Polynomials are expressions like $15x^3 - 14x^2 + 8$. Questions tagged with this concern common operations on polynomials, like adding, multiplying, polynomial long division, factoring and solving for roots.

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38 views

PSD matrix and non-negative polynomial

So I'm trying to prove that if there exists a $5 \times 5$ matrix $Q$ such that $$Q \succeq0,\,\, a_{l-1} = \sum\limits_{i+j=l} Q_{ij} , l=1,\ldots,5$$ then there exists a fourth degree polynomial ...
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1answer
35 views

How do I distribute this polynomial expansion?

Ok, so for some reason, I cannot seem to get this simple polynomial multiplication correct no matter how many times I do it. I am working in $\mathbb{Z}/13\mathbb{Z}$. $$ (4x+11)(5x+(3x^2+1)) $$ ...
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0answers
52 views

Roots of polynomial in $F_3[x]$

Let $\alpha$ be a root of $x^2 + x + 2 = 0$ in $F_3[x]$. I am asked to show that $x^3 + x + 1$ has roots $\alpha$, $\alpha^2$ and $\alpha^4$. I started by observing that $\alpha^2 + \alpha + 2 = 0 ...
2
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2answers
176 views

Uniform convergence of Lagrange polynomials

There is a well-known theorem that states that on a closed interval $[a,b]$ any continuous function is the limit of a uniformly convergent sequence of polynomials. Proofs for this theorem usually ...
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2answers
129 views

How find this system $a^2+b^2=3,a^2+c^2+ac=4,b^2+c^2+\sqrt{3}bc=7$

Find the this system real solution $$\begin{cases} a^2+b^2=3\\ a^2+c^2+ac=4\\ b^2+c^2+\sqrt{3}bc=7 \end{cases}$$ I think that one can use Geometry to solve this system. Maybe there exist an ...
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0answers
57 views

Number of integer roots possible of the following polynomial [duplicate]

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?
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4answers
882 views

Matching of polynomial coefficients

I am trying to find the proof/theorem that states: Given two polynomials in x, if they are equal to eachother, their coefficients must also be equal For example, in ax^3 + bx^2 + cx + d = ...
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3answers
50 views

Find all polynomials p with real coefficients

Find all polynomials $p$ with real coefficients such that $p(x+1)=p(x)+2x+1$. I feel like in this question you let $x+1=x'$.
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2answers
54 views

Addition in $\operatorname{GF}(2^4)$

How can I compute $A(x)+B(x) \mod P(x)$ in $\operatorname{GF}(2^4)$ using the irreducible polynomial $P(x)=x^4+x+1$. What is the influence of the choice of the reduction polynomial on the computation? ...
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3answers
30 views

Coefficients of even powers

Just a simple thought experiment I was running in my head. Say I have a nonnegative even degree polynomial such as $f(x) = ax^4 + bx^3 + cx^2 + dx + e$. Is it true that the coefficients of the even ...
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2answers
66 views

finding the equation of a polynomial given its graph

I have a graph of polynomial and I would like to know how to determine its equation. Please, this isn't homework. What I'd like to do is actually reproduce this graph. Thanks.
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1answer
46 views

Splitting a multivariable polynomial into homogeneous components

In Wikipedia's proof of the fundamental theorem of symmetric polynomials, it states that the proof focuses on the case where the polynomial is homogeneous, and that "The general case then follows by ...
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0answers
25 views

Divisibility by $z-z_0$ if $z_0\in \mathbb{C}$ [duplicate]

I have a problem I'm working on, and I'm just not getting it. Suppose that $z_0\in\mathbb{C}$ is fixed. Show that if $P(z)=c(z^k-z_0^k)$, then there exists a polynomial $Q(z)$ such that ...
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2answers
37 views

Non negativity condition for quartic polynomials?

Say I have a quartic polynomial $f(x) = ax^4 + bx^3 + cx^2 + d$. I am told that $f(x)$ is nonnegative iff it can be expressed as a sum of squares as follows. $f(x) = \sum_{i=1}^4 q_i(x)^2$. As an ...
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2answers
206 views

Help with Proof explanation

I need help in understanding a (topological) proof of Fundamental theorem of algebra. Here is the Proof: Suppose $f(z)=a_nz^n+...+a_0$ with $a_0 \neq 0, n \geq1.$ WLOG, assume that $a_n=1.$ We ...
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2answers
36 views

Polynomials division algebra problem

Find sum of coefficients of the quotient obtained in: $$\frac{2x^n+x^{n-1}+x^{n-2}+...+x^2+x+5}{x-\frac{1}{2}}$$ I got "n" as the answer but according to the book is wrong, I don't know what is ...
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426 views

