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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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Rational roots of polynomials

Can one construct a sequence $(a_k)_{k\geqslant 0}$ of rational numbers such that, for every positive integer $n$ the polynomial $a_nX^n+a_{n-1}X^{n-1}+\cdots +a_0$ has exactly $n$ distinct rational ...
101
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5answers
6k views

Why are the solutions of polynomial equations so unconstrained over the quaternions?

An $n$th-degree polynomial has at most $n$ distinct zeroes in the complex numbers. But it may have an uncountable set of zeroes in the quaternions. For example, $x^2+1$ has two zeroes in $\mathbb C$,...
86
votes
6answers
11k 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 ...
75
votes
4answers
7k views

Are these solutions of $2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}}$ correct?

Find $x$ in $$ \Large 2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}}$$ A trick to solve this is to see that $$\large 2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}} \quad\implies\quad 2 = x^{\Big(x^{x^{x^{...
71
votes
5answers
2k views

Polynomials such that roots=coefficients

Here is my question : Are there monic polynomials with degree $\geq 5$ such that they have the same real all non zero roots and coefficients ? Mathematically, prove or disprove the existence ...
58
votes
2answers
3k views

Denesting radicals like $\sqrt[3]{\sqrt[3]{2} - 1}$

The following result discussed by Ramanujan is very famous: $$\sqrt[3]{\sqrt[3]{2} - 1} = \sqrt[3]{\frac{1}{9}} - \sqrt[3]{\frac{2}{9}} + \sqrt[3]{\frac{4}{9}}\tag {1}$$ and can be easily proved by ...
55
votes
9answers
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Why does the discriminant in the Quadratic Formula reveal the number of real solutions?

Why does the discriminant in the quadratic formula reveal the number of real solutions to a quadratic equation? That is, we have one real solution if $$b^2 -4ac = 0,$$ we have two real solutions if $...
48
votes
5answers
2k views

AM-GM-HM Triplets

I want to understand what values can be simultaneously attained as the arithmetic (AM), geometric (GM), and harmonic (HM) means of finite sequences of positive real numbers. Precisely, for what points ...
44
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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 ...
41
votes
2answers
2k views

Why does this distribution of polynomial roots resemble a collection of affine IFS fractals?

Consider the following spectacular image, created by Sam Derbyshire and described in John Baez's article "The Beauty of Roots": In this image are plotted all the complex roots of all polynomials of ...
40
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3answers
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Why is it so hard to find the roots of polynomial equations?

The question that follows was inspired by this question: When trying to solve for the roots of a polynomial equation, the quadratic formula is much more simple than the cubic formula and the cubic ...
37
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11answers
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Why can a quadratic equation have only 2 roots?

It is commonly known that $ax^2+bx+c=0$ have two solutions $\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ but how to prove that another root couldn't exist? I think derivation of quadratic formula is not enough......
37
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3answers
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Monic polynomials whose roots are their remaining coefficients

I was bored in class one day and wondered to myself if there were any quadratics $x^2+ax+b$ such that $a$ and $b$ are the zeros. I found two: $x^2+x-2,$ and $x^2 -{1\over2}x -{1\over2}$. The comments ...
35
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11answers
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Is There A Polynomial That Has Infinitely Many Roots?

Is there a polynomial function $P(x)$ with real coefficients that has an infinite number of roots? What about if $P(x)$ is the null polynomial, $P(x)=0$ for all x?
35
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4answers
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Is there an intuitive way of visualising complex roots?

Consider the function $f(x)$ such that $f(x) = x^2-4x+13$. By considering the discriminant, it can immediately be seen that the function has no real roots, since $b^2-4ac = (-4)^2-4(13) = -36$ and $-...
34
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3answers
2k views

Relation between the roots of a function

I have this question from my exam where it's asked to find the sum: $$S=\sum_{k=1}^n \frac{1}{(1-r_k)^2}$$ where $r_k$ are the roots of $$f(x)=x^n-2x+2\quad,n\ge3$$ I recalled this relation $$\frac{f'(...
33
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2answers
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Something strange about $\sqrt{ 4+ \sqrt{ 4 + \sqrt{ 4-x}}}=x$ and its friends

This post can be generalized to, $$\begin{align} \sqrt{ 2+ \sqrt{ 2 + \sqrt{ 2-x}}}=x&,\quad\quad x = -2\cos\left(\frac{8\pi}{9}\right)=1.8793\dots\quad\quad\quad \\ \\ \sqrt{ 4+ \sqrt{ 4 + \sqrt{...
32
votes
4answers
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$p_n(x)=p_{n-1}(x)+p_{n-1}^{\prime}(x)$, then all the roots of $p_k(x)$ are real

$p_0(x)=a_mx^m+a_{m-1}x^{m-1}+\dotsb+a_1x+a_0(a_m,\dotsc,a_1,a_0\in\Bbb R)$ is a polynomial, and $$p_n(x)=p_{n-1}(x)+p_{n-1}^{\prime}(x),\qquad n=1,2,\dotsc$$ then, there exist $N\in\Bbb N$, such ...
32
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4answers
888 views

Patterns of the zeros of the Faulhaber polynomials (modified)

Faulhaber polynomial of order $p \in \Bbb{N}$ is defined as the unique polynomial of degree $p+1$ satisfying $$ S_{p}(n) = \sum_{k=1}^{n} k^p $$ for $n = 1, 2, 3, \cdots$. For example, \begin{align*...
31
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4answers
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Is it possible for a quadratic equation to have one rational root and one irrational root?

