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 (arithmetic) tag. For questions about roots of Lie algebras, use the (lie-algebra) tag instead.

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2
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1answer
24 views

How to know if the mth root of n is an integer?

If n can be represented in binary as a x bit integer, is there any algorithm such that we can determine if the mth root of n is an integer in time polynomial of x ?
0
votes
2answers
39 views

n'th and (n-1)'th derivative of $\frac{1}{x^n - a}$

I have a function of the form $f(x) = \frac{1}{x^n - a}$, and I need to programmatically find the n'th and (n-1)'th derivative of the function. Since the function has this specific form and that the ...
1
vote
1answer
24 views

Effect on roots of function on taking the derivative of the function

Suppose there is a function $$f(x)=(x-1)^{15}(x-2)^{20}(x-3)^{25}(x-4)^{30}$$ As we take the derivatives of the function, what will happen to the number of real roots and the number of distinct real ...
0
votes
0answers
13 views

Roots and Weights

I use a Mathematica package to compute roots and weights (and other things) but the package gives me only the expression of the roots in $\omega$-basis (basis of fundamental weights) and in the ...
3
votes
2answers
52 views

Solutions for quartic

Suppose I have an equation in the form $(x-a)^4 + (x-b)^4 = c$. What is a clever way to find all four solutions? I have tried expanding and then used long division. However, I believe a better way is ...
0
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0answers
37 views

Number of roots of a polynomial (Proof)

What might be a simple proof to show that the maximum number of roots of a polynomial is equal to the degree of the polynomial? For example a quadratic polynomial can have a maximum of 2 roots. Can ...
-4
votes
0answers
26 views

differentiate the given function. Simplify your answers [on hold]

In Exercise 1 through 28, differentiate the given function. Simplify your answers y=√2X
0
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4answers
78 views

Why all such polynomials have $-1$ as a root?

Why all polynomials of this form have $-1$ as a root? $ x^5+x^4+x^3+x^2+x+1 $ and similar polynomials like $ x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1$
1
vote
1answer
44 views

Number of integer roots possible of the following polynomial

Let $p(x)$ be polynomial with integer coefficients, such that $p(0)$ and $p(1)$ are both odd. What is the maximum possible number of integer roots this polynomial can have?
2
votes
0answers
40 views

How find the range value $a^2+b^2$ if $\cos{(a\sin{x})}=\sin{(b\cos{x})}$ have no solution

if the equation $$\cos{(a\sin{x})}=\sin{(b\cos{x})}$$ have no zero solution,then $a^2+b^2$ range of value $A:[0,\dfrac{\pi}{4})$,$B: [0,\dfrac{\pi^2}{2})$,$C: ...
2
votes
1answer
41 views

Solving a problem using Householder's method

For the following points on a plane: $(-1,1),(0,0),(1,1),(1,-1)$, we look for a polynomial $p(x)=a+bx$ such that: $$ \sum_{i=1}^4{(p(x_i)-y_i)^2} = min $$ How do I formulate this as problem as a ...
-2
votes
0answers
38 views

Determine the number of zeros of the upper half-plane [closed]

$$z^4 + 3iz^2 + z - 2 + i$$ Can anyone please help me????
1
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0answers
33 views

Find the number of zeros of the polynomial in the first quadrant [closed]

$$p(z) = z^6+9z^4+z^3+2z+4$$ Please help! I have an exam coming up and I don't completely understand!!!
2
votes
3answers
61 views

Determine the number of zeros in the first quadrant $f(z) = z^4- 3z^2 + 3$ [closed]

Determine the number of zeroes of the following function which are in the first quadrant: $$f(z) = z^4- 3z^2 + 3$$ Help please!!! I'm not that good at complex variables!
3
votes
2answers
54 views

Determine the number of zeros in the first quadrant

This is a homework question: $$f(z) = z^2 - z + 1$$ sorry for the poor code!
1
vote
1answer
27 views

How do I find zeros in D(0,2)

$p(z) = z^8 - 20z^4 + 7z^3 + 1$. I know there is 4 real roots, but how do i figure out how many zeroes are there in $D(0,2)$?
1
vote
2answers
37 views

How is the nature of the roots of a third degree polynomial determined?

Given a polynomial $p(x) = x^3-bx^2+cx-d = 0 $ such that all three roots are real positive integers. How does one figure out if the three roots are distinct? The coefficient of $x^3$ is 1. In the case ...
5
votes
1answer
79 views

Prove that the equation $1+x+\dfrac{x^2}{2!}+\cdots+\dfrac{x^n}{n!}$ cannot have a multiple root.

