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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0answers
24 views

Number of ordered pairs of $(p,q)$ [duplicate]

Let $x^2-px+q=0$ and $x^2-qx+p=0$ have unequal integral roots where $p$ and $q$ are natural numbers. Find the number of ordered pairs $(p,q)$ for which this is possible. I do understand that roots ...
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4answers
89 views

Find the roots of the polynomial? (Cardano's Method)

$y^3-\frac7{12}y-\frac7{216}$ This is part of Cardano's method, so I've gotten my first root to be: $y_1=\sqrt[3]{\frac7{432}+i\sqrt{\frac{49}{6912}}}+\sqrt[3]{\frac7{432}-i\sqrt{\frac{49}{6912}}}$ ...
0
votes
2answers
72 views

Showing that an equation has a root in an interval

Show that the equation $x^4 - 7x^3 + 1 = 0$ has a root in the interval $[0,1]$. How would I go about working this out in steps?
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votes
1answer
77 views

Prove $p(x)>0$ for $x>b$

This is a question from a past paper which I have no solution to. Let $p(x)=x^n + a_{1}x^{n-1}+\cdots+a_{n-1}x+a_{n}, n\geq 1$ be a polynomial of dgree n and let $b=1+|a_{1}|+\cdots ...
4
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1answer
54 views

$\int\limits_{0}^{32/9}\sqrt{1+\frac{9x}{4}}dx$

Question : Solve $\int\limits_{0}^{32/9}\sqrt{1+\frac{9x}{4}}dx$ My Try: Let u = $1+\frac{9x}{4}$ Then, $$du = \frac{9x}{4}dx$$ $$dx = \frac{4du}{9}$$ Substituting the above in the main ...
3
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2answers
87 views

Prove that $f(x)=x$ can have at most one solution if $f'(x)\ne1$

Prove that $f(x)=x$ can have at most one solution if $f'(x)\ne1$ What I did : Use $g(x) = f(x)-x$, then $g'(x) = f'(x)-1\ne0$ I suspect I have to use Rolle's theorem now, But I am having difficulty ...
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2answers
870 views

Is Vieta the only way out?

Let $a,b,c$ are the three roots of the equation $x^3-x-1=0$. Then find the equation whose roots are $\frac{1+a}{1-a}$,$\frac{1+b}{1-b}$,$\frac{1+c}{1-c}$. The only solution I could think of is by ...
1
vote
1answer
71 views

Location of the roots of $f'$ (Laguerre's theorem)

Let $f \in \mathbb{R}[X]$ be a polynomial of degree $n$ having $n$ distinct roots $a_1,...,a_n$. Let $b_1<...<b_{n-1}$ be the roots of its derivative $f'$ (note that $b_i \in ]a_{i}, a_{i+1}[$ ...
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0answers
41 views

Positive Zeroes within a Polynomial

Question: Let $a,b>0.$ Can the polynomial $$x^{10} − x^7 + 2x^5 + ax^3 − bx + 1$$ have exactly three (counting multiplicity) positive zeroes? Can it have three simple positive zeroes together with ...
2
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1answer
50 views

Confusion about exponents like ${x^m}^{(1/n)}$.

I've been reading this post. It says that $\sqrt[m]{x^n} = x^{n\frac 1m}=x^{\frac mn}=x$ if $m=n$. Let's take $x=-2$, and $m=n=2$. Now we have, $\sqrt[2]{(-2)^2}=\sqrt[2]{4}=2$ But according to that ...
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votes
2answers
252 views

Extreme point of quadratic equation

For the below question read here: Write a function quadratic that returns the interval of all values $f(t)$ such that $t$ is in the argument interval $x$ and $f(t)$ is a quadratic function: ...
0
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1answer
41 views

Exist another method to solve the problem?