Degree of a polynomial

If I have a polynomial, for example, $$ x^8 + x^2 + \dfrac{1}{x} $$ would this be considered to be of degree 8? I am working on a question involving the function $ \frac{1}{x} $ and I am wondering how ...
3
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0answers
95 views

Möbius transformation that permutes roots of a cubic polynomial

The roots of the polynomial $x^3-3x-1$ can be permuted by the function $z\mapsto \dfrac{-1}{1+z}$ which is easily checked by a direct calculation. Is there a simple formula for a Möbius ...
3
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4answers
42 views

Finding the Remainder

Given the polynomials $$P(x) = nx^n+(n-1)x^{n-1}+(n-2)x^{n-2}+\cdots+x+1$$ and $$Q(x)=x(x-1)^2$$ find the remainder of the division $\dfrac{P (x)}{Q (x)}$.
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248 views

Finding real, distinct eigenvalues for arbitrary constants

Let $A= \begin{bmatrix} 1 & 1 & 0 \\ -4 & -3 & 1 \\ k & 0 & 0 \end{bmatrix}$. Find all values of $k$ such that $A$ has three real distinct eigenvalues. I have obtained the ...
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1answer
87 views

Factors of integers of the form $k^2-k+1$

Factorisation of arbitrary integers is of course a computationally hard problem. But what if the integers I'm interested in factorising are all of the form $k^2-k+1$ ? Is there some way to compute ...
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0answers
81 views

Small generating set of third degree polynomials in $R=\mathbb{Z}_2[x_1,\dots,x_n]/\langle x_1^2-x_1,\dots,x_n^2-x_n\rangle$

Let $R=\mathbb{Z}_2[x_1,\dots,x_n]/\langle x_1^2-x_1,\dots,x_n^2-x_n\rangle$, i.e., we can think of $R$ as the ring of multivariate polynomials with the additional property that one can "linearize" ...
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0answers
93 views

Too many independent cubic polynomials in an ideal $I\subset \mathbb C[x,y,z]$

Let us consider the ideal $I=(x^2-x,y,xz)\subset \mathbb C[x,y,z]$. I want to prove that $I$ contains (exactly) $5$ linearly independent polynomials of degree $3$. In three variables, we have ...
6
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1answer
359 views

Why is the polynomial $S(\vec{x})$ with coefficients obeying a constraint homogeneous?

I have recently been working on a problem to prove that a particular polynomial is in fact homogeneous. Although I have found out that this is true, I am curious to see whether there might be a deeper ...
2
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0answers
33 views

Linear Independence of Powers of “roots vector” [duplicate]

Let us be working over the field of complex numbers. Suppose $f(x)= a_n x^n + \cdots +a_1 x + a_0$ is a degree $n$ polynomial with $n$ distinct roots $z_1,\ldots,z_n$. Is the following matrix always ...
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1answer
45 views

Show that $(x-a,x-b)=1$

Knowing that $K$ is a field, $a,b \in K$ different from each other,show that $x-a,x-b$ co-primes. We suppose that $\exists f(x) \in K(x)$ such that: $f(x)|x-a$ and $f(x)|x-b$ Then $\deg f(x) \leq ...
3
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2answers
148 views

Nice exercises on resultants

I would like to ask if some one knows a source (a book, or lecture notes ect) that contains several nice exercises on resultants of polynomials (it would be nice if there were some solutions as well ...
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1answer
29 views

Space of complex poynomials

Let $\mathbb{C}_n[z]$ be the space of polynomials (of degree $\le n$) with complex coefficients, let the inner product be $(p,q):=\int_{-1}^1p(t)\overline{q(t)}dt$. There is one and only one $K_{w} ...
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1answer
132 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 ...
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1answer
86 views

dimension of subspace - polynominals evaluated on f

I need to prove that the dimension of the subspace of endomorphisms is less or equal m, if m is the degree of a polynomial p of K[t] \ {0} with p(f) = 0 (f is endomorphism). In a second step I ...
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1answer
23 views

A Polynomial with square values

Can I find the number of values ​​of the variable X for which the value of the polynomial $100X^2+160X+M$ is a perfect square, depending on M
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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 ...
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1answer
25 views

$n$ degree polynomial $p(k)=\frac{k}{k+1}$ for integral $k=0$ to $n$.

