This tag is used for both basic and advanced questions on polynomials in any number of variables. Including, but not limited to: solving for roots, factoring, checking for irreducibility. This tag is rarely used as the only tag for a question.

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3
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3answers
145 views

Polynomial roots problems.

$$ X^5-55X+21$$ Prove that the given polynomial has 2 roots which satisfy the condition: $$X_1X_2=1$$ and find them. I have tried to make use of Viette's relations ,but couldnt get to a satisfying ...
1
vote
2answers
32 views

Finding parameters for $f(x) = x^2 + 2(m − a)x + 3am −2 = 0$ that satisfy a condition.

1.find a for $f(x) = x^2 + 2(m − a)x + 3am −2 = 0$ such that for every m real, f has real roots 2.find m such that for every a real, f has real roots My ideea is to demonstrate that $\delta=4(m-a)^2 ...
-2
votes
2answers
38 views

$p,q$ coprime polynomials - are $p^n,q^m$ coprime? [on hold]

Suppose that $p,q \in K[x]$ are coprime (there is no polynomial that divides both) and let $n,m \in\mathbb{N}$. Are $p^n$ and $q^n$ coprime and, if so, how to prove it?
0
votes
1answer
37 views

roots of a polynomial with zero coefficient summation [on hold]

Consider a polynomial, for which the summation of the coefficients is zero.What do we know about its roots?
2
votes
0answers
50 views

On a family of polynomials related to the expansion of $(1+\epsilon)^{x/\epsilon}$ as a series in $\epsilon$

Consider the sequence of polynomials $(P_n)_{n\geqslant0}$ uniquely defined by the recursion $$(P_n)'=\sum_{k=0}^{n-1}\frac{P_k}{n-k+1},$$ valid for every $n\geqslant0$, with the initial conditions ...
0
votes
1answer
19 views

Prove that $\text{det}(A)=p_1p_2-ba={bf(a)-af(b)\over b-a}$

Let $f(x)=(p_1-x)\cdots (p_n-x)$ $p_1,...p_n\in \mathbb R$ and let $a,b\in \mathbb R$ such that $a\neq b$ Prove that $\text{det} A={bf(a)-af(b)\over b-a}$ where $A$ is the matrix: ...
0
votes
0answers
22 views

Use Galois theory to find all complex roots of $T^4-2T^2-\sqrt{6}T+\frac{3}{4}$

I am currently studying Galois theory and a question that often comes up is "find all complex numbers which are roots of the polynomial $T^4+aT^2+bT+c$" where the coefficients are of the form ...
1
vote
1answer
42 views

Conics and conics of the form $ax^2+by^2+c=0$

The problem of finding rational points on conics is usually discussed (for example in the book of Silverman and Tate) for conics of the form $ax^2+by^2+c=0$. I assume that those conics are in ...
0
votes
0answers
13 views

Construction of polynomials in sagemath

This post is the mathematical part of a question I asked on Stackoverflow, which does not have $\LaTeX$. The question in here : ...
0
votes
1answer
15 views

Composition with polynomial/ same type of singularity

Let $f\in O(D_1(0){}-\{0\})$ and $ p $ a non constant polynomial. Then $f$ and $p(f)$ have the same type of singularity at $z_o=0 $. I think its fairtly easy to Show that if $f$ has a ...
1
vote
1answer
56 views

Square matrix over $\mathbb{Z}$ can't have $\frac{1}{4}(-3+ i \sqrt5)$ as an eigenvalue

Prove square matrix over $\mathbb{Z}$ can't have $\frac{1}{4}(-3+ i \sqrt5)$ as an eigenvalue. My proof: If matrix has eigenvalue z=$\frac{1}{4}(-3+ i \sqrt5)$, then it must has eigenvalue ...
-1
votes
1answer
14 views