We have $x_1,\:x_2,\:x_3\:\in \:\mathbb{C},\:\:f=x^3+x^2+mx+m,\:m\in \mathbb{R}$. We need to find $m\in\mathbb{R}$ such that $|x_1|=|x_2|=|x_3|$. Here is what I tried: $f=x^3+x^2+mx+m=(x^2+m)(x+1)$, ...
0
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1answer
59 views

How to evaluate real root of a polynomial equation? [closed]

If $\alpha$ is a real root of the polynomial equation $$300x^{299}+299x^4+343x^3+23x+300=0$$ Then how to find out the value of $[\alpha]\space $ where, '$[ \space]$' denotes greatest integer? I have ...
4
votes
3answers
126 views

Zeroes of sin(x)

Consider the function f = $\sin(x)$ defined as $$ \sin(x) = \frac{e^{ix}- e^{-ix}}{2i} $$ How to prove that the only zeroes of this function lie on the line $i = 0$ in the complex plane and ...
1
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0answers
31 views

Interval of Polynomial Root Finding

Let's say we have a polynomial of a given degree. You don't have any tools to figure out the amount of roots in this polynomial. All you know is the function and you cannot graph it. How would you ...
0
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1answer
57 views

Prove or disprove this relation between one root of the quadratic and the cubic equation of a certain form, and linear recurrences.

It is well known that the n-anacci (higher degree Fibonacci, that is Tribonacci and so on) numbers can be computed in closed form from roots of polynomials in the way Eric Weisstein at Mathworld ...
0
votes
2answers
52 views

Roots of quadratic equation

If the roots of $ax^2+bx+c$ are $\alpha$ and $\beta$, express $\frac1\alpha-\frac1\beta$ in terms of $a$, $b$ and $c$. I know how to express $\alpha+\beta$ or $\alpha\beta$ which is usually enough, ...
2
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1answer
38 views

Show that $z^n+nz-1$ has $n$ zeros in $D(0,R)$

Let $n\geq 3$. Show that the polynomial $z^n+nz-1$ has $n$ zeros in $D(0,R)$, where $$R=1+\left(\frac{2}{n-1}\right)^{1/2}.$$ I was hoping to use Induction and Rouche's Theorem. For the base case ...
0
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1answer
141 views

Proving that if the sequence $X_n$ converges to $x$, then ${X_n}^a$, where $a$ is a positive rational, converges to $x^a$.

I've been stuck on this problem for a while. I splitted a into $p/q$, so it would be $({X_n}^p)^{1/q}$, and I got the convergence of ${X_n}^p$ to be $x^p$ since it is just induction using the product ...
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1answer
94 views

Find the fixed points of a function $f(x) := exp(x - 2)$ using a recursive algorithm

I need to find the fixed points (i.e. when $f(x) = x$) of the following function $f(x) := exp(x - 2)$. I understood that the fixed points should be the intersecation points between $f(x)$ and a ...
2
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1answer
104 views

Effect on existing roots of polynomial when adding small higher-order term

How do existing roots of a polynomial change when adding higher-order term with a small coefficient? Given a sufficiently small coefficient of the new higher-order term, the existing roots shouldn't ...
0
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3answers
91 views

Find the real root $\alpha$ of the cubic equation $z^3-2z^2-3z+10=0$

Find the real root $\alpha$ of the cubic equation, $$z^3-2z^2-3z+10=0$$ The exam paper is giving just 2 marks for this and the mark scheme isn't very helpful. My idea is that you can use some of this ...
1
vote
1answer
750 views

Finding a root by bisection method in Excel

Working on a maths assignment and we're trying to use Excel for a bisection method. $$\frac12 e^{x/2}+\frac{1}{2x}-\frac32=0$$ Here is a pic, I can't get the formula to work with the exponent. This ...
41
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2answers
1k views

Only 12 polynomials exist with given properties

Prove that there are only 12 polynomials that have all real roots, and whose coefficients are all $-1$ or $1$. (zero coefficients are not allowed, and constant polynomials do not ...
0
votes
1answer
75 views

The roots of the cubic equation $z^3-2z^2+pz+10=0$ are $\alpha$, $\beta$ and $\gamma$. Show that $\alpha^2+\beta^2+\gamma^2=p+13$

$$z^3-2z^2+pz+10=0$$ $$ax^3+bx^2+cx+d=0$$ $$\Rightarrow\,\,\,\,\,\,\,\,\,a=1,\,\,\,\,\,\,\,\, b=-2,\,\,\,\,\,\,\,\, c=p,\,\,\,\,\,\,\,\, d=10$$ ...
2
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3answers
76 views

Prove $x^7+3x^5+1$ has exactly one real root using Bolzano's theorem and the MVT.

