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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Trigonometric Substitution Method to solve Cubic Equation.

Here are the questions. IN the wiki page, it says p has to be smaller than 0. But they didnt really explain why... Therefore, I assume it is impossible to have a complex number inside arcosine, is ...
11
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4answers
732 views

How to show that a polynomial does not have real roots

How to show generally that a polynomial does not have real roots. Well, for eg lets take the polynomial $x^8-x^7+x^2-x+15$ . Here the power($n=8$) is even so it can have real roots or it might not ...
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2answers
37 views

How to reduce the multiplicity of existing real roots without introducing new real roots?

Given a monic polyomial $P(x)=x^d+r_{d-1}x^{d-1}+\cdots+a_1r+a_0\in\mathbb{R}[x]$ is there a way to manipulate the coefficients of $P$ in an algebraic way such that the new polynomial has exactly as ...
2
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1answer
40 views

Conformal mapping and its application in finding roots of polynomial

So for a polynomial, if we want to find the roots in a complex plane. Rouche's theorem is the first tool in my head. However, I saw several problems of finding the roots in the first quadrant or upper ...
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2answers
47 views

Prove that the unique zeros of $f(x,y)=a x +(1-a)y+xy$ when $x,y\in[0,1]$, is $x=y=0$.

Prove that the unique zeros of the two-variables function: $$f(x,y)=a x +(1-a)y+xy$$ when $x,y\in[0,1]$, is $x=y=0$. Here, $a$ is a parameter between 0 and 1. I have no idea where to start. Any ...
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2answers
63 views

Relation between real roots of a polynomial and real roots of its derivative

I have this question which popped in my mind while solving questions of maxima and minima. Let $f(x)$ be an $n$ degree polynomial which has $r$ real roots. Using this can we say anything about the ...
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4answers
55 views

Roots of $f(x)=x-2+\frac{a-3}{x}$

I wanted to find the values of (a) for which the function $f(x)=x-2+\frac{a-3}{x}$ has more than one root. I know that the equation needs to be set equal to zero, from that step onward I have no idea ...
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3answers
40 views

Show the equation $x^2+(3a-2)x+a(a-1)=0$ has real roots for all values of a∈R and show that $x^2-x+1$ has same sign for all values of x [closed]

How to show the equation $x^2+(3a-2)x+a(a-1)=0$ has real roots for all values of a∈R How to show that $x^2-x+1$ has the same sign for all values of x.
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1answer
29 views

Why isn't the square root is cancelled in this formula?

$\sqrt{\sum\limits_{i=1}^M \vec{V^2_d}(d)}$ This is the formula of the Euclidean length of a vector in the vector space. The vector $V$ has a power of 2 so it is $V^2$. Why isn't the square root of ...
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0answers
21 views

Bairstow method improvements

I was reading about Bairstow method for polynomial root finding and I find very compelling that it uses just real numbers, as I'm interested in real roots of real polynomials only. However, couple of ...
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0answers
43 views

Is it possible to find solutions to polynomials purely by calculus and without iteration?

I know this may sound peculiar, but I was wondering if any mathematicians have found a method to finding roots purely through calculus without iteration. I can't imagine that such a method exists for ...
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2answers
28 views

Real Roots of Complex Quadratic Equation - (Kasana's first example)

I recently bought H.S. Kasana's Complex Variables. It seems quite interesting, and a little harder for me than I had expected, though I should be able to get through it if I take my time. ...
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1answer
29 views

Equilibrium Points for 8th Degree Polynomial

I have an 8th degree polynomial that I need the zeros for. Is there even a way to explicitly solve one? Its for a diff equations review. I need to sketch the phase line, which is a breeze once I get ...
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5answers
525 views

Solving following quartic equation

Solve in $\mathbb{R}$ : $$(x^2+2)^2+8x^2=6x (x^2+2) $$ My try: I tried to make the graph by calculating values for $x=1, 2, 3, 4$ and I found that the function is positive at $x=0$ but negative at ...
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3answers
48 views

(Discriminant) For which values of k will the equation g(x) = x + k have two real roots that are of opposite signs?

I am currently in Grade 12 and came across the following question in a past paper: $$g(x) = \frac{2}{x+1}+1$$ The question asks: For which values of k will the equation $g(x) = x + k$ have two real ...
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3answers
64 views
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3answers
305 views

Number of real roots

Find number of real roots of the equation $$3^{|x|}-|2-|x||=1$$ My try:I have tried to remove the modulas by assuming x in some intervals and moved the linear part to RHS and drawn the rough graph ...
4
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2answers
63 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 ...
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2answers
35 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 ...
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0answers
53 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 ...
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1answer
25 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
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1answer
46 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 ...
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0answers
42 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
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2answers
66 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 ...
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2answers
78 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 ...
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2answers
82 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 ...
3
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3answers
63 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 ...
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2answers
49 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. ...
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1answer
58 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 $f$ ...
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2answers
58 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 ...
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4answers
80 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 ...
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2answers
42 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 ...
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1answer
31 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 ...
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1answer
160 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 ...
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5answers
122 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?
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3answers
55 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.
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2answers
65 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{ ...
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5answers
51 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 ...
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4answers
657 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
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3answers
342 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
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1answer
447 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 ...
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0answers
59 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 ...
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0answers
15 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 $x^* \in \Omega$ ...
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3answers
36 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) - ...
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1answer
26 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.
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1answer
104 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
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1answer
57 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
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2answers
74 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?
2
votes
1answer
88 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
41 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 ...