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Questions tagged [roots]

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

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1answer
36 views

Prove that if $ab < 0$ then the equation $ax^{3} + bx + c = 0$ has at most three real roots.

Prove that if $ab < 0$ then the equation $ax^{3} + bx + c = 0$ has at most three real roots. I would need verification on the proof below, thanks! Proof: Let $f(x) = ax^{3} + bx + c.$ Assume ...
2
votes
1answer
29 views

Show that $f_a(z)=z+a-e^z$ has only 1 zero in $Re(z)<0$ and this zero is $<0$. $(a>1)$

I am trying to use Rouche's Theorem somehow but I can't seem to be able to find a proper function to compare $f_a(z)$ with. I tried $g(z)=z+a$ but then I can't deal with the $e^z$ term. Any ...
1
vote
1answer
35 views

How many possible rational roots are there for $2x^4 + 4x^3 - 6x^2 + 15x - 12 = 0$?

The question is as follows: How many possible rational roots are there for $2x^4 + 4x^3 - 6x^2 + 15x - 12 = 0$? A. 4 B. 6 C. 8 D. 12 E. 16 I was taught that a ...
0
votes
3answers
34 views

Is $i$ a 3rd root of $-i$

I was asked to find the 3rd root of -i. Now, since $i^3 = -i$ I thought that I can just extract the 3rd root out of both sides and get $\root3\of{i^3} = \root3\of{-i} \implies i = \root3\of{-i}$. ...
0
votes
3answers
61 views

Show that $ Ax^2 + x + 1 = 0 $ has roots greater than $1$ given $-1<A<0$

Not sure how to answer this question, i tried putting the numbers in the quadratic formula and got $x = 2-2\sqrt{A}/2a$? Any help would be much appreciated.
1
vote
1answer
21 views

Limits Rational Fn missing p and q [on hold]

$f(x)=\frac{4z^3x^2-3x-7z}{3x^2+2x-1}$ Find all values of $z$ for which the limit of $x$ as $x$ goes to $-1$ of $f(x)$ exists?
0
votes
1answer
41 views

Rational $F_n$ Limit $\to$ Find Denominator Polynomial

Given: $P(x)$ is a polynomial such that the following apply: I) $\lim_\limits{x \to 1} \dfrac{P(x)}{(x-1)^3} = 6$ II). $\lim_\limits{x \to -2} \dfrac{P(x)}{x^2+x-2} = 9$ III) $\lim_\...
0
votes
3answers
37 views

If natural number $n$ doesn't has the form of $k^{m}$, for natural $k$ and $m$, then $x^{m}=n$ has no rational root.

Prove: If a natural number $n$ doesn't has the form of $k^{m}$, where $k$ and $m$ are natural numbers, then the equation $x^{m}=n$ has no rational root. How do I start to prove with contradiction (or ...
1
vote
0answers
56 views

Calculate $\lim_{a \to 0}\frac{2^n-(2-a)(2-2^2a)(2-3^2a)…(2-n^2a)}{2^na}$

I have tried every way I know to calculate this end (divide $a^n$, divide $2^n$, derivation) but without any result. Can you help me calculate it? and thank you very much. $$ \lim_{a \to 0}\frac{2^n-...
4
votes
2answers
71 views

Roots of polynomial equation $x^6+2x^5+4x^4+8x^3+16x^2+32x+64=0$

If $x_1,x_2,...,x_6$ be the roots of $x^6+2x^5+4x^4+8x^3+16x^2+32x+64=0$ then I have to show that $|x_i|=2\space\space\space\forall i$ I get that the roots of the equation must be complex, of the ...
3
votes
2answers
38 views

A proof on multinomial roots

If $x_1,x_2,...,x_{n-1},x_n$ be the roots of the equation $$1 + x + x^2 + ... + x^n = 0$$ and $y_1,y_2,...,y_{n},y_{n+1}$ be those of equation $$1 + x + x^2 + ... + x^{n+1} = 0$$ show that $$(1-x_1)(1-...
0
votes
1answer
42 views