Prove $f(x)=x^7+3x^5+1$ has exactly one real root using Bolzano's theorem and the MVT. What I did: $f(-1)=-3$ $f(0)=1$ As $f$ is continuous, there exists a $c \in (-1,0) /f(c)=0$ Then computed ...
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3answers
187 views

Solve $ \left(\sqrt[3]{4-\sqrt{15}}\right)^x+\left(\sqrt[3]{4+\sqrt{15}}\right)^x=8 $ [closed]

I don't know what can I substitute for $x$ so that equation becomes satisfied. Any assistance will be greatly valued. Thanks!
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0answers
164 views

Solving quartic equation? (Cardano/Ferrari)

today I've written a little Code-Snippet that is based upon the steps that are mentionned in this wikipedia-Article to solve a general quartic polynom. Here's my matlab-implementation: ...
0
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1answer
50 views

general theorem on roots of a polynomial needed to show it's identically zero.

Polynomial degree k, one variable, if it's zero at k+1 values, then it's identically zero. Can someone point me to a proof of this? I know derivatives at points can count as these roots (if k-degree ...
0
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2answers
58 views

Entire functions of order 0

Sorry, this may be a stupid question, but I am just beginning to learn about this and cannot find the answer anywhere I have looked so far. Clearly if we have any polynomial $P(z)$, then it is easy to ...
0
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1answer
46 views

I have to show $p=p(x-\lambda)$ if and only if they have the same zeros in $F$

Suppose $F$ is a field, $|F|\geq n \geq 2$. Given $\lambda \in F$ I know $p,p(x-\lambda)\in F[x]$ are irreducible monic polynomials. I have to show $p=p(x-\lambda)$ if and only if they have the same ...
0
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1answer
58 views

Help required! Polynomials

Let $D(p) = p^{20} - p^{18} - p^{16} - \dots - p^2 - 2$ Prove that the sum of fourth powers of all the real roots of $D(p) = 8.$ Please help.
1
vote
1answer
49 views

What are 'Regular Products'?

When looking at the functional equation for the Riemann zeta function, I came across the statement: For $s$ an even positive integer, the product $\sin{(\frac{\pi s}{2})}\Gamma({1-s})$ is regular. ...
3
votes
1answer
116 views

How to remove duplicate roots from a polynomial?

Given a polynomial equation (with real coefficients of any degree with any number of repeating roots), let say $x^5 + 6x^4 - 18x^3 - 10x^2 + 45x - 24 = 0$, ... (A) it can be written as $(x-1)^2 ...
1
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0answers
94 views

Root locus vs Matlab's root locus function

Today during a lecture out teacher was demonstrating Matlab's rlocus function, but before that he decided to do it on chalkboard. Below we have a simple transfer ...
1
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1answer
31 views

Algebraic vs. analytic definition of the multiplicity of a polynomial's root

Let $f(x) = a(x - c_1)^{d_1}(x - c_2)^{d_2} \dots (x - c_n)^{d_n}$ be a polynomial over the complex numbers ($n, d_i \in \{1, 2, \dots\}$, $a \in \mathbb{C}\setminus \{0\}$), where the roots $c_1, ...
2
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2answers
123 views

Solving recurrences whose characteristic equations have complex roots

In my Discrete Mathematics lecture notes, there is a section regarding solutions for linear recurrences whose characteristic polynomials have complex roots. There is a particular statement which I am ...
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1answer
26 views

Finding roots of complex polynomial with conjugates

I am having problem with the following question... I know that I should use De Moivre's formula somewhere... but can't quite get to it $$ (-15w + 34\bar{w})^4 = -1 $$ will be happy to get some help, ...
2
votes
2answers
64 views

Roots of a Polynomial Minus It's Constant Term

Suppose we have a sequence of integers $a_1,\dots,a_n$. Is there any way to determine the roots of the polynomial $$P(x) = (x+a_1)\dots(x+a_n) - a_1\dots a_n$$ Clearly $P(0) = 0$, but can anything ...
2
votes
4answers
191 views

Find the square root of $404.11$.