Here's the problem statement: Given an $n$ degree polynomial $p(k)$ such that: $$p(k)=\frac{k}{k+1}$$ for all integer $k$ from $0$ to $n$, determine $p(n+1)$. Any ideas on how to solve it?
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Creating Polynomial

By relative prime factor theorem $$R = (Zm,+,.)$$ where R is the ring structure the input is $e_0 = 0$ and $e_1=1$ output is $$S_0 = { k : \gcd(m,k)>1 }$$ $$S_1 = { k : \gcd(m,k) = 1}$$ Now ...
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3answers
86 views

An irreducible polynomial cannot share a root with a polynomial without dividing it

There is a lemma of Galois stating, "An irreducible equation can have no common root with a rational equation without dividing it". His definitions are a little bit imprecise, but I think he means: ...
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201 views

Regarding a Basis for Infinite Dimensional Vector Spaces

In my linear algebra class, during the discussion of vector spaces, our instructor mentioned infinite dimensional spaces, including the polynomial space over Q and the space of all continuous ...
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gcd of $x^4 + 2x^3 + x + 1$ and $x^5 + 2x^3 + x^2 + x + 1$ in $F_3[x]$

I am asked to find the gcd of $x^4 + 2x^3 + x + 1$ and $x^5 + 2x^3 + x^2 + x + 1$ in $F_3[x]$ (polynomials in $\mathbb Z/3\mathbb Z)$. Using Euclid's Algorithm, I first divided $x^4 + 2x^3 + x + 1$ ...
4
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1answer
291 views

Capelli Lemma for polynomials

I have seen this lemma given without proof in some articles (see example here), and I guess it is well known, but I couldn't find an online reference for a proof. It states like this: Let $K$ be ...
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2answers
119 views

Is the composition of irreducible polynomials again irreducible

I've been pondering this since yesterday. Is it true that given two irreducible polynomials $f(x)$ and $ g(x)$ will $f(g(x))$ or $g(f(x))$ be irreducible?
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262 views

Intuition behind Descartes' Rule of Signs

I have read several places that Descartes' Rule of Signs was familiar to both Descartes and Newton, and that both considered it too "obvious" to merit a proof. I know how to prove it, but I would like ...
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1answer
66 views

Is it possible that the zeroes of a polynomial form an infinite field?

Let $K/F$ be a finite field extension and suppose that $F$ is infinite. Is it possible to have a nonzero polynomial $p \in K[x_1,...,x_n]$ that vanishes in $F^n$?
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Expressing a Polynomial as a sum of cube roots of integers

How do you prove $x^3-3x^2-6x-4$ has a zero of the form $\sqrt[3]a+\sqrt[3]b+\sqrt[3]c$, for distinct positive integers a,b,c
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1answer
112 views

Irreducible Polynomials over $\mathbb{R}$ or $\mathbb{Q}$

I am interested in generating irreducible polynomials of a given, arbitrary degree over either the reals or rationals using integer coefficients. They don't necessarily have to be arbitrary ...
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1answer
84 views

A Cubic Equation

$2x^3+ax^2+bx+4=0$, $(a,b \in R^+)$ has three real roots. Then : A. $a\geqslant 4.2^{\frac 1 3}$ B. $a\geqslant 1.2^{\frac 1 3}$ C. $a\geqslant 6.2^{\frac 1 3}$ D. $a\geqslant 2.2^{\frac 1 3}$ ...
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1answer
54 views

Show there exists another polynomial with specified roots.

Let $\alpha$ be a complex number. Suppose there exists a a monic polynomial $f(x) \in \mathbb{Z}[x]$ such that $f(\alpha)=0$. Show that there exists a monic polynomial $g(x) \in \mathbb{Z}[x]$ such ...
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153 views

Calculate the primary decomposition

Consider the polynomial ring $R=K[x_1,\ldots, x_8]$ over field $K$. Set $\mathfrak{p}_1=(x_1, x_2, x_5, x_6)$, $\mathfrak{p}_2=(x_3, x_4, x_7, x_8)$ and $I=\mathfrak{p}_1\cap \mathfrak{p}_2$, ...
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1answer
42 views

multiplication in finite fields irreducible polynomial

I just started doing some reading about multiplication in finite fields and i keep stumbling over one point: in the field G(2^8) how does x^8 + x^4 + x^3 + x + 1 = 0 imply that x^8 = x^4 + x^3 + x + ...
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1answer
27 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 ...
4
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3answers
74 views

Invalid subtraction when solving system of equations?

I'm trying to solve these two equations: $$\begin{cases} 1-4x(x^2+y^2)=0 \\ 1-4y(x^2+y^2)=0 \end{cases}$$ and I tried to do it by subtracting the first equation from the second, yielding ...
0
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2answers
34 views

Proving that an eigenvalue is a root of a polynomial

Let $A$ be an $n \times n$ matrix, and let $\lambda$ be an eigenvalue of A. Prove that if $p$ is a polynomial such that $p(A)=\mathbb{0}$ then $\lambda$ is a root of $p$.