$ be polynomials such that their derivatives $f'(x),g'(x)$ have $n$ and $m$ real roots. What is the possible minimal/maximal numbers of real roots for the polynomial $(f(g(x))'$? My ...