Find the square root of $404.11$ without using calculator accurate upto $2$ decimal places . It is clear that $20<\sqrt{404.11}<21$ so it will be $20.ab$ without trial and error what ...
1
vote
1answer
48 views

The number of zeros of a polynomial that almost changes signs

Let $p$ be a polynomial, and let $x_0, x_1, \dots, x_n$ be distinct numbers in the interval $[-1, 1]$, listed in increasing order, for which the following holds: $$ (-1)^ip(x_i) \geq 0,\hspace{1cm}i ...
1
vote
1answer
39 views

Is Gershgorin bound of roots sharp?

Gershgorin circle theorem tells that the eigenvalues of a matrix $A$ lie in the union of the associated Gershgorin circles. $A=\begin{pmatrix} 0 & 0 & \dots & 0 & -a_0 \\ 1 & 0 ...
2
votes
0answers
277 views

nth-root of continued fraction with Raney transducers

There are some algorithms for doing basic arithmetic by using regular continued fraction expansions. These algorithms are mainly due to Gosper (1972) and Raney (1973). These two approaches use ...
0
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1answer
39 views

Is there closed form solution for this infinite polynomial or high-order polymonial?

The equation is as follows \begin{align} \sum_{N=1}^{\infty}P(N)x^N=Z, \end{align} where $P(N)$'s are real number satisfying $0\leq P(N)\leq 1$. Another equation is \begin{align} \sum_{N=1}^{\bar ...
0
votes
1answer
35 views

Roots of trigonometric equation

In the following trigonometric equation $$1 + \alpha^2 \cos^2 (n \theta) = 0$$ The complex solutions are $$\cos (n \theta) = \pm i/\alpha$$ So I thought that the correspondant angles were $$n ...
1
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1answer
106 views

Numerically find the minimum distance between given point and the curve of given function.

Let $f\colon\Bbb{R}^n\to\Bbb{R}$ and $\mathbf{x}_0\in\Bbb{R}^n$. How could I (numerically) find the minimum Euclidean distance between the curve $f(\mathbf{x})=0$ and $\mathbf{x}_0$, granted that $f$ ...
3
votes
5answers
186 views

Prove that equation $x^6+x^5-x^4-x^3+x^2+x-1=0$ has two real roots

Prove that equation $$x^6+x^5-x^4-x^3+x^2+x-1=0$$ has two real roots and $$x^6-x^5+x^4+x^3-x^2-x+1=0$$ has two real roots I think that: ...
1
vote
2answers
82 views

Given some zeroes of a real polynomial of a given degree, how can one find the remaining zeroes?

Here is what the problem says: If $2$, $-\sqrt{5}$, and $3+i$ are three zeroes of a $5$th degree polynomial function with real coefficients, find the other zeroes of multiplicity $1$. I don't ...
3
votes
2answers
55 views

Elementary bound theorem of a monic real polynomial

An elementary bound theorem on the roots of a real monic polynomial states that $$M := \operatorname{max} (1, |a_0| + \cdots + |a_{n-1}|) := \operatorname{max} (1, B)$$ is an upper and lower ($-M$) ...
1
vote
1answer
61 views

Can a quartic equation be reduced to a cubic/quadratic knowing that two roots are real?

I have a quartic equation that is the determinant of a 4-by-4 matrix that looks like: $det(M-\lambda I) = det \left( \matrix{m_{11}-\lambda & m_{12} & m_{13} & 0 \\ m_{21} & ...