Prove that the equation $$1+x+\dfrac{x^2}{2!}+\cdots+\dfrac{x^n}{n!}$$cannot have a multiple root. Using induction and the result that $f(x)=0$ have a root $\alpha$ of multiplicity $r\implies ...
1
vote
0answers
25 views

Find all integers $m$ and positive integers $n > 1$ so that $m + \sum_{k=1}^n x^k/k!$ has a rational root

If $m = 1$, then $m + \sum_{k=1}^n x^k/k!$ has no rational root for $n > 1$. And clearly the polynomial has a rational foot for all integers $m$ if $n = 1$. So, besides those cases, for what ...
1
vote
3answers
98 views

Prove that $f(x)=(e^x-1)(x^2+3x-2)+x$ has exactly one positive root, exactly one negative root and one root at $x=0$.

Prove that the function $f(x)=(e^x-1)(x^2+3x-2)+x$ has exactly one positive root, exactly one negative root and one root at $x=0$. My work so far: $f(0)=0$ Thus, $x=0$ is a root. For the ...
1
vote
3answers
53 views

A polynomial's roots

Let $Q_n(x) = (x^2-1)^n$ and $P_n(x) = Q_n^{(n)}(x)$. Using Rolle's theorem, prove that $P_n$ has exactly $n$ roots.
0
votes
1answer
20 views

Show that a Polynomial has certain factorization

$P(x)$ is a polynomial in $x$ of degree $\leq n-1$. Show that $P(x)$ has $n-1$ distinct roots and thus has the factorization $$k\Pi_{i=2}^n(x-a_i)$$, where the constant $k$ is the coefficient of ...
0
votes
1answer
33 views

Inverse Quadratic Interpolation and the secant method

I am currently completing a maths project that aims to approximate the roots of functions using MATLAB. The two root finding methods that I have used are inverse quadratic interpolation and the ...
4
votes
2answers
78 views

If $a,b,c(a,b,c\in\mathbb{R} )$ satisfy $b^2-4ac<0$ then equation $f(x)=0$ has complex root

I would appreciate if somebody could help me with the following problem: Q: show that ($n>2, n\in\mathbb{N}$) Let $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+ax^2+bx+c, ...
2
votes
3answers
87 views

Numerical Solution of $\frac{x}{1-e^{-x}} -5 = 0$

I am working on a problem at the moment which cuts down to the following question: How do I get a numerical solution for: $$\frac{x}{1-e^{-x}} -5 = 0?$$ I've been thinking about using Newton's ...
2
votes
0answers
25 views

Finding product of roots of equation of unknown degree when a root is given

If $7^{\frac13} + 7^{\frac23}$ is a root of equation of minimum possible degree with rational coefficients, then what is the product of roots of this equation? How do I solve it?
1
vote
1answer
24 views

Breakaway Point in Root-Locus

Can anyone explain me why the breakaway points in Root-Locus are only on the real axis?
1
vote
1answer
30 views

Why does the Uniqueness Principle imply real identities are true in the complex analogue?

Uniqueness principle theorem :If $f$ and $g$ are analytic functions on a domain $D$, and if $f(z)=g(z)$ for $z$ belonging to a set that has a non isolated point, then $f(z)=g(z)$ for all $z\in D$. ...
1
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0answers
44 views

Analytic solutions of an equation

I am trying to find analytic solutions of this equation for $x$ with parameter $a$ ($x>0, a>0$): ...
2
votes
1answer
22 views

Open mapping principle complex?

Shows that if $f(z)$ is a non-constant analytic function on a domain D, then the image under f(z) of any open set is an open set. What I have so far: Since $f(z)$ is non-constant and and analytic, it ...
1
vote
1answer
28 views

Is $n=2$ the only root of $M(n!)$…?

Wolfram can help till $n=9$, but are there other value larger than $2$ for which $$ M(n!)=0, $$ where $M(n)$ is Merten's function.
1
vote
0answers
35 views

Why doesn't Logz/z have zeros?

Our book claims that $\frac {Logz}{z}$ has no zeros, where Logz is the principle branch of the complex natural logarithm. However, $Logz=log|z|+iArg(z)$, correct? So $Log1=log|1|+iArg(1)=0+i0=0.$ ...
1
vote
1answer
36 views

How do I find the roots of a quartic, without guessing?

