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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2
votes
2answers
59 views

Is “the nth root of x” well-defined without further qualification?

(No, I'm not asking if $\sqrt{-1} = +i$ or if $\sqrt{-1} = -i$. Yes, I know $+i$ is the principal square root.) Consider the cube root of -8. If asked to evaluate it, I would say -2, and I think we ...
0
votes
0answers
43 views

Integral roots of a polynomial

I have one doubt. Suppose, $f_{n}(x)=a_0x^n+a_1x^{n-1}+a_2x^{n-2}+,...,+a_{n-1}x+a_n=0$ be a polynomial with an integral coefficients. If for some $n$ ( say $n=2 \ or \ 3$) , $f_{n}(t)=0,$ where, $t \...
0
votes
2answers
92 views

Finding real roots of a Polynomial Equation without graphs.

I am interested in finding the number of real roots of the polynomial equation $$ x^9 + \frac{9}{8}x^6 + \frac{27}{64}x^3 - x + \frac{219}{512} = 0. $$ I know that graphing it would tell me how ...
2
votes
2answers
39 views

Zeros of Trigonometric Equation

I'm studying the function $$ f(x) = \log(x + 1) + \cos(x)/2 $$ The first derivative is: $$ f'(x) = 1/(x + 1) − \sin(x)/2. $$ To find the first two positive critical points (without Wolfram and the ...
0
votes
0answers
36 views

Is this true :${(a+ib)}^{(k+ij)}=0$ iff $0<a=k<1$ and $b<j$?

let $z=a+ib ,s=k+ij$ are two complex numbers and let $f(z,s)$ be a complex function defined as follow :$$f(z,s)=z^s={(a+ib)}^{(k+ij)}$$ and $a,b,j, k$ are non -nul real numbers . .After some ...
0
votes
0answers
54 views

Explanation of symmetric sum in a solution

Can someone explain me why $x+y=5$ in $\text{E8}$ clearly.
1
vote
1answer
140 views

drawing $\sqrt[3]{2}$

I know that drawing cubic root of "2" ($\sqrt[3]{2}$) is not possible with just a ruler and a compass. But is there a way or a tool to draw this? I mean, a segment line with a length of $\sqrt[3]{2}$.
3
votes
0answers
112 views

Finding exact roots

I know of the rational root theorem to find all rational zeros and Newtons method of approximating zeros, but what if all the solutions are irrational/imaginary and you need exact answers for the ...
1
vote
4answers
202 views

Sum of non-real roots of equation?

What is the sum of all non-real, complex roots of this equation - $$x^5 = 1024$$ Also, please provide explanation about how to find sum all of non real, complex roots of any $n$ degree polynomial. ...
6
votes
3answers
159 views

Prove : The polynomial has no integral roots. [duplicate]

Q. Prove that a polynomial $f(x)$,with integer coefficients has no integral roots if $f(0)$ and $f(1)$ are both odd integers. My attempt: Let $$f(x)=a_0+a_1x+a_2x^2+\dots+a_nx^n$$ now $f(0)=a_0$ ...
0
votes
1answer
77 views

Number of real roots of polynomial derivative

Let $W(x)$ be a polynomial with n real roots and $P(x) = \alpha W(x) + W'(x)$. Prove that for any $\alpha \in \mathbb{R}$: $P(x)$ have at least $n-1$ real roots. I know that the degree of the ...
0
votes
0answers
23 views

Reduce multivariate polynomials by known roots?

Consider three multivariate polynomials $p_1(x,y,z)$, $p_2(x,y,z)$ and $p_3(x,y,z)$ with $x,y,z\in\mathbb{C}$. Imagine that the set of polynomials above is constructed such that they have exactly $6$ ...
3
votes
4answers
142 views

Condition for common roots of two Quadratic equations: $px^2+qx+r=0$ and $qx^2+rx+p=0$

The question is: Show that the equation $px^2+qx+r=0$ and $qx^2+rx+p=0$ will have a common root if $p+q+r=0$ or $p=q=r$. How should I approach the problem? Should I assume three roots $\alpha$, $...
1
vote
3answers
86 views

Finding all the values of $\sqrt[3]{7-4i}$

I'm reading about De Moivre's Formula and the Roots of Unity, and one of the exercises is to find all the different values of $$ \sqrt[3]{7-4i} $$ I know that you can find the $n$th root of 1 with $...
2
votes
0answers
51 views

