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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1answer
18 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
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3answers
57 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 ...
0
votes
2answers
314 views

Graeffe's root finding method

What are the practical applications of Graeffe's root finding method?I searched a lot but couldn't find.I found that it is used in aerodynamics and electric circuit analysis.But don't know much about ...
2
votes
0answers
19 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 ...
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2answers
50 views

Mathematics Radical Numbers Problem [on hold]

If, $$\frac{\sqrt 5+1}{\sqrt 2-1} = x $$ then, $$\frac{\sqrt 5-1}{\sqrt 2+1} = ? $$
1
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1answer
99 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.
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2answers
61 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 ...
1
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2answers
71 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 ...
0
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1answer
44 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). ...
0
votes
1answer
90 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
32 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} ...
2
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3answers
1k views

Polynomial root finding

I have an univariate polynomial of some degree - how do I numerically find all of its real roots? I never thought I would ask this question - everyone knows how to find polynomial roots, right..? ...
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1answer
51 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 ...
42
votes
6answers
4k views

Continuity of the roots of a polynomial in terms of its coefficients

It's commonly stated that the roots of a polynomial are a continuous function of the coefficients. How is this statement formalized? I would assume it's by restricting to polynomials of a fixed ...
0
votes
1answer
48 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 ...
4
votes
1answer
66 views

How many iterations of the Newton's method are needed to achieve a given precision

There is a formula for bisection method to estimate number of iterations that are needed to achieve a given precision (desired significant figures) in the interval $[a,b]$ $$ n\ge ...
1
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2answers
62 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 ...
12
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2answers
231 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, ...
10
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4answers
709 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 ...
1
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0answers
30 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
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0answers
29 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 ...
0
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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 ...
0
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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
votes
1answer
33 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 ...
2
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0answers
12 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 ...
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2answers
420 views

Find the cube roots of $ -8 i $ and plot them on a plane.

I can’t figure out the angle of this equation. I set it up like this: $$ z^{3} = 0 - 8 i. $$ I find that the $ r $-value is $ 2 $, but when I try to find the angle, I’m stuck. I can’t divide by $ 0 ...
0
votes
2answers
37 views

Explaining this inequality

In a proof I was working on today, I assumed this equation was true which lead to devastating results $$ \sqrt{\bar{x^2}} =\bar{\lvert x\rvert} $$ For instance, given the data 0 and 2, the left hand ...
0
votes
3answers
94 views

How to prove that the roots of a quartic equation are not ALL real [closed]

Given this equation: $$x^4 + x^3 - 3x^2 + 4x - 2 = 0$$ I wanna prove that not all roots are real. How can I go about achieving this?
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2answers
56 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 ...
2
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4answers
54 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
33 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.
1
vote
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 ...
0
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0answers
12 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
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0answers
42 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 ...
4
votes
2answers
57 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
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2answers
27 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
28 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
4answers
634 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 ...
1
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3answers
42 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 ...
6
votes
5answers
518 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
1answer
25 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.
3
votes
3answers
300 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 ...
2
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1answer
378 views

Working with casus irreducibilis

I read about casus irreducibilis here. As an example of casus irreducibilis, it says we can factor $x^3 - 15x - 4$ to find $4$ as a root and it also has two other real roots. Using Cardano's method we ...
1
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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 ...
1
vote
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
votes
0answers
44 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 ...
0
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1answer
39 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 ...
3
votes
2answers
65 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 ...
0
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0answers
40 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 ...