I'm given a quartic function to sketch, and one of the things to find is the zeros/x-roots/x-intercepts. After a lot of guessing (and no success) I pulled it up on my trusty TI, to find the roots are ...
2
votes
5answers
126 views

How to solve $x^4-8x^3+24x^2-32x+16=0$

How can we solve this equation? $x^4-8x^3+24x^2-32x+16=0.$
3
votes
4answers
167 views

Find all roots of $x^{6} + 1$

I'm studying for my linear algebra exam and I came across this exercise that I can't solve. Find all roots of polynomial $x^{6} + 1$. Hint: use De Moivre's formula. I guessed that two roots are $i$ ...
5
votes
1answer
76 views

Analyzing a fourth degree polynomial

Let $a,b$ and $c$ be real numbers. Then prove that the fourth degree polynomial in $x$ $acx^4+b(a+c)x^3+(a^2+b^2+c^2)x^2+b(a+c)x+ac$ has either 4 real roots or 4 complex roots. I have never solved a ...
4
votes
0answers
78 views

All roots of a polynomial lie on a circle.

I'm stuck in the following problem and I need your help to solve it. Given a number $\alpha$, $0 < \alpha < 1$. $A_j(x)$ is a sequence of polynomials of $x^{-1}$ such that: $A_0(x) = 1; \\ ...
1
vote
0answers
46 views

Number of zeros of Wronskian

Is there some relation between the number of zeros of a Wronskian and properties of given functions? Having Wronskian (e.g. $2$ x $2$) $$W(x)=\left|\begin{array}{c}f_1(x) & f_2(x)\\f'_1(x) & ...
4
votes
6answers
157 views

$f(x)=x^3+ax^2+bx+c$ where $1\ge a\ge b\ge c\ge 0$. If $\lambda$ is any root of the polynomial, show that $|\lambda|\le 1$

$f(x)=x^3+ax^2+bx+c$ where $1\ge a\ge b\ge c\ge 0$. If $\lambda$ is any root of the polynomial, show that $|\lambda|\le 1$. My attempt: As the polynomial is a cubic, it must have atleast one real ...
2
votes
2answers
69 views

Find the order of magnitude of the equation solution

Find the order of magnitude of the following equation solution: $$ x(\ln x)^{2001}=n $$
1
vote
1answer
34 views

Stuck on perturbation theory for finding a root of polynomial, with rescaling

I have been given the polynomial $$\epsilon x^3+x-2=0,$$ where epsilon is very small and I need to find the roots using perturbation theory. So far I have found the first root, 2, using the direct ...
1
vote
2answers
35 views

Sum of fifth power of roots

What is the sum of fifth power of roots of the equation $$x^3+qx+r$$.I tried expanding $$(a+b+c)^5$$ but it didn't work instead it is becoming more and more complex.
1
vote
1answer
71 views

Find the solutions of the equation…

How can I solve this equation? $$ \begin{equation*} \sqrt[3]{x-2}+\sqrt{x-1}=5 \end{equation*} $$ Frankly, I just have no idea at all!!! Thank you in advance!
1
vote
1answer
56 views

Working out the discriminant to a polynomial and using for working out “a”

For an equation: $$ x-b^2/x^3+a=0 \\$$ i.e. $$ x^4-b^2+ax^3=0 \\$$ If the discriminant is positive (i.e. $> or =0$) for real roots, what is the discriminant for these equations? Can you use the ...
0
votes
0answers
19 views

Padé approximant of transfer function with gain and time delay.

$$ H(\omega) = A e^{-j \omega \tau} $$ I'm trying to use Padé approximation to generate a numerator and denominator polynomial for the above transfer function but genuinely struggling with how to ...
4
votes
2answers
91 views

Can you find a Polynomial of Degree 7 that has 2 complex roots and 5 real?

Can you find a Polynomial of Degree 7 that has 2 complex roots and 5 real? The polynomial, call it $f(x)$ must be irreducible over $\mathbb{Q}$ (or over $\mathbb{Z}$ as Gauss' lemma can be used.) ...
3
votes
1answer
71 views

Finding an asymptotic expansion for a transcedental equation

I am new around here and was hoping you will be able to help me with the following. I have the equation: $x^3 - 3x^2 +(3-\epsilon ) x + \epsilon = sin(\frac{\pi}{2} x +\frac{\pi \epsilon}{2} ) $ and ...
0
votes
1answer
50 views

Improvement to regula falsi method?

The regula falsi algorithm is based on a linear interpolation between the points $a$ and $b$, which bracket a root we want to find. Would it be any improvement to use a parabolic interpolation ...
1
vote
0answers
78 views

Solving an 8th degree polynomial

I know that through the Abel Ruffini Theorem the general solution to a polynomial of degree five or more cannot be found explicitly. But are there are any other ways to find the roots of such a ...
0
votes
0answers
17 views

The number of roots of the system of equations in finite field

Let q be a prime power, $GF(q)$ be a finite field and $GF(q)[x]$ the polynomial ring over GF(q). For $m \in \mathbb{Z}_{>0}$: $$f_1(x),f_2(x),\dots,f_m(x) \in GF(q)[x]$$ and each degree is at ...