Help with proving or disproving

True or false? If $\frac{a_0}{1}+\frac{a_1}{2}+\ldots+\frac{a_n}{n+1}=0$ then there is $x\in(0,1)$ that solves $a_0+a_1 x +\ldots +a_n x^n =0$.
2
votes
2answers
33 views

Determining a constant from polynomial

Given that $x^4= (x-c)^2$ which c is a real constant number. If the above solution has four real roots. Hence, c must be a. $$-\frac{1}{4}\le c \le \frac{1}{4}$$ b.$$c \le -\frac{1}{4}$$ c.$$ c ...
1
vote
0answers
14 views

Number of terms in multivariate polynomial

We know that the number of terms in a univariate polynomial of degree n is n+1. But what about if there are multiple variables: for eg: for variables $x,y$ polynomial of degree 2 will have: ...
1
vote
0answers
20 views

Polynomial division for identifying an expression in terms of complex numbers.

This question is blatantly copied from here, for the sake of learning more I specify it a bit more: $$f(z)= (3x^2 + 2x - 3y^2 - 1) + i(6xy + 2y)$$ $$z = a+bi$$ I want to write $f(z)$ in terms of ...
2
votes
0answers
44 views

How can I solve for the roots of a polynomial that has no algebraic solution?

I've learned that polynomials of degree >= 5 (i.e. $x^5 - x -1$) are not necessarily solvable in the radicals, due to the Abel-Ruffini Theorem. My question is: given that you can't solve a polynomial ...
0
votes
0answers
16 views

How to find critical points in a cubic function in two variables?

Given a cubic function $f$ in two variables $x$ and $y$ $$ f(x,y)=\sum_{i=0}^3 \sum_{j=0}^3 k_{i,j}x^i y^j, $$ I would like to find the points ($x,y$ pairs) where $\nabla f = \mathbf{0}$. Since $f$ ...
0
votes
3answers
40 views

Is there any way to calculate the roots of this polynom?

I need to calculate the roots of the real function $f$: $$ f(x)=\frac{-{x}^{3}+2{x}^{2}+4}{{x}^{2}} $$ But I am not able to decompose the numerator. There should be only one real solution and two ...
2
votes
2answers
48 views

Irreducibility of polynomials in $\mathbf{Z}_p[x]$ - understanding proofs

I am reading through some irreducibility proofs and there's something I don't quite understand: $x^3+2x+1$ is irreducible in $\mathbf{Z}_3[x]:$ no roots in $\mathbf{Z}_3$ and degree $3$ so ...
1
vote
4answers
30 views

Find the condition such that the roots of the polynomial are in AP

$f(x)=x^3+3px^2+3qx+r$ has roots in AP.Find the relation between $p,q$ and $r$. [Answer:$-2p^2-3pq+r=0$] My attempt:- Taking $d$ as the common difference of the roots in AP ...
1
vote
0answers
56 views

proving that $8x^3-6x-1$ is irreducible over $\mathbb{Q}$

When considering the impossibility of trisecting the 60 degree angle, one comes across the polynomial $f(x)=8x^3-6x-1$, which I want to prove is irreducible over $\mathbb{Q}$. I reduced the ...
0
votes
0answers
41 views

How many elements are in the field of fractions $\Bbb Z_3(t)$?

As in exercise for my Galois Theory course I am supposed to find the number of elements in the field of fractions $\Bbb Z_3(t)$. I am very confused as to how to approach this question because I ...
0
votes
0answers
14 views

Determining positivity of a blackbox multivariate polynomial

Is there a way to check the positivity (or non-negativity) of a multivariate polynomial $f: \mathbb{R}^n \to \mathbb{R}$, of a given degree $d$, by querying the value of $f$ at finitely many points?
1
vote
1answer
25 views

Higher degree polynomial with complex roots

I'm working on the following problem: $$ r^4 - 3r^2 -4r = 0 $$ I factor out one $r$ and leaving me $ r(r^3 - 3r -4) = 0 $. One real root is $r=0$, and I'm unable to find the other ones. I tried ...
1
vote
1answer
36 views