Zero locus of 2-variate real polynomial are smooth curves

This seems like it should be an easy question, and probably already has already had answer in advanced mathematics, but I only know some basic calculus, so I would like to know how do I go about doing ...
-2
votes
2answers
57 views

Mathematics Radical Numbers Problem [closed]

If, $$\frac{\sqrt 5+1}{\sqrt 2-1} = x $$ then, $$\frac{\sqrt 5-1}{\sqrt 2+1} = ? $$
1
vote
1answer
29 views

Simplifying transfer functions in Z domain

I have difficulties to check whether the below transfer function is recursive or non-recursive: $$H(z)=\frac{1-z^{-1}+z^{-2}-3z^{-3}}{z^{-2}(1-z^{-1})}$$ I know that I have to multiply the num and ...
1
vote
1answer
114 views

Solve $x^3+x+3=0$ by Galois's theory

Solve with radicals the following equation $x^3+x+3=0$, using Galois Theory. I'm just starting learning this and I do not have many ideas.
0
votes
1answer
82 views

Lowest root of a quintic equation with 5 positive roots

I have a quintic equation $$ x^5-a_4 x^4+a_3 x^3-a_2 x^2+a_1 x - a_0=0 $$ with $a_n>0$ real coefficients, and I know that all 5 roots are real and positive (it is a characteristic polynomial). I'...
0
votes
2answers
106 views

Finding roots of $2x^3-5x^2+18x+45$

solve $2x^3-5x^2+18x+45$ not exactly sure where to start on finding the zeros complex or real. There is one real zero and two complex I know that from graphing just cannot do it on paper to understand ...
0
votes
1answer
95 views

Find $r$ in the next formula

Lets suppose I have the next values $$0, 7, 8, 5, 6$$ And the next formula $$4250 = \frac{0}{(1+r)} + \frac{7}{(1+r)^2} + \frac{8}{(1+r)^3} + \frac{5}{(1+r)^4} + \frac{6}{(1+r)^5}.$$ What is the ...
2
votes
1answer
89 views

Weird square root disappearing and flipping fraction upside down?

So here I was, making 2 math problems, I was able to solve them, but 2 operations seem a bit intractable to me. Maybe you can help me understand why this is true: The first problem: $$x = \frac{1}{5} ...
-1
votes
1answer
60 views

Cubic Depressed Form ! What can we deduce form it?

Cubic depressed form with equation $f(x) = x^3 + px + q$ The question is, when $p$ is positive, will the function have $3$ real roots ? or does it have to have $1$ real and $2$ complex roots? My ...
0
votes
1answer
66 views

False positives with Descartes rule of signs

Descartes rule of sign can be used to isolate the intervals containing the real roots of a real polynomial. The rule bounds the number of roots from above, that is, it is exact only for intervals ...
1
vote
2answers
83 views

Square root equation

I have the equation $\sqrt{(7-x)} - \sqrt {(x+13)} = 2 $ The square root should be expanded so it is square root of $7-x$ - square root of $x+13 = 2$. When i square both sides i get: $7-x - x-13 = 4 $...
2
votes
0answers
47 views

Finding roots of $4$th degree conjugate reciprocal polynomial

I am developing a computer program and the following polynomial, of which I need to obtain the roots, turned up $$Ax^4 + Bx^3 + Cx^2 + \overline{B}x + \overline{A}, \quad \text{where } A, B,x \in \...
0
votes
0answers
95 views

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 ...
13
votes
4answers
1k 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 ...
0
votes
2answers
39 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
votes
1answer
58 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 ...
0
votes
2answers
58 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 ...
0
votes
2answers
480 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. First Case:Let $f(x)$ be an $n$ degree polynomial which has $r$ real roots. Using this can we say anything ...
2
votes
4answers
56 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 ...
1
vote
1answer
37 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 ...
0
votes
0answers
31 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 ...
1
vote
0answers
57 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 ...
1
vote
2answers
43 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. ...
0
votes
1answer
32 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 ...
6
votes
5answers
536 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 ...
1
vote
3answers
233 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 ...
0
votes
3answers
166 views

Finding Zeros of cubic without using fzero or roots in Matlab [closed]

okay, so I've modifies my code a bit. ...
3
votes
3answers
330 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
votes
2answers
92 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 Routh-...
1
vote
2answers
67 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
201 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 not ...
0
votes
1answer
313 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
58 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
119 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
112 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 ...