Possible factorizations of a polynomial in $\mathbb{Z}_{12}$

I am given the equation $x^3-2x^2-3x=0 \in \mathbb{Z}_{12}$ and I have found the roots of the polynomial to be {$0,3,5,8,9,11$}$\subset \mathbb{Z}_{12}$. Now, I am trying to find all the different ...
0
votes
1answer
59 views

If $\alpha$ and $\beta$ are roots of $(x^2)-(4x)-1=0$, find $\sqrt[3]{\alpha}$+ $\sqrt[3]{\beta}$

My question in handwriting https://i.stack.imgur.com/4vPBs.jpg If $\alpha$ and $\beta$ are roots of this equation $$(x^2)-(4x)-1=0$$ Then find $$\sqrt[3]{\alpha}+\sqrt[3]{\beta}$$ Please do not ...
0
votes
1answer
17 views

Can $\frac{zR(z)-aR(a)}{z-a}$ have an inside root if $R$ does not?

Let $R:\mathbb C \to \mathbb C$ be a rational function, and for some $a\in \mathbb C$ let $$S(z)\equiv\frac{zR(z)-aR(a)}{z-a}.$$ Can $S$ have a root inside the unit circle if $R$ does not?
2
votes
2answers
52 views

Define constant a in way that $x^2+(3a+1)x+81=0$ solutions are complex

Problem Define constant $a$ in way that $x^2+(3a+1)x+81=0$ solutions are complex. After that define $a$ in way that the solutions are strictly imaginary (when real part is $0$) Attempt to solve ...
6
votes
5answers
48 views

Find all complex numbers $z$ that satisfy equation $z^3=-8$

Problem Find all complex numbers $z$ that satisfy equation $z^3=-8$ Attempt to solve The real solution is quite easily computable or more specifically complex solution where imaginary part is ...
0
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0answers
12 views

Finding roots of a specific equation of higher degree.

I want to find the roots of $(1-\frac jk)(x^{k-1}+x^{k-2}+...+x^{k-j})-{\frac jk}(x^{k-j-1}+x^{k-j-2}+...+x+1)=0$ ,where $k,j$ are constant and $j\le k$. Is there any method to find all roots? Is it ...
2
votes
1answer
94 views
+50

Best (fastest) method for root finding of 6th degree polynomial

Given a polynomial on the form: $$0=x^{6}+k_1{1}x^{5}+k_2x^{4}+k_3x^{3}+k_4x^{2}+k_5x+k_6$$ Whats the best(read: fastest) numerical root finding algorithm ? And one which should include complex ...
0
votes
2answers
47 views

I am trying to find all of the roots (real and imaginary) of this polynomial.

I am trying to find all of the real and imaginary roots of this polynomial. $$y=9x^7-x^6-4x^5+2x^4-2x.$$
1
vote
1answer
28 views

Find number of elements $z$, $1<|z|<2$ satisfying $f(z)=0$ where $f(z)=z^5+z^3+5z^2+2$.

Let $g(z)=z^5$ then $\forall z$ s.t $|z|=2$ $|g(z)-f(z)|\leq |2|^3+5|2|^2+2< |2|^5=|g(z)|$ Therefore, By Rouché's $f(z) $ has 5 zeroes in $D(0,2)$. I do not know how to proceed any further. ...
1
vote
0answers
22 views

Satisfying a specified accuracy in a numerical method

In an iterative root finding numerical method, with an assumption of convergence, with every iteration, we attain more and more correct digits of the true root. If I obtain three correct decimal ...
2
votes
3answers
160 views

Proving that a polynomial has no roots on the unit circle

I want to prove that if $|z|=1 $ then $z^8-3z^2+1 \neq 0$. I tried to prove the reciprocal by taking norms in $z^8-3z^2+1= 0$ and then solving for $ |z|$ but it does not work. I also assume $| z|=1 $...
3
votes
2answers
43 views

Approximation of square roots

Recently, I've seen a YouTube video where they approximate square roots real quick. They use this approximation : $$\sqrt{x} \approx \lfloor \sqrt{x} \rfloor+\dfrac{x-(\lfloor \sqrt{x} \rfloor)^2}{2\...
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votes
4answers
55 views

I have a problem solving a system of equations. The problem is that I don't know how to continue the process.