Coprime polynomials in $k[x,y]$ are also coprime in $k(y)[x]$

Let $f,g \in k[x,y]$ be polynomials with no common factor. Prove that when viewed as elements of $k(y)[x]$ they still do not have a common factor. Say we have $f=\sum a_{ij}x^iy^j,\ g=\sum ...
0
votes
0answers
10 views

Proof of $x ≤ \max { (k | c_{m1} |^{1/m1} , k | c_{m2} |^{1/m2} , . . . ,k | c_{m2} |^{1/mk})} .$

Can you please give a proof of this lemma : Let $P (X)$ be an univariate polynomial of degree n $: P (X) = X^{n} + c_{1} X^{n-1} + . . . + c_{n}$ with $ c_{n} \neq 0$. Let $c_{m1} , c_{m2} , . . . , ...
4
votes
1answer
33 views

Discriminant of a trinomial $x^n+ax^m+b$

I am trying to compute the discriminant of the trinomial $x^n+ax^m+b$. I have tried using resultants but cannot see how to approach it. Any hints?
1
vote
1answer
35 views

When does a binomial have repeated roots mod p?

Given a polynomial $f(x)=x^n+a$, and I have that $p$ does not divide $an$, can I show that $f(x)\pmod p$ has no repeated roots? I'm not sure how to proceed.
0
votes
1answer
29 views

Find a recurrence relation and the Fourier-Legendre Series

Rodrique's Formula for the $n$th Legendre Polynomial is $$P_n\left(x\right)=\dfrac{1}{2^nn!}\dfrac{d^n}{dx^n}\left(\left(x^2-1\right)^n\right)$$ The Fourier-Legendre series of a function f is ...
2
votes
1answer
37 views

If $P$ is an integer polynomial with $P(1)=P(2)=0$, then some coefficient is less than $-1$

Let $P (x)$ be a polynomial with integer coefficients. It is known that the numbers $1$ and $2$ are its roots. Prove that there exists a coefficient that is less than $-1$. My work so far: Let ...
0
votes
0answers
21 views

Integral computation with Mathematica and Sympy differ

To compute the integral: $I = \int_{0}^{+oo} ue^{Au^{2}+Bu}du$ where $A<0$ and $B>0$ I have tried both Mathematica and Sympy but they yield different results: Mathematica yields: $ I = ...
1
vote
0answers
20 views

Polynomial division in factored form

Given two factored polynomials of the same degree $N$: $$ \begin{align} P(x) &= \prod_{k=1}^{k=N} (x - p_k) \\ Q(x) &= \prod_{k=1}^{k=N} (x - q_k) \end{align} $$ Due to $P$ and $Q$ having ...
1
vote
1answer
48 views

Stirling's approximation from Euler-Maclaurin formula

I try to derive Stirling's approximation from Euler-Maclaurin formula with form: ...
1
vote
0answers
29 views

Lower bound on the difference between max. and min. values of a polynomial over $[-1, 1]$

Problem: $P(x)$ be a monic, n-degree polynomial with real coefficients. Prove that it is not possible that for all $t \in [-1, 1]$, $$\frac{-1}{2^n} < P(x) < \frac{1}{2^n}$$. I tried it to put ...
4
votes
1answer
37 views

Graphically solving for complex roots — how to visualize?

So recently we've been doing the complex roots of quadratics, cubics and polynomials in general in school. But my question is, is there a way to see where these roots are, just like you can see where ...
0
votes
1answer
29 views

Kernel of a polynomial with matrix, $ker(p(A))$

Let $A\in Mat(3,3,\mathbb R)$ a matrix and $\chi_A(x)=p_1(x)\cdot p_2(x)$ the characteristic polynomial. Evaluate $ker(p_1(A))$.$$A=\begin{pmatrix} 0 & 0 & 2 \\ 1 & 0 & 1\\ 0 & ...
6
votes
1answer
29 views