Is it possible for a quadratic equation to have one rational root and one irrational root? Yes, a pretty straightforward question. Is it possible?
31
votes
1answer
875 views

Distribution of roots of complex polynomials

I generated random quadratic and cubic polynomials with coefficients in $\mathbb{C}$ uniformly distributed in the unit disk $|z| \le 1$. The distribution of the roots of 10000 of these polynomials are ...
29
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3answers
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Galois groups of polynomials and explicit equations for the roots

Lets say I have calculated the galois group of some polynomial and I also have the subgroup structure. What's an effective procedure to turn the group into equations for the actual roots of the ...
29
votes
2answers
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Countability of the zero set of a real polynomial

This is the question from my calculus homework: Is it possible for a polynomial $f\colon\, \mathbb{R}^{n}\to \mathbb{R}$ to have a countable zero-set $f^{-1}(\{0\})$? (By countable I mean countably ...
28
votes
4answers
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Why does Wolfram Alpha say the roots of a cubic involve square roots of negative numbers, when all three roots are real?

Can someone please explain Wolfram Alpha's response to the equation $x^3-3x-1=0$? The roots are real and graphic Wolfram itself shows them (they are approximately equal to $-1.5321, -0.3473$ and $1....
27
votes
6answers
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Function with no roots

Given a non-constant function $f(x)$, is it possible for it to have no zeroes (neither real nor complex)? Say for example, $f(x)=\cos x-2$, does a complex solution exist for this because for real $x$,...
27
votes
4answers
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Is there a simple explanation why degree 5 polynomials (and up) are unsolvable?

We can solve (get some kind of answer) equations like: $$ ax^2 + bx + c=0$$ $$ax^3 + bx^2 + cx + d=0$$ $$ax^4 + bx^3 + cx^2 + dx + e=0$$ But why is there no formula for an equation like $$ax^5 + ...
27
votes
1answer
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Showing that the roots of a polynomial with descending positive coefficients lie in the unit disc.

Let $P(z)=a_0+a_1z+\cdots+a_nz^n$ be a polynomial whose coefficients satisfy $$0<a_0<a_1<\cdots<a_n.$$ I want to show that the roots of $P$ live in unit disc. The obvious idea is to use ...
27
votes
1answer
799 views

The Modulus of all the roots of a Polynomial are equal to $1$

Suppose the real number $\lambda \in (0,1)$, and let $n$ be a positive integer. Prove that all roots of the polynomial $$f\left ( x \right )=\sum_{k=0}^{n}\binom{n}{k}\lambda^{k\left ( n-k \right )}x^{...
26
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3answers
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Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least n+1 zeros on $(-1,1)$

Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$ such that $f(x)=0$ on $\mathbb{R}$\ $(-1,1)$. Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least n+1 zeros on $(-1,1)$ My attempt : ...
25
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8answers
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How to prove that a polynomial of degree $n$ has at most $n$ roots?

How can I prove, that a polynomial function $$f(x) = \sum_{0\le k \le n}a_k x^k\qquad n\in\mathbb N,\ a_k\in\mathbb C$$ is zero for at most $n$ different values of $x$? (Except $n=0$ where $f(x)$ is $...
25
votes
2answers
2k views

Cubic formula gives the wrong result (triple checked)

I'd like to solve $ax^3 + bx^2 + cx + d = 0$ using the cubic formula. I coded three versions of this formula, described in three sources: MathWorld, EqWorld, and in the book, "The Unattainable ...
24
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6answers
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What does this converge to and why?

What does the below expression converge to and why? $$ \cfrac{2}{3 -\cfrac{2}{3-\cfrac{2}{3-\cfrac2\ddots}}}$$ Setting it equal to $ x $, you can rewrite the above as $ x = \dfrac{2}{3-x} $, which ...
23
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3answers
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Influence of small constant term on roots of polynomial

Let $p(x)=a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0,\enspace a_i\in\mathbb{C}$ some polynomial. Suppose that $|a_0|$ is very small (compared to the other coefficients' magnitude). Is there any ...
23
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2answers
964 views

How to show that a root of the equation $x (x+1)(x+2) … (x+2009) = c $ can have multiplicity at most 2?