I need help with a problem based on system of equations: $$ \left\{ \begin{array}{c} -1=(-b-\sqrt{b^2-4ac})/2a \\ 3=(-b+\sqrt{b^2-4ac})/2a \\ \end{array} \right. $$ I tried to solve it and I ...
4
votes
3answers
123 views

Roots of $2x^3-4x+1$

I'm having difficulty getting the solution to the cubic equation $2x^3-4x+1=0$ and from http://www2.trinity.unimelb.edu.au/~rbroekst/MathX/Cubic%20Formula.pdf it claims that the general solution to $...
8
votes
1answer
114 views

Visualization of p-adic numbers

I try to understand and get a feeling which gaps p-adic numbers fill to complete $\mathbb{Q}$. In the course of this I depicted (for $p = 2$) the "base" $\{p^k\}_{k\in\mathbb{Z}}$ with respect to ...
0
votes
1answer
23 views

What does “$f(x)$ has a root of order $k$” mean?

I'm confused by the wording of this, the question states: Assume $f(x)$ has a root of order $k=3$ at $x(n)$. What does the "root of order $k=3$" part mean? Thanks
0
votes
1answer
63 views

If $x-\sqrt{\frac7x}=8$, find $x-\sqrt{7x}$. [closed]

If $x-\sqrt{\frac7x}=8$, find $x-\sqrt{7x}$.
15
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1answer
109 views

Number of real roots of an iterated quadratic: $x^2-3/2$

I was messing around with polynomials and their real roots when I, as recreational mathematicians do, asked myself the following random question: Suppose I am given a polynomial $P(x)$. How can I ...
0
votes
3answers
86 views

How do I prove that a function has real roots?

I want to prove the existence of real roots of a function, not solve the function for the roots. I am aware of discriminant, but that is restricted to quadratic functions. I am aware of the ...
0
votes
3answers
68 views

Find a polynomial in terms of cos $θ$

The question gives a hint; first, 'find the general solution for $cos$ $5θ = cos$ $4θ$' So far my progress with this hint is; $ 5θ = 2πk + 4θ $ or $ 5θ = 2πn - 4θ $ $ θ = 2πk$ or $ 9θ = 2πn $ , ...
1
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4answers
880 views

Understanding the proof of Cauchy-Schwartz inequality

While reading the proof of Cauchy-schwarz inequality, I didn't get one step. The step is as below, by positivity axiom, for any real number $t$ $0≤⟨tu+v,tu+v⟩=⟨u,u⟩t^2+ 2⟨u,v⟩t+⟨v,v⟩$ This imply $$...
0
votes
2answers
88 views

Roots of the equation $x^3+15x^2+cx+860=0$

If $-5+i\beta$ , $-5+i\gamma$ ,$\beta^2\ne\gamma^2$; $\beta,\gamma \in R$ are the roots of the equation $x^3+15x^2+cx+860=0$, $c\in R$, then find the three roots of the equation. My approach is as ...
1
vote
1answer
29 views

Verify that $x$ is a fixed point

Given the function $f(x) = {-x^4 \over 4} + x^3 -4x + 4$ I have graphically localized two roots $\alpha$ and $\beta$ (with $\alpha < \beta$). After analyzing them with Newton's algorithm I'm given ...
2
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0answers
22 views

Real roots of Chromatic Polynomial

This question consist of two parts: The first one seems much easier. $1.$ The only real $x<1$ which can be a root of a chromatic polynomial is $0$ $2.$ No real root of a chromatic polynomial can ...
2
votes
1answer
44 views

Problems with Taylor Series to Approximate Square Roots

I am currently looking into comparing different methods to compute the value of square roots, and I've decided to compare two relatively well known methods - the famous Babylonian Method and the ...
0
votes
5answers
58 views

How to solve the roots of following cubic equation $a^3-6a^2+9a-4=0$ [closed]

How to solve the roots of following cubic equation $$a^3-6a^2+9a-4=0$$ I am solving roots of characteristics equation so I use casio 991ms calculator so it gave me roots $-0.355301,3.177650,3....
8
votes
4answers
228 views

Finding the sum of squares of the real roots

Let $r_1,r_2,r_3,\cdots,r_n$ be the distinct real zeroes of the equation $$x^8-14x^4-8x^3-x^2+1=0.$$ Then $r_1^2+r_2^2+r_3^2+\cdots+r_n^2$ is $$(A)\,3\quad(B)\,14\quad(C)\,8\quad(D)\,16$$ I can get ...
0
votes
0answers
35 views

Limits of square root of sum

Does the square root have an "averaging" property? That is, for a square root of a sum, will outliers be more "irrelevant", the more summands we add? My intuition tells me yes, but I can't figure out ...
10
votes
3answers
584 views

Finding the sum of squares of roots of a quartic polynomial.