Iteration of polynomial has only positive roots

Let $P(x)$ be a real polynomial with a positive leading coefficient, and $k\geq 2$ an integer. Suppose that $Q(x)=P(P(\dots(P(x))\dots))$, where there are $k$ iterations of $P$'s, has at least one ...
4
votes
5answers
115 views

Roots of $x^{101}-100x^{100}+100=0$

I do not know how to prove that $x^{101}-100x^{100}+100=0$ has exactly two positive roots. Some can give me hint for solving this please. Thanks for your time.
0
votes
0answers
18 views

Loops around 0 of polynomial restricted to the unit circle [duplicate]

Given a polynomial with coefficients in C, consider the image of the polynomial restricted to the unit circle (That is plugging in only things with absolute value one). How many loops around 0 can ...
13
votes
3answers
531 views

What is the degree of the zero polynomial and why is it so? [duplicate]

My teacher says- The degree of the zero polynomial is undefined. My book says- The degree of the zero polynomial is defined to be zero. Wikipedia says- The degree of the zero ...
1
vote
0answers
13 views

finding generating function of orthogonal polynomials through their moments

I was studying a method to find the generating function of Orthogonal Polynomials through its moments. Please refer to the paper Use of Hermite's method to obtain generating functions for classical ...
0
votes
0answers
25 views

A simple Lagrange interpolation-type identity

I am unable to prove an identity that looks very much like the Lagrange interpolation identity, Problem: Given $f(x)$ is a monic, $n-1$ degree polynomial and $a_1, a_2, \cdots a_n$ distinct real ...
0
votes
1answer
13 views

How does variable ordering in expressions work when creating functions from an equation?

I'm having a really hard time understanding some aspects of functions, i've tried looking around on Khan academy and haven't quite found something to answer my question, i'm sure i'm overlooking ...
3
votes
3answers
67 views

Need help solving $x^4-3x^3-11x^2+3x+10=0$

Solve $x^4-3x^3-11x^2+3x+10=0$ I have tried to solve this equation using 'general formula from roots' from https://en.wikipedia.org/wiki/Quartic_function. $$ax^4+bx^3+cx^2+dx+e=0$$ $$x_{1,2}=-\frac ...
1
vote
1answer
39 views

Is expression $(1/x)/(2/x^2)$ is fraction expression or rational expression?

A rational expression is nothing more than a fraction in which the numerator and/or the denominator are polynomials. Here are some examples of rational expressions. $$\dfrac{6}{x-1}, ...
0
votes
0answers
54 views

Find a polynomial such that this proposed root finding algorithm fails.

Is this polynomial root finding algorithm below known, and under what conditions for the choice of polynomial coefficients does it find at least one root? Description of the algorithm: Consider the ...
1
vote
1answer
31 views

Multiplicity of roots of polynomial with rational coefficients decidable?

From the standpoint of intuitionistic logic, multiplicity of roots of generic polynomial is uncomputable due to the inability to compare two real numbers. Even though the roots themselves are ...
1
vote
0answers
20 views

Constant of Holder-type Inequality for Polynomial Function

Is anybody aware of an inequality in the following form $$ \Vert f \Vert_{L_p(\Omega)} \leq C(p) \Vert f \Vert_{L_q(\Omega)} $$ where $f$ is a polynomial function of degree $p$ on $\Omega \subset ...
0
votes
4answers
39 views

Determine whether the set of vectors is a linear subspace

Let $V$ be the vector space of all polynomials $f\left(t\right)$ over $\mathbb{R}$ of degree at most $3$. I am trying to show that whether all polynomials $a+bt+ct^2+dt^3$ with ...
2
votes
2answers
47 views

To show that the variables in the system are same in magnitude

I am stuck with this interesting problem, If for non-negative integers $a, b, \text{and} c$, $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ and $\frac{b}{a}+\frac{c}{b}+\frac{a}{c}$ are both integers then ...