How to show that a root of the equation $$x (x+1)(x+2) ....... (x+2009) = c $$ can have multiplicity at most 2 , and to find the value of $ c $ for which this is possible. I proceeded by using the ...
23
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3answers
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Square roots — positive and negative

It is perhaps a bit embarrassing that while doing higher-level math, I have forgot some more fundamental concepts. I would like to ask whether the square root of a number includes both the positive ...
22
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5answers
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Roots of Legendre Polynomial

I was wondering if the following properties of the Legendre polynomials are true in general. They hold for the first ten or fifteen polynomials. Are the roots always simple (i.e., multiplicity $1$)? ...
22
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2answers
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Must imaginary roots come in conjugate pairs?

I am aware that for any polynomial with real coefficients, the imaginary roots (if there are any) must come in complex pairs. However, I always believed that you could still have an imaginary number - ...
22
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2answers
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Why does the discriminant of a cubic polynomial being less than $0$ indicate complex roots?

The discriminant $\Delta = 18abcd - 4b^3d + b^2 c^2 - 4ac^3 - 27a^2d^2$ of the cubic polynomial $ax^3 + bx^2 + cx+ d$ indicates not only if there are repeated roots when $\Delta$ vanishes, but also ...
20
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3answers
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Probability of $ax^2 + bx + c = 0$ having real solutions

$a$, $b$, $c$ are random integer numbers between $1$ and $100$ (including $1$ and $100$, and uniformly distributed). What is the probability that the equation $ax^2 + bx + c = 0$ has real solutions? ...
19
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6answers
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Can $x^3+3x^2+1=0$ be solved using high school methods?

I encountered the following problem in a high-school math text, which I wasn't able to solve using factorization/factor theorem: Solve $x^3+3x^2+1=0$ Am I missing something here, or is indeed a more ...
19
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3answers
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On applying the quadratic formula to a first-degree equation

You're probably thinking, "Why?" Please let me explain... It is (very) well-known that $$ \forall (a,b,c,x) \in \mathbb{C}^* \times \mathbb{C}^3: ax^2 + bx + c = 0 \Leftrightarrow x = \frac{-b \pm \...
19
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1answer
878 views

Can we permute the coefficients of a polynomial so that it has NO real roots?

Let $P(x)=a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\ldots+a_{0}$ be an even degree polynomial with positive coefficients. Is it possible to permute the coefficients of $P(x)$ so that the resulting polynomial ...
19
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1answer
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Using Vieta's theorem for cubic equations to derive the cubic discriminant

Background: Vieta's Theorem for cubic equations says that if a cubic equation $x^3 + px^2 + qx + r = 0$ has three different roots $x_1, x_2, x_3$, then $$\begin{eqnarray*} -p &=& x_1 + x_2 +...
19
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1answer
298 views

Roots of polynomials with bounded integer coefficients

For $a\in\mathbb N$, let $R_a\subseteq\mathbb{R}$ be the set of real roots of polynomials whose coefficients are integers with absolute value at most $a$. $$R_a=\left\{r\in\mathbb{R}\middle|\sum_{i=0}...
18
votes
3answers
928 views

Find all roots of $\,(x + 1)(x + 2)(x + 3)^2(x + 4)(x + 5) = 360$

The question is to find all complex roots of $$(x + 1)(x + 2)(x + 3)^2(x + 4)(x + 5) = 360$$ and it is meant to be solved by hand. Is there any quick way to solve this using some trick that I'm not ...
18
votes
1answer
342 views

Number of real zeroes of iterated polynomial: $x^3-2x+1$

If $P(x)=x^3-2x+1$, define $z_n$ as the number of real roots of the polynomial $P^{\circ n}(x)$, where the superscript denotes $n$-fold composition. Can we find a general formula for $z_n$, or perhaps ...
18
votes
1answer
295 views

Find all polynomials $\sum_{k=0}^na_kx^k$, where $a_k=\pm2$ or $a_k=\pm1$, and $0\leq k\leq n,1\leq n<\infty$, such that they have only real zeroes

Find all polynomials $\sum_{k=0}^na_kx^k$, where $a_k=\pm2$ or $a_k=\pm1$, and $0\leq k\leq n,1\leq n<\infty$, such that they have only real zeroes. I've been thinking about this question, but I'...
17
votes
4answers
10k 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 ...
17
votes
3answers
917 views

How would you find the exact roots of $y=x^3+x^2-2x-1$?

My friend asked me what the roots of $y=x^3+x^2-2x-1$ was. I didn't really know and when I graphed it, it had no integer solutions. So I asked him what the answer was, and he said that the $3$ roots ...
17
votes
1answer
5k views

Derivation of asymptotic solution of $\tan(x) = x$.

An equation that seems to come up everywhere is the transcendental $\tan(x) = x$. Normally when it comes up you content yourself with a numerical solution usually using Newton's method. However, ...