What is the sum of the squares of the roots of $ x^4 - 8x^3 + 16x^2 - 11x + 5 $ ? This question is from the 2nd qualifying round of last year's Who Wants to be a Mathematician high school competition ...
0
votes
1answer
36 views

How to prove the following statement from the solution manual of Convex Optimization by Boyd & Vandenberghe?

This question is related to Problem 3.51 of the book. Consider a polynomial $p(x)$ which has $n$ roots i.e. $$p(x)=(x-s_1)(x-s_2)\cdots(x-s_n)$$ where we assume w.l.o.g. that $s_1\leq s_2\cdots \leq ...
0
votes
3answers
60 views

For what values of $m$ is there a common root to $mx^2+2x+1=0$ and $x^2+2x+m=0$? [closed]

If the equation $mx^2+2x+1 = 0$ and $x^2+2x+m = 0$ have a common root, find the possible values of $m$ and the value of the common root in each case.
4
votes
3answers
78 views

Solving roots of a polynomial on $\mathbb{Z}_p$

This is probably a simple question. and I would like to work out an example. How do we solve $x^2 + x + [1] = 0$ over the field $\mathbb{Z}_7$? I tried a simple case, for example: $[4] x - [3] = 0$, ...
0
votes
3answers
40 views

Number of Real Roots of a polynomial equation

Let $f(x) = x^5-x^4+x^3-4x^2-12x $ How many real zeros are there? I tried factoring it but couldn't find a way. I tried using Descartes rules of signs and it said that it has 1 negative root and 4 ...
0
votes
3answers
31 views

The set of numbers of the following form that are cubes

I have found this problem in a 10th grade textbook and it's given me headaches trying to solve it. It says, determine the set: $$ A = \left \{ x \in \mathbb Z| \root3\of{\frac{7x+2}{x+5}} \in \mathbb ...
5
votes
1answer
57 views

How to show a specific function has a unique root?

Let $f(x)=\frac{\left(\sqrt{x}+3\right) \log \left(\frac{x+3}{4}\right)-\sqrt{x} \log (x)}{\left(\sqrt{x}-1\right)^3 \sqrt{x}}$, where $x>0$ and the $\log$ is the natural logarithm (with base $e$). ...
0
votes
1answer
54 views

How do I find roots of $x^2 + bx^\frac{1}{2} + c$?

I thought I might substitute $u = x^\frac{1}{2}$, to get $u^4 + bu + c$, but I don't really know what to do with that either. I plugged both into Wolfram Alpha, which provided some really long ...
1
vote
1answer
56 views

Newton's method to find an update rule to compute $\frac{1}{y}$ given $f(x)=\frac{1}{x}-y$

I have to use Newton's method to derive an update rule for finding a root of the form $\frac{1}{y}$ given a specific $f(x)$, where $f(x)=\frac{1}{x}-y$. From the given, $\frac{1}{y}$ is a valid root ...
0
votes
0answers
63 views

What is the formula for $a_m$ in the cosine like and sine like functions for nested radical constants?

What is the general formula for a_m in the following cosine-like and sin-like functions? eta 9/27: haha.. so now I can mention what I'm fishing for, since another Riemanned already. Are all the ...
0
votes
3answers
73 views

Complex Roots $\alpha$ and $\beta$ satisfy the equation $(x-\alpha)(x-\beta) = 0$ but not $(x+\alpha)(x+\beta) = 0$

I would like to ask a question about Complex Roots in Further Mathematics. I am new to the subject and one of the statements given in the book without further explanation is If the roots of the ...